Rigorous Results for the Periodic
Oscillation of an Adiabatic Piston
Paul Wright
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
May 2007
Lai-Sang Young — Advisor
http://arxiv.org/abs/0704.0698v1
To Elizabeth
Acknowledgements
This dissertation would not have been possible without the help and support
of many people. First and foremost, I would like to thank my advisor, Lai-Sang
Young. I have benefited greatly from both the breadth of her interests and the
depth of her understanding. She has been an excellent teacher and mentor, and I
am indebted to her for much of my understanding of mathematics.
I would also like to thank a number of other people at the Courant Institute.
Professors Henry McKean and Jalal Shatah guided me through much of my early
time at Courant. Oscar Lanford patiently listened to and critiqued many of my
arguments, and George Zaslavsky contributed to my understanding of physics.
Postdoc’s Kevin Lin and Will Ott gave me constant encouragement and support,
and both helped to critique much of my writing. Péter Bálint generously proofread
a manuscript of my piston results in higher dimensions.
Of course, graduate school would never have been the same without my fellow
travelers, the other graduate students, to whom I am grateful. I would especially
like to thank the other dynamics students, José Koiller, Stan Mintchev, and Tanya
Yarmola, for being good friends and study partners. I would also like to thank
Tom Alberts, Gil Ariel, and Paris Pender, and my many great officemates, Hantaek
Bae, Umberto Hryniewicz, Helga Schaffrin, and Ross Tulloch.
I must also thank Dmitry Dolgopyat of the University of Maryland, who first
suggested the adiabatic piston problem to me, and who generously shared with
me his unpublished notes on averaging. In addition, I am grateful to Marco Lenci
and Luca Bussolari of the Stevens Institute of Technology for their participation
in Courant’s dynamical systems seminar, and to Albert Fathi of the ENS Lyon
for arranging for me to be a visitor there during the fall of 2005. During graduate
school, I was partially supported by a National Science Foundation Graduate
Research Fellowship.
Finally, I would like to thank my family, who have supported me throughout
my studies and enabled me to become who I am today. My mother shared with me
her love of learning, and my father shared the beauty of mathematics. My wife
has strengthened me in countless ways. I cannot thank her enough for sharing
with me her writing skills and her expert advice. I dedicate this work to her.
Abstract
We study a heavy piston of mass M that moves in one dimension. The piston
separates two gas chambers, each of which contains finitely many ideal, unit mass
gas particles moving in d dimensions, where d ≥ 1. Using averaging techniques, we
prove that the actual motions of the piston converge in probability to the predicted
averaged behavior on the time scale M1/2 whenM tends to infinity while the total
energy of the system is bounded and the number of gas particles is fixed. Neishtadt
and Sinai previously pointed out that an averaging theorem due to Anosov should
extend to this situation.
When d = 1, the gas particles move in just one dimension, and we prove
that the rate of convergence of the actual motions of the piston to its averaged
behavior is O(M−1/2) on the time scale M1/2. The convergence is uniform over
all initial conditions in a compact set. We also investigate the piston system when
the particle interactions have been smoothed. The convergence to the averaged
behavior again takes place uniformly, both over initial conditions and over the
amount of smoothing.
In addition, we prove generalizations of our results to N pistons separating
N + 1 gas chambers. We also provide a general discussion of averaging theory
and the proofs of a number of previously known averaging results. In particular,
we include a new proof of Anosov’s averaging theorem for smooth systems that is
primarily due to Dolgopyat.
Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
1.1 The adiabatic piston . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Physical motivation for the results . . . . . . . . . . . . . . . . . . . 4
2 Background Averaging Material 8
2.1 The averaging framework . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Some classical averaging results . . . . . . . . . . . . . . . . . . . . 12
2.3 A proof of Anosov’s theorem . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Moral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Results for piston systems in one dimension 27
3.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Heuristic derivation of the averaged equation for the hard core piston 34
3.3 Proof of the main result for the hard core piston . . . . . . . . . . . 35
3.4 Proof of the main result for the soft core piston . . . . . . . . . . . 41
3.5 Appendix to Section 3.4 . . . . . . . . . . . . . . . . . . . . . . . . 50
4 The periodic oscillation of an adiabatic piston in two or three
dimensions 54
4.1 Statement of the main result . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Preparatory material concerning a
two-dimensional gas container with only one gas particle on each side 58
4.3 Proof of the main result for two-dimensional gas containers with
only one gas particle on each side . . . . . . . . . . . . . . . . . . . 66
4.4 Generalization to a full proof
of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Inducing maps on subspaces . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Derivative bounds for the billiard map
in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography 84
List of Figures
1.1 A gas container D in d = 2 dimensions separated by an adiabatic
piston. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 An effective potential. . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 A schematic of the phase space M. Note that although the level
set Mc = {h = c} is depicted as a torus, it need not be a torus. It
could be any compact, co-dimension m submanifold. . . . . . . . . . 11
3.1 The piston system with n1 = 3 and n2 = 4. Note that the gas
particles do not interact with each other, but only with the piston
and the walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 A gas container D ⊂ R2 separated by a piston. . . . . . . . . . . . . 55
4.2 A choice of coordinates on phase space. . . . . . . . . . . . . . . . . 59
4.3 An analysis of the divergences of orbits when ε > 0 and the left gas
particle collides with the moving piston to the right of Q0. Note
that the dimensions are distorted for visual clarity, but that εL and
εL/γ are both o(γ) as ε→ 0. . . . . . . . . . . . . . . . . . . . . . 79
Chapter 1
Introduction
What can be rigorously understood about the nonequilibrium dynamics of chaotic,
many particle systems? Although much progress has been made in understanding
the infinite time behavior of such systems, our understanding on finite time scales
is still far from complete. Systems of many particles contain a large number of
degrees of freedom, and it is often impractical or impossible to keep track of their
full dynamics. However, if one is only interested in the evolution of macroscopic
quantities, then these variables form a small subset of all of the variables. The evo-
lution of these quantities does not itself form a closed dynamical system, because
it depends on events happening in all of the (very large) phase space. We must
therefore develop techniques for describing the evolution of just a few variables in
phase space. Such descriptions are valid on limited time scales because a large
amount of information about the dynamics of the full system is lost. However, the
time scales of validity can often be long enough to enable a good prediction of the
observable dynamics.
Averaging techniques help to describe the evolution of certain variables in
some physical systems, especially when the system has components that move on
different time scales. The primary results of this thesis involve applying averaging
techniques to chaotic microscopic models of gas particles separated by an adiabatic
piston for the purposes of justifying and understanding macroscopic laws.
This thesis is organized as follows. In Section 1.1 we briefly introduce the the
adiabatic piston problem and our results. In Section 1.2 we review the physical
motivations for our results. The following three chapters may each be read inde-
pendently. Chapter 2 presents an introduction to averaging theory and the proofs
of a number of averaging theorems for smooth systems that motivate our later
proofs for the piston problem. Chapter 3 contains our results for piston systems
in one dimension, and Chapter 4 contains our results for the piston system in
dimensions two and three.
1.1 The adiabatic piston
Consider the following simple model of an adiabatic piston separating two gas
containers: A massive piston of mass M ≫ 1 divides a container in Rd, d =
1, 2, or 3, into two halves. The piston has no internal degrees of freedom and can
only move along one axis of the container. On either side of the piston there are a
finite number of ideal, unit mass, point gas particles that interact with the walls
of the container and with the piston via elastic collisions. When M = ∞, the
piston remains fixed in place, and each gas particle performs billiard motion at a
constant energy in its sub-container. We make an ergodicity assumption on the
behavior of the gas particles when the piston is fixed. Then we study the motions
of the piston when the number of gas particles is fixed, the total energy of the
system is bounded, but M is very large.
Heuristically, after some time, one expects the system to approach a steady
state, where the energy of the system is equidistributed amongst the particles
and the piston. However, even if we could show that the full system is ergodic,
an abstract ergodic theorem says nothing about the time scale required to reach
such a steady state. Because the piston will move much slower than a typical gas
particle, it is natural to try to determine the intermediate behavior of the piston
by averaging techniques. By averaging over the motion of the gas particles on a
time scale chosen short enough that the piston is nearly fixed, but long enough
that the ergodic behavior of individual gas particles is observable, we will show
that the system does not approach the expected steady state on the time scale
M1/2. Instead, the piston oscillates periodically, and there is no net energy transfer
between the gas particles.
The results of this thesis follow earlier work by Neishtadt and Sinai [Sin99,
NS04]. They determined that for a wide variety of Hamiltonians for the gas par-
ticles, the averaged behavior of the piston is periodic oscillation, with the piston
moving inside an effective potential well whose shape depends on the initial po-
sition of the piston and the gas particles’ Hamiltonians. They pointed out that
an averaging theorem due to Anosov [Ano60, LM88], proved for smooth systems,
should extend to this case. The main result of the present work, Theorem 4.1.1,
is that Anosov’s theorem does extend to the particular gas particle Hamiltonian
described above. Thus, if we examine the actual motions of the piston with respect
to the slow time τ = t/M1/2, then, as M → ∞, in probability (with respect to
Liouville measure) most initial conditions give rise to orbits whose actual motion is
accurately described by the averaged behavior for 0 ≤ τ ≤ 1, i.e. for 0 ≤ t ≤M1/2.
A recent study involving some similar ideas by Chernov and Dolgopyat [CD06a]
considered the motion inside a two-dimensional domain of a single heavy, large gas
particle (a disk) of mass M ≫ 1 and a single unit mass point particle. They as-
sumed that for each fixed location of the heavy particle, the light particle moves
inside a dispersing (Sinai) billiard domain. By averaging over the strongly hy-
perbolic motions of the light particle, they showed that under an appropriate
scaling of space and time the limiting process of the heavy particle’s velocity is a
(time-inhomogeneous) Brownian motion on a time scale O(M1/2). It is not clear
whether a similar result holds for the piston problem, even for gas containers with
good hyperbolic properties, such as the Bunimovich stadium. In such a container
the motion of a gas particle when the piston is fixed is only nonuniformly hyper-
bolic because it can experience many collisions with the flat walls of the container
immediately preceding and following a collision with the piston.
The present work provides a weak law of large numbers, and it is an open
problem to describe the sizes of the deviations for the piston problem [CD06b].
Although our result does not yield concrete information on the sizes of the devia-
tions, it is general in that it imposes very few conditions on the shape of the gas
container. Most studies of billiard systems impose strict conditions on the shape
of the boundary, generally involving the sign of the curvature and how the corners
are put together. The proofs in this work require no such restrictions. In particu-
lar, the gas container can have cusps as corners and need satisfy no hyperbolicity
conditions.
If the piston divides a container in R2 or R3 with axial symmetry, such as a
rectangle or a cylinder, then our ergodicity assumption on the behavior of the gas
particles when the piston is fixed does not hold. In this case, the interactions of the
gas particles with the piston and the ends of the container are completely specified
by their motions along the normal axis of the container. Thus, this system projects
onto a system inside an interval consisting of a massive point particle, the piston,
which interacts with the gas particles on either side of it. These gas particles make
elastic collisions with the walls at the ends of the container and with the piston,
but they do not interact with each other. For such one-dimensional containers,
the effects of the gas particles are quasi-periodic and can be essentially decoupled,
and we recover a strong law of large numbers with a uniform rate, reminiscent
of classical averaging over just one fast variable in S1: The convergence of the
actual motions to the averaged behavior is uniform over all initial conditions, with
the size of the deviations being no larger than O(M−1/2) on the time scale M+1/2.
See Theorem 3.1.1. Gorelyshev and Neishtadt [GN06] independently obtained this
result.
For systems in d = 1 dimension, we also investigate the behavior of the system
when the interactions of the gas particles with the walls and the piston have been
smoothed, so that Anosov’s theorem applies directly. Let δ ≥ 0 be a parameter
of smoothing, so that δ = 0 corresponds to the hard core setting above. Then
the averaged behavior of the piston is still a periodic oscillation, which depends
smoothly on δ. We show that the deviations of the actual motions of the piston
from the averaged behavior are again not more than O(M−1/2) on the time scale
M1/2. The size of the deviations is bounded uniformly, both over initial conditions
and over the amount of smoothing, Theorem 3.1.2.
Our results for a single heavy piston separating two gas containers generalize
to the case of N heavy pistons separating N +1 gas containers. Here the averaged
behavior of the pistons has them moving like an N -dimensional particle inside an
effective potential well. Compare Section 3.1.3.
The systems under consideration in this work are simple models of an adiabatic
piston. The general adiabatic piston problem [Cal63], well-known from physics,
consists of the following: An insulating piston separates two gas containers, and
initially the piston is fixed in place, and the gas in each container is in a separate
thermal equilibrium. At some time, the piston is no longer externally constrained
and is free to move. One hopes to show that eventually the system will come to a
full thermal equilibrium, where each gas has the same pressure and temperature.
Whether the system will evolve to thermal equilibrium and the interim behavior of
the piston are mechanical problems, not adequately described by thermodynam-
ics [Gru99], that have recently generated much interest within the physics and
mathematics communities following Lieb’s address [Lie99]. One expects that the
system will evolve in at least two stages. First, the system relaxes deterministically
toward a mechanical equilibrium, where the pressures on either side of the piston
are equal. In the second, much longer, stage, the piston drifts stochastically in the
direction of the hotter gas, and the temperatures of the gases equilibrate. See for
example [GPL03, CL02, Che04] and the references therein. Previously, rigorous
results have been limited mainly to models where the effects of gas particles rec-
olliding with the piston can be neglected, either by restricting to extremely short
time scales [LSC02, CLS02] or to infinite gas containers [Che04].
1.2 Physical motivation for the results
In this section, we briefly review the physical motivations for our results on the
adiabatic piston.
Consider a massive, insulating piston of massM that separates a gas container
D in Rd, d = 1, 2, or 3. See Figure 1.1. Denote the location of the piston by Q
and its velocity by dQ/dt = V . If Q is fixed, then the piston divides D into two
subdomains, D1(Q) = D1 on the left and D2(Q) = D2 on the right. By |Di| we
denote the area (when d = 2, or length, when d = 1, or volume, when d = 3) of
Di. Define
∂ |D1(Q)|
= −∂ |D2(Q)|
so that ℓ is the piston’s cross-sectional length (when d = 2, or area, when d = 3).
If d = 1, then ℓ = 1. By Ei we denote the total energy of the gas inside Di.
D1(Q) D2(Q)
E1 E2
D = D1(Q) ⊔ D2(Q)
✲ V = εW
M = ε−2 ≫ 1
s✟✟✙ℓ
Figure 1.1: A gas container D in d = 2 dimensions separated by an adiabatic
piston.
We are interested in the dynamics of the piston when the system’s total energy
is bounded and M → ∞. When M = ∞, the piston remains fixed in place, and
each energy Ei remains constant. When M is large but finite, MV
2/2 is bounded,
and so V = O(M−1/2). It is natural to define
ε =M−1/2,
so that W is of order 1 as ε → 0. This is equivalent to scaling time by ε, and so
we introduce the slow time
τ = εt.
If we let Pi denote the pressure of the gas inside Di, then heuristically the
dynamics of the piston should be governed by the following differential equation:
= V, M
= P1ℓ− P2ℓ,
= P1ℓ− P2ℓ.
(1.1)
To find differential equations for the energies of the gases, note that in a short
amount of time dt, the change in energy should come entirely from the work done
on a gas, i.e. the force applied to the gas times the distance the piston has moved,
because the piston is adiabatic. Thus, one expects that
= −V P1ℓ,
= +V P2ℓ,
= −WP1ℓ,
= +WP2ℓ.
(1.2)
To obtain a closed system of differential equations, it is necessary to insert an
expression for the pressures. Piℓ should be the average force from the gas particles
in Di experienced by the piston when it is held fixed in place. Whether such an
expression, depending only on Ei and Di(Q), exists and is the same for (almost)
every initial condition of the gas particles depends strongly on the microscopic
model of the gas particle dynamics. Sinai and Neishtadt [Sin99, NS04] pointed
out that for many microscopic models where the pressures are well defined, the
solutions of Equations (1.1) and (1.2) have the piston moving according to a model-
dependent effective Hamiltonian.
Because the pressure of an ideal gas in d dimensions is proportional to the
energy density, with the constant of proportionality 2/d, we choose to insert
d |Di|
Later, we will make assumptions on the microscopic gas particle dynamics to justify
this substitution. However, if we accept this definition of the pressure, we obtain
the following ordinary differential equations for the four macroscopic variables of
the system:
d |D1(Q)|
− 2E2ℓ
d |D2(Q)|
− 2WE1ℓ
d |D1(Q)|
2WE2ℓ
d |D2(Q)|
. (1.3)
For these equations, one can see the effective Hamiltonian as follows. Since
d ln(Ei)
d ln(|Di(Q)|)
Ei(τ) = Ei(0)
|Di(Q(0))|
|Di(Q(τ))|
Hence
d2Q(τ)
E1(0) |D1(Q(0))|2/d
|D1(Q(τ))|1+2/d
E2(0) |D2(Q(0))|2/d
|D2(Q(τ))|1+2/d
effective potential
P1 = P2
Figure 1.2: An effective potential.
and so (Q,W ) behave as if they were the coordinates of a Hamiltonian system
describing a particle undergoing motion inside a potential well. The effective
Hamiltonian may be expressed as
W 2 +
E1(0) |D1(Q(0))|2/d
|D1(Q)|2/d
E2(0) |D2(Q(0))|2/d
|D2(Q)|2/d
. (1.4)
The question is, do the solutions of Equation (1.3) give an accurate description
of the actual motions of the macroscopic variables when M tends to infinity? The
main result of this thesis, Theorem 4.1.1, is that, for an appropriately defined
system, the answer to this question is affirmative for 0 ≤ t ≤ M1/2, at least for
most initial conditions of the microscopic variables. Observe that one should not
expect the description to be accurate on time scales much longer than O(M1/2) =
O(ε−1). The reason for this is that, presumably, there are corrections of size O(ε)
in Equation (1.3) that we are neglecting. For τ = εt > O(1), these corrections
should become significant. Such higher order corrections for the adiabatic piston
were studied by Crosignani et al. [CDPS96].
Chapter 2
Background Averaging Material
In this chapter, we present a number of well-known classical averaging results
for smooth systems, as well as a proof of Anosov’s averaging theorem, which is
the first general multi-phase averaging result. All of these theorems are at least
45 years old. However, we present them here because our proofs of the classical
results are at least slightly novel, and the ideas in them lend themselves well to
certain higher-dimensional generalizations. In particular, they are fairly close to
the ideas in the proof we give for our piston results in one dimension. The proof of
Anosov’s theorem is a new and unpublished proof due mainly to Dolgopyat, with
some further simplifications made. The ideas in this proof underly the ideas we
will use to prove the weak law of large numbers for our piston system in dimensions
two and three.
We begin by giving a discussion of a framework for general averaging theory
and some averaging results. A number of classical averaging theorems are then
proved, followed by the proof of Anosov’s theorem.
2.1 The averaging framework
In this section, consider a family of ordinary differential equations
= Z(z, ε) (2.1)
on a smooth, finite-dimensional Riemannian manifold M, which is indexed by the
real parameter ε ∈ [0, ε0]. Assume
• Regularity: the functions Z and ∂Z/∂ε are both C1 on M× [0, ε0].
We denote the flow generated by Z(·, ε) by zε(t, z) = zε(t). We will usually
suppress the dependence on the initial condition z = zε(0, z). Think of zε(·)
as being a random variable whose domain is the space of initial conditions for
the differential equation (2.1) and whose range is the space of continuous paths
(depending on the parameter t) in M.
• Existence of smooth integrals: z0(t) has m independent C2 first integrals
h = (h1, . . . , hm) : M → Rm.
Then h is conserved by z0(t), and at every point the linear operator ∂h/∂z has
full rank. It follows from the implicit function theorem that each level set
Mc := {h = c}
is a smooth submanifold of co-dimension m, which is invariant under z0(t). Fur-
ther, assume that there exists an open ball U ⊂ Rm satisfying:
• Compactness: ∀c ∈ U , Mc is compact.
• Preservation of smooth measures: ∀c ∈ U , z0(t)|Mc preserves a smooth
measure µc that varies smoothly with c, i.e. there exists a C1 function
g : M → R>0 such that g|Mc is the density of µc with respect to the
restriction of Riemannian volume.
hε(t, z) = hε(t) := h(zε(t)).
Again, think of hε(·) as being a random variable that takes initial conditions z ∈ M
to continuous paths (depending on the parameter t) in U . Since dh0/dt ≡ 0,
Hadamard’s Lemma allows us to write
= εH(zε, ε)
for some C1 function H : M× [0, ε0] → U . Observe that
(t) = Dh(zε(t))Z(zε(t), ε) = Dh(zε(t))
Z(zε(t), ε)− Z(zε(t), 0)
so that
H(z, 0) = L ∂Z
|ε=0h.
Here L denotes the Lie derivative.
Define the averaged vector field H̄ by
H̄(h) =
H(z, 0)dµh(z). (2.2)
Then H̄ is C1. Fix a compact set V ⊂ U , and introduce the slow time
τ = εt.
Let h̄(τ, z) = h̄(τ) be the random variable that is the solution of
= H̄(h̄), h̄(0) = hε(0).
We only consider the dynamics in a compact subset of phase space, so for initial
conditions z ∈ h−1U , define the stopping time
Tε(z) = Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Heuristically, think of the phase space M as being a fiber bundle whose base
is the open set U and whose fibers are the compact sets Mh. See Figure 2.1.
Then the vector field Z(·, 0) is perpendicular to the base, so its orbits z0(t) flow
only along the fibers. Now when 0 < ε ≪ 1, the vector field Z(·, ε) acquires a
component of size O(ε) along the base, and so its orbits zε(t) have a small drift
along the base, which we can follow by observing the evolution of hε(t). Because
of this, we refer to h as consisting of the slow variables. Other variables, used
to complete h to a parameterization of (a piece of) phase space, are called fast
variables. Note that hε(t) depends on all the dimensions of phase space, and so it
is not the flow of a vector field on the m-dimensional space U . However, because
the motion along each fiber is relatively fast compared to the motion across fibers,
we hope to be able to average over the fast motions and obtain a vector field
on U that gives a good description of hε(t) over a relatively long time interval,
independent of where the solution zε(t) started on Mhε(0). Because our averaged
vector field, as defined by Equation (2.2), only accounts for deviations of size O(ε),
we cannot expect this time interval to be longer than size O(1/ε). In terms of
the slow time τ = εt, this length becomes O(1). In other words, the goal of the
first-order averaging method described above should be to show that, in some
sense, sup0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣→ 0 as ε → 0. This is often referred to as the
averaging principle.
Note that the assumptions of regularity, existence of smooth integrals, com-
pactness, and preservation of smooth measures above are not sufficient for the
averaging principle to hold in any form. As an example of just one possible
obstruction, the level sets Mc could separate into two completely disjoint sets,
Mc = M+c ⊔ M−c . If this were the case, then it would be implausible that the
solutions of the averaged vector field defined by averaging over all of Mc would
accurately describe hε(t, z), independent of whether z ∈ M+c or z ∈ M−c .
Some averaging results
So far, we are in a general averaging setting. Frequently, one also assumes that the
invariant submanifolds, Mh, are tori, and that there exists a choice of coordinates
z = (h, ϕ)
M = {(h, ϕ)}
✲ h ∈ U ⊂ Rm
“slow variables”
ϕ =“fast variables”
✄✗Z(·, 0) Z(·, ε)
Figure 2.1: A schematic of the phase space M. Note that although the level set
Mc = {h = c} is depicted as a torus, it need not be a torus. It could be any
compact, co-dimension m submanifold.
on M in which the differential equation (2.1) takes the form
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε).
Then if ϕ ∈ S1 and the differential equation for the fast variable is regular,
i.e. Φ(h, ϕ, 0) is bounded away from zero for h ∈ U ,
initial conditions
s.t. hε(0)∈V
0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε → 0.
See for example Chapter 5 in [SV85], Chapter 3 in [LM88], or Theorem 2.2.3 in
the following section.
When the differential equation for the fast variable is not regular, or when
there is more than one fast variable, the typical averaging result becomes much
weaker than the uniform convergence above. For example, consider the case when
ϕ ∈ Tn, n > 1, and the unperturbed motion is quasi-periodic, i.e. Φ(h, ϕ, 0) =
Ω(h). Also assume that H ∈ Cn+2 and that Ω is nonvanishing and satisfies a
nondegeneracy condition on U (for example, Ω : U → Tn is a submersion). Let P
denote Riemannian volume on M. Neishtadt [LM88, Nei76] showed that in this
situation, for each fixed δ > 0,
0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
ε/δ),
and that this result is optimal. Thus, the averaged equation only describes the
actual motions of the slow variables in probability on the time scale 1/ε as ε→ 0.
Neishtadt’s result was motivated by a general averaging theorem for smooth
systems due to Anosov. This theorem requires none of the additional assumptions
in the averaging results above. Under the conditions of regularity, existence of
smooth integrals, compactness, and preservation of smooth measures, as well as
• Ergodicity: for Lebesgue almost every c ∈ U , (z0(·), µc) is ergodic,
Anosov showed that sup0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ → 0 in probability (w.r.t. Rie-
mannian volume on initial conditions) as ε→ 0, i.e.
Theorem 2.1.1 (Anosov’s averaging theorem [Ano60]). For each T > 0 and for
each fixed δ > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
as ε→ 0.
We present a recent proof of this theorem in Section 2.3 below.
If we consider hε(·) and h̄(·) to be random variables, Anosov’s theorem is a
version of the weak law of large numbers. In general, we can do no better: There
is no general strong law in this setting. There exists a simple example due to
Neishtadt (which comes from the equations for the motion of a pendulum with
linear drag being driven by a constant torque) where for no initial condition in a
positive measure set do we have convergence of hε(t) to h̄(εt) on the time scale 1/ε
as ε → 0 [Kif04b]. Here, the phase space is R× S1, and the unperturbed motion
is (uniquely) ergodic on all but one fiber.
2.2 Some classical averaging results
In this section we present some simple, well-known averaging results. See for
example Chapter 5 in [SV85] or Chapter 3 in [LM88].
2.2.1 Averaging for time-periodic vector fields
Consider a family of time dependent ordinary differential equations
= εH(h, t, ε), (2.3)
indexed by the real parameter ε ≥ 0, where h ∈ Rm. Fix V ⊂⊂ U ⊂ Rm, and
suppose
• Regularity: H ∈ C1(U × R× [0,∞)).
• Periodicity: There exists T > 0 such that for each h ∈ U , H(h, t, 0) is
T -periodic in time.
= εH(h, t, 0) +O(ε2).
Let hε(t) denote the solution of Equation (2.3). We seek a time independent
vector field whose solutions approximate hε(t), at least for a long length of time.
It is natural to define the averaged vector field H̄ by
H̄(h) =
H(h, s, 0)ds.
Then H̄ ∈ C1(U). Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
It is reasonable to hope that h̄(εt) and hε(t) are close together for 0 ≤ t ≤ ε−1.
We only consider the dynamics in a compact subset of phase space, so for initial
conditions in U , we define the stopping time
Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.1 (Time-periodic averaging). For each T > 0,
hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Proof. We divide our proof into three essential steps.
Step 1: Reduction using Gronwall’s Inequality. Now, h̄(τ) satisfies the
integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(hε(s), s, ε)ds
= O(ε) + ε
∫ τ/ε
H(hε(s), s, 0)ds
= O(ε) + ε
∫ τ/ε
H(hε(s), s, 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(hε(s), s, 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄|V)T .
Step 2: A sequence of times adapted for ergodization. Ergodization refers
to the convergence along an orbit of a function’s time average to its space average.
We define a sequence of times tk for k ≥ 0 by tk = kT . This sequence of times is
motivated by the fact that
tk+1 − tk
∫ tk+1
H(h0(s), s, 0)ds = H̄(h0).
Note that h0(t) is independent of time. Thus,
0≤τ≤T∧Tε
|eε(τ)| ≤ O(ε) + ε
tk+1≤T∧Tεε
∫ tk+1
H(hε(s), s, 0)− H̄(hε(s))ds
. (2.4)
Step 3: Control of individual terms by comparison with solutions of the
ε = 0 equation. The sum in Equation (2.4) has no more than O(1/ε) terms,
and so it suffices to show that each term
∫ tk+1
H(hε(s), s, 0) − H̄(hε(s))ds is no
larger than O(ε). We can accomplish this by comparing the motions of hε(t) for
tk ≤ t ≤ tk+1 with hk,ε(t), which is defined to be the solution of the ε = 0 ordinary
differential equation satisfying hk,ε(tk) = hε(tk), i.e. hk,ε(t) ≡ hε(tk).
Lemma 2.2.2. If tk+1 ≤ T∧Tεε , then suptk≤t≤tk+1 |hk,ε(t)− hε(t)| = O(ε).
Proof. dhε/dt = O(ε).
Using that H and H̄ are Lipschitz continuous, we conclude that
∫ tk+1
H(hε(s), s, 0)− H̄(hε(s))ds
∫ tk+1
H(hε(s), s, 0)−H(hk,ε(s), s, 0)ds
∫ tk+1
H(hk,ε(s), s, 0)− H̄(hk,ε(s))ds
∫ tk+1
H̄(hk,ε(s))− H̄(hε(s))ds
=O(ε) + 0 +O(ε)
=O(ε).
Thus we see that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≤ O(ε), independent of the
initial condition hε(0) ∈ V.
Remark 2.2.1. Note that the O(ε) control in Theorem 2.2.1 on a time scale t =
O(ε−1) is generally optimal. For example, take H(h, t, ε) = cos(t) + ε.
2.2.2 Averaging for vector fields with one regular fast vari-
For h ∈ Rm and ϕ ∈ S1 = [0, 1]/0 ∼ 1, consider the family of ordinary differential
equations
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε), (2.5)
indexed by the real parameter ε ≥ 0. With z = (h, ϕ), we write this family of
differential equations as dz/dt = Z(z, ε).
Fix V ⊂⊂ U ⊂ Rm, and suppose
• Regularity: Z ∈ C1(U × S1 × [0,∞)).
• Regular fast variable: Φ(h, ϕ, 0) is bounded away from 0 for h ∈ U , i.e.
(h,ϕ)∈U×S1
|Φ(h, ϕ, 0)| > 0.
Without loss of generality, we assume that Φ(h, ϕ, 0) > 0.
Let zε(t) = (hε(t), ϕε(t)) denote the solution of Equation (2.5). Then z0(t)
leaves invariant the circles Mc = {h = c} in phase space. In fact, z0(t) preserves
an uniquely ergodic invariant probability measure on Mc, whose density is given
dµc =
Φ(c, ϕ, 0)
where Kc =
Φ(c,ϕ,0)
is a normalization constant.
The averaged vector field H̄ is defined by averaging H(h, ϕ, 0) over ϕ:
H̄(h) =
H(h, ϕ, 0)dµh(ϕ) =
H(h, ϕ, 0)
Φ(h, ϕ, 0)
Then H̄ ∈ C1(U). Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
For initial conditions in U × S1, we have the usual stopping time Tε = inf{τ ≥ 0 :
h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.3 (Averaging over one regular fast variable). For each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Remark 2.2.2. This result encompasses Theorem 2.2.1 for time-periodic averaging.
For example, if T = 1, simply take ϕ = t mod 1 and Φ(h, ϕ, ε) = 1.
Remark 2.2.3. Many of the proofs of the above theorem of which we are aware
hinge on considering ϕ as a time-like variable. For example, one could write
H(h, ϕ, 0)
Φ(h, ϕ, 0)
+O(ε2),
and this looks very similar to the time-periodic situation considered previously.
However, it does take some work to justify such arguments rigorously, and the
traditional proofs do not easily generalize to averaging over multiple fast variables.
Our proof essentially uses ϕ to mark off time, and it will immediately generalize
to a specific instance of multiphase averaging.
Proof. Again, we have three steps.
Step 1: Reduction using Gronwall’s Inequality. Now
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(zε(s), ε)ds = O(ε) + ε
∫ τ/ε
H(zε(s), 0)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄|V)T .
Step 2: A sequence of times adapted for ergodization. Now for each
initial condition in our phase space and for each fixed ε, we define a sequence of
times tk,ε and a sequence of solutions zk,ε(t) inductively as follows: t0,ε = 0 and
z0,ε(t) = z0(t). For k > 0, tk,ε = inf{t > tk−1,ε : ϕk−1,ε(t) = ϕε(0)}, and zk,ε(t) is
defined as the solution of
dzk,ε
= Z(zk,ε, 0) = (0,Φ(zk,ε, 0)), zk,ε(tk,ε) = zε(tk,ε).
This sequence of times is motivated by the fact that
tk+1,ε − tk,ε
∫ tk+1,ε
tk ,ε
H(zk,ε(s), 0)ds = H̄(hk,ε).
Recall that hk,ε(t) is independent of time. The elements of this sequence of times
are approximately uniformly spaced, i.e. if we fix ω > 0 such that z ∈ V × S1 ⇒
1/ω < Φ(z, 0) < ω, then if tk+1,ε ≤ (T ∧ Tε)/ε, 1/ω < tk+1,ε − tk,ε < ω.
Thus,
0≤τ≤T∧Tε
|eε(τ)| ≤ O(ε) + ε
tk+1,ε≤T∧Tεε
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
where the sum in in this equation has no more than O(1/ε) terms.
Step 3: Control of individual terms by comparison with solutions along
fibers. It suffices to show that each term
∫ tk+1,ε
H(zε(s), 0) − H̄(hε(s))ds is no
larger than O(ε). We can accomplish this by comparing the motions of zε(t) for
tk,ε ≤ t ≤ tk+1,ε with zk,ε(t).
Lemma 2.2.4. If tk+1,ε ≤ T∧Tεε , then suptk,ε≤t≤tk+1,ε |zk,ε(t)− zε(t)| = O(ε).
Proof. Without loss of generality, we take k = 0, so that zk,ε(t) = z0(t). Since
h0(t) = hε(0) and dhε/dt = O(ε), supt0,ε≤t≤t1,ε |h0(t)− hε(t)| = O(ε).
Now ϕε(t) − ϕε(0) =
Φ(hε(s), ϕε(s), ε)ds, and because Φ is Lipschitz, we
find that
|ϕε(t)− ϕ0(t)| ≤ O(ε) + Lip (Φ)
|ϕε(s)− ϕ0(s)| ds
for 0 ≤ t ≤ ω. The result follows from Gronwall’s Inequality.
Using that H and H̄ are Lipschitz continuous, we conclude that
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
∫ tk+1,ε
H(zε(s), 0)−H(zk,ε(s), 0)ds
∫ tk+1,ε
H(zk,ε(s), 0)− H̄(hk,ε(s))ds
∫ tk+1,ε
H̄(hk,ε(s))− H̄(hε(s))ds
=O(ε) + 0 +O(ε)
=O(ε).
Thus we see that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε), independent of the
initial condition (hε(0), ϕε(0)) ∈ V × S1.
2.2.3 Multiphase averaging for vector fields with separa-
ble, regular fast variables
As explained in Section 2.1, when the differential equation for the fast variable is
not regular, or when there is more than one fast variable, the typical averaging
result becomes much weaker than the uniform convergence in Theorems 2.2.1
and 2.2.3 above. Nonetheless, if the differential equations under consideration
satisfy some very specific hypotheses, the proof in the previous section immediately
generalizes to yield uniform convergence.
For h ∈ Rm and ϕ = (ϕ1, · · · , ϕn) ∈ Tn = ([0, 1]/0 ∼ 1)n, consider the family
of ordinary differential equations
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε), (2.6)
indexed by the real parameter ε ≥ 0. We also write z = (h, ϕ) and dz/dt = Z(z, ε).
Fix V ⊂⊂ U ⊂ Rm, and suppose
• Regularity: Z ∈ C1(U × Tn × [0,∞)).
• Separable fast variables: H(h, ϕ, 0) and Φ(h, ϕ, 0) have the following specific
forms:
– There exist C1 functionsHj(h, ϕj) such thatH(h, ϕ, 0) =
j=1Hj(h, ϕ
This can be thought of as saying that, to first order in ε, each fast vari-
able affects the slow variables independently of the other fast variables.
– The components Φj of Φ satisfy Φj(h, ϕ, 0) = Φj(h, ϕj, 0), i.e. the un-
perturbed motion has each fast variable moving independently of the
other fast variables. Note that this assumption is satisfied if the unper-
turbed motion is quasi-periodic, i.e. Φ(h, ϕ, 0) = Ω(h).
• Regular fast variables: For each j,
(h,ϕj)∈U×S1
∣Φj(h, ϕj, 0)
∣ > 0.
Let zε(t) = (hε(t), ϕε(t)) denote the solution of Equation (2.6). Then z0(t)
leaves invariant the tori Mc = {h = c} in phase space. In fact, z0(t) preserves a
(not necessarily ergodic) invariant probability measure on Mc, whose density is
given by
dµc =
|Φj(c, ϕj, 0)| ,
where Kjc =
|Φj(c,ϕj ,0)| .
The averaged vector field H̄ is defined by
H̄(h) =
H(h, ϕ, 0)dµh(ϕ) =
Hj(h, ϕ
j)dµh(ϕ)
Hj(h, ϕ
|Φj(h, ϕj, 0)|dϕ
H̄j(h).
Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0),
and the stopping time Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.5 (Averaging over multiple separable, regular fast variables). For
each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Proof. The proof is essentially the same as the proof of Theorem 2.2.3. As before,
we need only show that sup0≤τ≤T∧Tε |eε(τ)| = O(ε), where
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
But by our separability assumptions, it suffices to show that for each j,
0≤τ≤T∧Tε
|ej,ε(τ)| = O(ε),
where ej,ε(τ) is defined by
ej,ε(τ) = ε
∫ τ/ε
Hj(hε(s), ϕ
ε(s))− H̄j(hε(s))ds.
Thus, we have effectively separated the effects of each fast variable, and now the
proof can be completed by essentially following steps 2 and 3 in the proof of
Theorem 2.2.3.
2.3 A proof of Anosov’s theorem
Anosov’s original proof of Theorem 2.1.1 from 1960 may be found in [Ano60]. An
exposition of the theorem and Anosov’s proof in English may be found in [LM88].
Recently, Kifer [Kif04a] proved necessary and sufficient conditions for the averaging
principle to hold in an averaged with respect to initial conditions sense. He also
showed explicitly that his conditions are met in the setting of Anosov’s theorem.
The proof of Anosov’s theorem given here is mainly due to Dolgopyat [Dol05],
although some further simplifications have been made.
Proof of Anosov’s theorem. We begin by showing that without loss of generality
we may take Tε = ∞. This is just for convenience, and not an essential part of
the proof. To accomplish this, let ψ(h) be a smooth bump function satisfying
• ψ(h) = 1 if h ∈ V,
• ψ(h) > 0 if h ∈ interior(Ṽ),
• ψ(h) = 0 if h /∈ Ṽ,
where Ṽ is a compact set chosen such that V ⊂⊂ interior(Ṽ) ⊂⊂ U . Next,
set Z̃(z, ε) = ψ(h(z))Z(z, ε). Because the bump function was chosen to depend
only on the slow variables, our assumption about preservation of measures is still
satisfied; on each fiber, Z̃(z, 0) is a scaler multiple of Z(z, 0). Furthermore, the
flow of Z̃(·, 0)|Mh is ergodic for almost every h ∈ Ṽ. Then it would suffice to prove
our theorem for the vector fields Z̃(z, ε) with the set Ṽ replacing V. We assume
that this reduction has been made, although we will not use it until Step 5 below.
Step 1: Reduction using Gronwall’s Inequality. Observe that h̄(τ) satisfies
the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(zε(s), ε)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε. Here we have used the fact that h−1V × [0, ε0] is compact to
achieve uniformity over all initial conditions in the size of the O(ε) term above. We
use this fact repeatedly in what follows. In particular, H , H̄, and Z are uniformly
bounded and have uniform Lipschitz constants on the domains of interest.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄ |V)T . (2.7)
Step 2: Introduction of a time scale for ergodization. Choose a real-valued
function L(ε) such that L(ε) → ∞, L(ε) = o(log ε−1) as ε → 0. Think of L(ε) as
being a time scale which grows as ε→ 0 so that ergodization, i.e. the convergence
along an orbit of a function’s time average to a space average, can take place.
However, L(ε) doesn’t grow too fast, so that on this time scale zε(t) essentially
stays on one fiber, where we have our ergodicity assumption. Set tk,ε = kL(ε), so
0≤τ≤T∧Tε
|eε(τ)| ≤ O(εL(ε)) + ε
εL(ε)
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
. (2.8)
Step 3: A splitting for using the triangle inequality. Now we let zk,ε(s)
be the solution of
dzk,ε
= Z(zk,ε, 0), zk,ε(tk,ε) = zε(tk,ε).
Set hk,ε(t) = h(zk,ε(t)). Observe that hk,ε(t) is independent of t. We break up the
integral
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds into three parts:
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
∫ tk+1,ε
H(zε(s), 0)−H(zk,ε(s), 0)ds
∫ tk+1,ε
H(zk,ε(s), 0)− H̄(hk,ε(s))ds
∫ tk+1,ε
H̄(hk,ε(s))ds− H̄(hε(s))ds
:=Ik,ε + IIk,ε + IIIk,ε.
The term IIk,ε represents an “ergodicity term” that can be controlled by our
assumptions on the ergodicity of the flow z0(t), while the terms Ik,ε and IIIk,ε
represent “continuity terms” that can be controlled using the following control on
the drift from solutions along fibers.
Step 4: Control of drift from solutions along fibers.
Lemma 2.3.1. If 0 < tk+1,ε ≤ T∧Tεε ,
tk,ε≤t≤tk+1,ε
|zk,ε(t)− zε(t)| ≤ O(εL(ε)eLip(Z)L(ε))
Proof. Without loss of generality we may set k = 0, so that zk,ε(t) = z0(t). Then
for 0 ≤ t ≤ L(ε),
|z0(t)− zε(t)| =
Z(z0(s), 0)− Z(zε(s), ε)ds
≤ Lip (Z)
|ε|+ |z0(s)− zε(s)| ds
= O(εL(ε)) + Lip (Z)
|z0(s)− zε(s)| ds.
The result follows from Gronwall’s Inequality.
From Lemma 2.3.1 we find that Ik,ε, IIIk,ε = O(εL(ε)2eLip(Z)L(ε)).
Step 5: Use of ergodicity along fibers to control IIk,ε. From Equations
(2.7) and (2.8) and the triangle inequality, we already know that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
≤ O(ε) +O(εL(ε)) + ε T
εL(ε)
O(εL(ε)2eLip(Z)L(ε)) +O
εL(ε)
|IIk,ε|
= O(εL(ε)eLip(Z)L(ε)) +O
εL(ε)
|IIk,ε|
(2.9)
Fix δ > 0. Recalling that Tε = ∞, it suffices to show that
εL(ε)
|IIk,ε| ≥ δ
as ε→ 0.
For initial conditions z ∈ M and for 0 ≤ k ≤ T
εL(ε)
define
Bk,ε =
|IIk,ε| >
Bz,ε = {k : z ∈ Bk,ε} .
Think of these sets as describing “bad ergodization.” For example, roughly speak-
ing, z ∈ Bk,ε if the orbit zε(t) starting at z spends the time between tk,ε and tk+1,ε
in a region of phase space where the function H(·, 0) is “poorly ergodized” on the
time scale L(ε) by the flow z0(t) (as measured by the parameter δ/2T ). As IIk,ε
is clearly never larger than O(L(ε)), it follows that
εL(ε)
|IIk,ε| ≤
+O(εL(ε)#(Bz,ε)).
Therefore it suffices to show that
#(Bz,ε) ≥
const εL(ε)
as ε→ 0. By Chebyshev’s Inequality, we need only show that
E(εL(ε)#(Bz,ε)) = εL(ε)
εL(ε)
P (Bk,ε)
tends to 0 with ε.
In order to estimate the size of P (Bk,ε), it is convenient to introduce a new
measure P f that is uniformly equivalent to the restriction of Riemannian volume
P to h−1V. Here the f stands for “factor,” and P f is defined by
dP f = dh · dµh,
where dh represents integration with respect to the uniform measure on V.
Observe that B0,ε = zε(tk,ε)Bk,ε. In words, the initial conditions giving rise
to orbits that are “bad” on the time interval [tk,ε, tk+1,ε], moved forward by time
tk,ε, are precisely the initial conditions giving rise to orbits that are “bad” on the
time interval [t0,ε, t1,ε]. Because the flow z0(·) preserves the measure P f , we expect
P f(B0,ε) and P f(Bk,ε) to have roughly the same size. This is made precise by the
following lemma.
Lemma 2.3.2. There exists a constant K such that for each Borel set B ⊂ M
and each t ∈ [−T/ε, T/ε], P f(zε(t)B) ≤ eKTP f(B).
Proof. Assume that P f(B) > 0, and set γ(t) = ln
P f(zε(t)B)/P
. Then
γ(0) = 0, and
(t) =
zε(t)B
f̃(z)dz
zε(t)B
f̃(z)dz
zε(t)B
divP fZ(z, ε)dz
zε(t)B
f̃(z)dz
where f̃ > 0 is the C1 density of P f with respect to Riemannian volume on
h−1V, dz represents integration with respect to that volume, and divP fZ(z, ε) =
divzf̃(z)Z(z, ε). Because z0(t) preserves P
f , divP fZ(z, 0) ≡ 0. By Hadamard’s
Lemma, it follows that divP f Z(z, ε) = O(ε) on the compact set h−1V. Hence
dγ(t)/dt = O(ε), and the result follows.
Returning to our proof of Anosov’s theorem, it suffices to show that
P f(B0,ε) =
dh · µh
∫ L(ε)
H(z0(s), 0)− H̄(h0(0))ds
tends to 0 with ε. By our ergodicity assumption, for almost every h,
∫ L(ε)
H(z0(s), 0)− H̄(h0(0))ds
→ 0 as ε→ 0.
Finally, an application of the Bounded Convergence Theorem finishes the proof.
2.4 Moral
From the proofs of the theorems in this chapter, it should be apparent that there
are at least two key steps necessary for proving a version of the averaging principle
in the setting presented in Section 2.1.
The first step is estimating the continuity between the ε = 0 and the ε > 0
solutions of
= Z(z, ε).
In particular, on some relatively long timescale L = L(ε) ≪ ε−1, we need to show
0≤t≤L
|z0(t)− zε(t)| → 0
as ε → 0. As long as L is sub-logarithmic in ε−1, such estimates for smooth
systems can be made using Gronwall’s Inequality.
The second step is estimating the rate of ergodization of H(·, 0) by z0(t), i.e. es-
timating how fast
H(z0(s), 0) ds→ H̄(h0)
(generally as L → ∞). Note that the estimates in this step compete with those
in the first step in that, if L is small we obtain better continuity, but if L is large
we usually obtain better ergodization. Also, we do not need the full force of the
assumption of ergodicity of (z0(t), µh) on the fibers Mh. We only need z0(t) to
ergodize the specific function H(·, 0). Compare the proof of Theorem 2.2.5.
Note that in the setting of Anosov’s theorem, uniform ergodization leads to
uniform convergence in the averaging principle. Returning to the proof of Theo-
rem 2.1.1 above, suppose that
∫ L(ε)
H(z0(s), 0)ds→ H̄(h0)
uniformly over all initial conditions as L(ε) → ∞. Then for all ε sufficiently small
and each k, Bk,ε = ∅, and hence for all ε sufficiently small and each z, #(Bz,ε) = 0.
From Equation (2.9), it follows that sup0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
∣ → 0 as ε → 0,
uniformly over all initial conditions z ∈ h−1V. However, uniform convergence in
Birkhoff’s Ergodic Theorem is extremely rare and usually comes about because of
unique ergodicity, so it is unreasonable to expect this sort of uniform convergence
in most situations where Anosov’s theorem applies.
Chapter 3
Results for piston systems in one
dimension
In this chapter, we present our results for piston systems in one dimension. These
results may also be found in [Wri06].
3.1 Statement of results
3.1.1 The hard core piston problem
Consider the system of n1 + n2 + 1 point particles moving inside the unit interval
indicated in Figure 3.1. One distinguished particle, the piston, has position Q and
mass M . To the left of the piston there are n1 > 0 particles with positions q1,j
and masses m1,j, 1 ≤ j ≤ n1, and to the right there are n2 > 0 particles with
positions q2,j and masses m2,j, 1 ≤ j ≤ n2. These gas particles do not interact
with each other, but they interact with the piston and with walls located at the
end points of the unit interval via elastic collisions. We denote the velocities by
dQ/dt = V and dxi,j/dt = vi,j. There is a standard method for transforming
this system into a billiard system consisting of a point particle moving inside an
(n1 + n2 + 1)-dimensional polytope [CM06a], but we will not use this in what
follows.
We are interested in the dynamics of this system when the numbers and masses
of the gas particles are fixed, the total energy is bounded, and the mass of the
piston tends to infinity. When M = ∞, the piston remains at rest, and each
gas particle performs periodic motion. More interesting are the motions of the
system when M is very large but finite. Because the total energy of the system is
bounded, MV 2/2 ≤ const, and so V = O(M−1/2). Set
ε =M−1/2,
t t t t t t t
Figure 3.1: The piston system with n1 = 3 and n2 = 4. Note that the gas particles
do not interact with each other, but only with the piston and the walls.
and let
so that
with W = O(1).
When ε = 0, the system has n1 + n2 + 2 independent first integrals (conserved
quantities), which we take to be Q, W , and si,j = |vi,j |, the speeds of the gas
particles. We refer to these variables as the slow variables because they should
change slowly with time when ε is small, and we denote them by
h = (Q,W, s1,1, s1,2, · · · , s1,n1, s2,1, s2,2, · · · , s2,n2) ∈ Rn1+n2+2.
We will often abbreviate by writing h = (Q,W, s1,j, s2,j). Let hε(t, z) = hε(t)
denote the dynamics of these variables in time for a fixed value of ε, where z
represents the dependence on the initial condition in phase space. We usually
suppress the initial condition in our notation. Think of hε(·) as a random variable
which, given an initial condition in the 2(n1 + n2 + 1)-dimensional phase space,
produces a piecewise continuous path in Rn1+n2+2. These paths are the projection
of the actual motions in our phase space onto a lower dimensional space. The goal
of averaging is to find a vector field on Rn1+n2+2 whose orbits approximate hε(t).
The averaged equation
Sinai [Sin99] derived
= H̄(h) :=
j=1 m1,js
j=1 m2,js
−s1,jW
s2,jW
(3.1)
as the averaged equation (with respect to the slow time τ = εt) for the slow
variables. We provide a heuristic derivation in Section 3.2. Sinai solved this
equation as follows: From
d ln(s1,j)
= −d ln(Q)
s1,j(τ) = s1,j(0)Q(0)/Q(τ). Similarly, s2,j(τ) = s2,j(0)(1−Q(0))/(1−Q(τ)). Hence
j=1m1,js1,j(0)
2Q(0)2
j=1m2,js2,j(0)
2(1−Q(0))2
(1−Q)3 ,
and so (Q,W ) behave as if they were the coordinates of a Hamiltonian system
describing a particle undergoing periodic motion inside a potential well. If we let
s2i,j
be the kinetic energy of the gas particles on one side of the piston, the effective
Hamiltonian may be expressed as
W 2 +
E1(0)Q(0)
E2(0)(1−Q(0))2
(1−Q)2 . (3.2)
Hence, the solutions to the averaged equation are periodic for all initial conditions
under consideration.
Main result in the hard core setting
The solutions of the averaged equation approximate the motions of the slow vari-
ables, hε(t), on a time scale O(1/ε) as ε → 0. Precisely, let h̄(τ, z) = h̄(τ) be the
solution of
= H̄(h̄), h̄(0) = hε(0).
Again, think of h̄(·) as being a random variable that takes an initial condition in
our phase space and produces a path in Rn1+n2+2.
Next, fix a compact set V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂
R, and si,j ⊂⊂ (0,∞) for each i and j.1 For the remainder of this discussion we
will restrict our attention to the dynamics of the system while the slow variables
remain in the set V. To this end, we define the stopping time
Tε(z) = Tε := inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 3.1.1. For each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε =M−1/2 → 0.
This result was independently obtained by Gorelyshev and Neishtadt [GN06].
Note that the stopping time does not unduly restrict the result. Given any
c such that h = c ⇒ Q ∈ (0, 1), si,j ∈ (0,∞), then by an appropriate choice of
the compact set V we may ensure that, for all ε sufficiently small and all initial
conditions in our phase space with hε(0) = c, Tε ≥ T . We do this by choosing
V ∋ c such that the distance between ∂V and the periodic orbit h̄(τ) with h̄(0) = c
is positive. Call this distance d. Then Tε can only occur before T if hε(τ/ε) has
deviated by at least d from h̄(τ) for some τ ∈ [0, T ). Since the size of the deviations
tends to zero uniformly with ε, this is impossible for all small ε.
3.1.2 The soft core piston problem
In this section, we consider the same system of one piston and gas particles inside
the unit interval considered in Section 3.1.1, but now the interactions of the gas
particles with the walls and with the piston are smooth. Let κ : R → R be a C2
function satisfying
• κ(x) = 0 if x ≥ 1,
• κ′(x) < 0 if x < 1.
Let δ > 0 be a parameter of smoothing, and set
κδ(x) = κ(x/δ).
1 We have introduced this notation for convenience. For example, h ∈ V ⇒ Q ⊂⊂ (0, 1)
means that there exists a compact set A ⊂ (0, 1) such that h ∈ V ⇒ Q ∈ A, and similarly for
the other variables.
Then consider the Hamiltonian system obtained by having the gas particles inter-
act with the piston and the walls via the potential
κδ(q1,j) + κδ(Q− q1,j) +
κδ(q2,j −Q) + κδ(1− q2,j).
As before, we set ε =M−1/2 and W = V/ε. If we let
E1,j =
m1,jv
1,j + κδ(q1,j) + κδ(Q− q1,j), 1 ≤ j ≤ n1,
E2,j =
m2,jv
2,j + κδ(q2,j −Q) + κδ(1− q2,j), 1 ≤ j ≤ n2,
(3.3)
then Ei,j may be thought of as the energy associated with a gas particle, and
W 2/2 +
j=1E1,j +
j=1E2,j is the conserved energy.
When ε = 0, the Hamiltonian system admits n1 + n2 + 2 independent first
integrals, which we choose this time as h = (Q,W,E1,j , E2,j). While discussing
the soft core dynamics we use the energies Ei,j rather than the variables si,j =
2Ei,j/mi,j, which we used for the hard core dynamics, for convenience.
For comparison with the hard core results, we formally consider the dynamics
described by setting δ = 0 to be the hard core dynamics described in Section 3.1.1.
This is reasonable because we will only consider gas particle energies below the
barrier height κ(0). Then for any ε, δ ≥ 0, hδε(t) denotes the actual time evolution
of the slow variables. While discussing the soft core dynamics we often use δ as a
superscript to specify the dynamics for a certain value of δ. We usually suppress
the dependence on δ, unless it is needed for clarity.
Main result in the soft core setting
We have already seen that when δ = 0, there is an appropriate averaged vector
field H̄0 whose solutions approximate the actual motions of the slow variables,
h0ε(t). We will show that when δ > 0, there is also an appropriate averaged vector
field H̄δ whose solutions still approximate the actual motions of the slow variables,
hδε(t). We delay the derivation of H̄
δ until Section 3.4.1.
Fix a compact set V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂ R,
and Ei,j ⊂⊂ (0, κ(0)) for each i and j. For each ε, δ ≥ 0 we define the functions
h̄δ(·) and T δε on our phase space by letting h̄δ(τ) be the solution of
= H̄δ(h̄δ), h̄δ(0) = hδε(0), (3.4)
T δε = inf{τ ≥ 0 : h̄δ(τ) /∈ V or hδε(τ/ε) /∈ V}.
Theorem 3.1.2. There exists δ0 > 0 such that the averaged vector field H̄
δ(h) is
C1 on the domain {(δ, h) : 0 ≤ δ ≤ δ0, h ∈ V}. Furthermore, for each T > 0,
0≤δ≤δ0
initial conditions
s.t. hδε(0)∈V
0≤τ≤T∧T δε
∣hδε(τ/ε)− h̄δ(τ)
∣ = O(ε) as ε =M−1/2 → 0.
As in Section 3.1.1, for any fixed c there exists a suitable choice of the compact
set V such that for all sufficiently small ε and δ, T δε ≥ T whenever hδε(0) = c.
As we will see, for each fixed δ > 0, Anosov’s theorem 2.1.1 applies to the soft
core system and yields a weak law of large numbers, and Theorem 2.2.5 applies and
yields a strong law of large numbers with a uniform rate of convergence. However,
neither of these theorems yields the uniformity over δ in the result above.
3.1.3 Applications and generalizations
Relationship between the hard core and the soft core piston
It is not a priori clear that we can compare the motions of the slow variables on
the time scale 1/ε for δ > 0 versus δ = 0, i.e. compare the motions of the soft core
piston with the motions of the hard core piston on a relatively long time scale. It
is impossible to compare the motions of the fast-moving gas particles on this time
scale as ε → 0. As we see in Section 3.4, the frequency with which a gas particle
hits the piston changes by an amount O(δ) when we smooth the interaction. Thus,
on the time scale 1/ε, the number of collisions is altered by roughly O(δ/ε), and
this number diverges if δ is held fixed while ε → 0.
Similarly, one might expect that it is impossible to compare the motions of the
soft and hard core pistons as ε → 0 without letting δ → 0 with ε. However, from
Gronwall’s Inequality it follows that if h̄δ(0) = h̄0(0), then
0≤τ≤T∧T δε ∧T 0ε
∣h̄δ(τ)− h̄0(τ)
∣ = O(δ).
From the triangle inequality and Theorems 3.1.1 and 3.1.2 we obtain the following
corollary, which allows us to compare the motions of the hard core and the soft
core piston.
Corollary 3.1.3. As ε =M−1/2, δ → 0,
initial conditions
s.t. hδε(0)=c=h
0≤t≤(T∧T δε ∧T 0ε )/ε
∣hδε(t)− h0ε(t)
∣ = O(ε) +O(δ).
This shows that, provided the slow variables have the same initial conditions,
0≤t≤1/ε
∣hδε(t)− h0ε(t)
∣ = O(ε) +O(δ).
Thus the motions of the slow variables converge on the time scale 1/ε as ε, δ → 0,
and it is immaterial in which order we let these parameters tend to zero.
The adiabatic piston problem
We comment on what Theorem 3.1.1 says about the adiabatic piston problem. The
initial conditions of the adiabatic piston problem require thatW (0) = 0. Although
our system is so simple that a proper thermodynamical pressure is not defined,
we can define the pressure of a gas to be the average force received from the gas
particles by the piston when it is held fixed, i.e. P1 =
j=1 2m1,js1,j
= 2E1/Q
and P2 = 2E2/(1 − Q). Then if P1(0) > P2(0), the initial condition for our
averaged equation (3.1) has the motion of the piston starting at the left turning
point of a periodic orbit determined by the effective potential well. Up to errors not
much bigger than M−1/2, we see the piston oscillate periodically on the time scale
M1/2. If P1(0) < P2(0), the motion of the piston starts at a right turning point.
However, if P1(0) = P2(0), then the motion of the piston starts at the bottom of
the effective potential well. In this case of mechanical equilibrium, h̄(τ) = h̄(0),
and we conclude that, up to errors not much bigger thanM−1/2, we see no motion
of the piston on the time scale M1/2. A much longer time scale is required to see
if the temperatures equilibrate.
Generalizations
A simple generalization of Theorem 3.1.1, proved by similar techniques, follows.
The system consists of N − 1 pistons, that is, heavy point particles, located inside
the unit interval at positions Q1 < Q2 < . . . < QN−1. Walls are located at Q0 ≡ 0
and QN ≡ 1, and the piston at position Qi has mass Mi. Then the pistons divide
the unit interval into N chambers. Inside the ith chamber, there are ni ≥ 1 gas
particles whose locations and masses will be denoted by xi,j and mi,j , respectively,
where 1 ≤ j ≤ ni. All of the particles are point particles, and the gas particles
interact with the pistons and with the walls via elastic collisions. However, the
gas particles do not directly interact with each other. We scale the piston masses
as Mi = M̂i/ε
2 with M̂i constant, define Wi by dQi/dt = εWi, and let Ei be
the kinetic energy of the gas particles in the ith chamber. Then we can find an
appropriate averaged equation whose solutions have the pistons moving like an
(N − 1)-dimensional particle inside a potential well with an effective Hamiltonian
Ei(0)(Qi(0)−Qi−1(0))2
(Qi −Qi−1)2
If we write the slow variables as h = (Qi,Wi, |vi,j|) and fix a compact set V such
that h ∈ V ⇒ Qi+1 − Qi ⊂⊂ (0, 1),Wi ⊂⊂ R, and |vi,j| ⊂⊂ (0,∞), then the
convergence of the actual motions of the slow variables to the averaged solutions
is exactly the same as the convergence given in Theorem 3.1.1.
Remark 3.1.1. The inverse quadratic potential between adjacent pistons in the
effective Hamiltonian above is also referred to as the Calogero-Moser-Sutherland
potential. It has also been observed as the effective potential created between two
adjacent tagged particles in a one-dimensional Rayleigh gas by the insertion of one
very light particle inbetween the tagged particles [BTT07].
3.2 Heuristic derivation of the averaged equa-
tion for the hard core piston
We present here a heuristic derivation of Sinai’s averaged equation (3.1) that is
found in [Dol05].
First, we examine interparticle collisions when ε > 0. When a particle on the
left, say the one at position q1,j , collides with the piston, s1,j andW instantaneously
change according to the laws of elastic collisions:
v+1,j
m1,j +M
m1,j −M 2M
2m1,j M −m1,j
v−1,j
. (3.5)
If the speed of the left gas particle is bounded away from zero, and W = M1/2V
is also bounded, it follows that for all ε sufficiently small, any collision will have
v−1,j > 0 and v
1,j < 0. In this case, when we translate Equation (3.5) into our new
coordinates, we find that
s+1,j
1 + ε2m1,j
1− ε2m1,j −2ε
2εm1,j 1− ε2m1,j
s−1,j
, (3.6)
so that
∆s1,j = s
1,j − s−1,j = −2εW− +O(ε2),
∆W =W+ −W− = +2εm1,js−1,j +O(ε2).
The situation is analogous when particles on the right collide with the piston.
For all ε sufficiently small, s2,j and W instantaneously change by
∆W = W+ −W− = −2εm2,js−2,j +O(ε2),
∆s2,j = s
2,j − s−2,j = +2εW− +O(ε2).
We defer discussing the rare events in which multiple gas particles collide with the
piston simultaneously, although we will see that they can be handled appropriately.
Let ∆t be a length of time long enough such that the piston experiences many
collisions with the gas particles, but short enough such that the slow variables
change very little, in this time interval. From each collision with the particle at
position q1,j , W changes by an amount +2εm1,js1,j + O(ε2), and the frequency
of these collisions is approximately
. Arguing similarly for collisions with the
other particles, we guess that
2m1,js1,j
2m2,js2,j
2(1−Q) +O(ε
Note that not only does the position of the piston change slowly in time, but its
velocity also changes slowly, i.e. the piston has inertia. With τ = εt as the slow
time, a reasonable guess for the averaged equation for W is
m1,js
m2,js
1−Q .
Similar arguments for the other slow variables lead to the averaged equation (3.1).
3.3 Proof of the main result for the hard core
piston
3.3.1 Proof of Theorem 3.1.1 with only one gas particle on
each side
We specialize to the case when there is only one gas particle on either side of
the piston, i.e. we assume that n1 = n2 = 1. We then denote x1,1 by q1, m2,2
by m2, etc. This allows the proof’s major ideas to be clearly expressed, without
substantially limiting their applicability. At the end of this section, we outline the
simple generalizations needed to make the proof apply in the general case.
A choice of coordinates on the phase space for a three particle system
As part of our proof, we choose a set of coordinates on our six-dimensional phase
space such that, in these coordinates, the ε = 0 dynamics are smooth. Complete
the slow variables h = (Q,W, s1, s2) to a full set of coordinates by adding the
coordinates ϕi ∈ [0, 1]/ 0 ∼ 1 = S1, i = 1, 2, defined as follows:
ϕ1 = ϕ1(q1, v1, Q) =
if v1 > 0
1− q1
if v1 < 0
ϕ2 = ϕ2(q2, v2, Q) =
2(1−Q) if v2 < 0
1− 1−q2
2(1−Q) if v2 > 0
When ε = 0, these coordinates are simply the angle variable portion of action-
angle coordinates for an integrable Hamiltonian system. They are defined such
that collisions occur between the piston and the gas particles precisely when ϕ1
or ϕ2 = 1/2. Then z = (h, ϕ1, ϕ2) represents a choice of coordinates on our phase
space, which is homeomorphic to (a subset of R4) × T2. We abuse notation and
also let h(z) represent the projection onto the first four coordinates of z.
Now we describe the dynamics of our system in these coordinates. When
ϕ1, ϕ2 6= 1/2,
ϕ1 if 0 ≤ ϕ1 < 1/2
(1− ϕ1) if 1/2 < ϕ1 ≤ 1
2(1−Q) +
1−Qϕ2 if 0 ≤ ϕ2 < 1/2
2(1−Q) −
1−Q(1− ϕ2) if 1/2 < ϕ2 ≤ 1
Hence between interparticle collisions, the dynamics are smooth and are described
= εW,
+O(ε),
2(1−Q) +O(ε).
(3.7)
When ϕ1 reaches 1/2, while ϕ2 6= 1/2, the coordinates Q, s2, ϕ1, and ϕ2 are
instantaneously unchanged, while s1 andW instantaneously jump, as described by
Equation (3.6). As an aside, it is curious that s+1 +εW
+ = s−1 −εW−, so that dϕ1/dt
is continuous as ϕ1 crosses 1/2. However, the collision induces discontinuous jumps
of size O(ε2) in dQ/dt and dϕ2/dt. Denote the linear transformation in Equation
(3.6) with j = 1 by A1,ε. Then
A1,ε =
1 −2ε
2εm1 1
+O(ε2).
The situation is analogous when ϕ2 reaches 1/2, while ϕ1 6= 1/2. Then W and
s2 are instantaneously transformed by a linear transformation
A2,ε =
1 −2εm2
+O(ε2).
We also account for the possibility of all three particles colliding simultane-
ously. There is no completely satisfactory way to do this, as the dynamics have an
essential singularity near {ϕ1 = ϕ2 = 1/2}. Furthermore, such three particle colli-
sions occur with probability zero with respect to the invariant measure discussed
below. However, the two 3× 3 matrices
A1,ε 0
0 A2,ε
have a commutator of size O(ε2). We will see that this small of an error will make
no difference to us as ε → 0, and so when ϕ1 = ϕ2 = 1/2, we pretend that the
left particle collides with the piston instantaneously before the right particle does.
Precisely, we transform the variables s1, W, and s2 by
0 A2,ε
A1,ε 0
We find that
∆s1 = s
1 − s−1 = −2εW− +O(ε2),
∆W = W+ −W− = +2εm1s−1 − 2εm2s−2 +O(ε2),
∆s2 = s
2 − s−2 = +2εW− +O(ε2).
The above rules define a flow on the phase space, which we denote by zε(t).
We denote its components by Qε(t), Wε(t), s1,ε(t), etc. When ε > 0, the flow is
not continuous, and for definiteness we take zε(t) to be left continuous in t.
Because our system comes from a Hamiltonian system, it preserves Liouville
measure. In our coordinates, this measure has a density proportional to Q(1 −
Q). That this measure is preserved also follows from the fact that the ordinary
differential equation (3.7) preserves this measure, and the matrices A1,ε, A2,ε have
determinant 1. Also note that the set {ϕ1 = ϕ2 = 1/2} has co-dimension two,
and so
t zε(t){ϕ1 = ϕ2 = 1/2} has co-dimension one, which shows that only a
measure zero set of initial conditions will give rise to three particle collisions.
Argument for uniform convergence
Step 1: Reduction using Gronwall’s Inequality. Define H(z) by
H(z) =
2m1s1δϕ1=1/2 − 2m2s2δϕ2=1/2
−2Wδϕ1=1/2
2Wδϕ2=1/2
Here we make use of Dirac delta functions. All integrals involving these delta
functions may be replaced by sums. We explicitly deal with any ambiguities arising
from collisions occurring at the limits of integration.
Lemma 3.3.1. For 0 ≤ t ≤ T∧Tε
hε(t)− hε(0) = ε
H(zε(s))ds+O(ε),
where any ambiguity about changes due to collisions occurring precisely at times 0
and t is absorbed in the O(ε) term.
Proof. There are four components to verify. The first component requires that
Qε(t)−Qε(0) = ε
Wε(s)ds+O(ε). This is trivially true because Qε(t)−Qε(0) =
Wε(s)ds.
The second component states that
Wε(t)−Wε(0) = ε
2m1s1,ε(s)δϕ1,ε(s)=1/2−2m2s2,ε(s)δϕ2,ε(s)=1/2ds+O(ε). (3.8)
Let rk and qj be the times in (0, t) such that ϕ1,ε(rk) = 1/2 and ϕ2,ε(qj) = 1/2,
respectively. Then
Wε(t)−Wε(0) =
∆Wε(rk) +
∆Wε(qj) +O(ε).
Observe that there exists ω > 0 such that for all sufficiently small ε and all h ∈ V,
1/ω < dϕi
< ω. Thus the number of collisions in a time interval grows no faster
than linearly in the length of that time interval. Because t ≤ T/ε, it follows that
Wε(t)−Wε(0) = ε
2m1s1,ε(rk)− ε
2m2s2,ε(qj) +O(ε),
and Equation (3.8) is verified. Note that because V is compact, there is uniformity
over all initial conditions in the size of the O(ε) terms above. The third and fourth
components are handled similarly.
Next, h̄(τ) satisfies the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = O(ε) + ε
∫ τ/ε
H(zε(s))ds
= O(ε) + ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄ |V)T . (3.9)
Gronwall’s Inequality is usually stated for continuous paths, but the standard
proof (found in [SV85]) still works for paths that are merely integrable, and
∣h̄(τ)− hε(τ/ε)
∣ is piecewise smooth.
Step 2: A splitting according to particles. Now
H(z)− H̄(h) =
2m1s1δϕ1=1/2 −m1s21/Q
−2Wδϕ1=1/2 + s1W/Q
−2m2s2δϕ2=1/2 +m2s22/(1−Q)
2Wδϕ2=1/2 − s2W/(1−Q)
and so, in order to show that sup0≤τ≤T∧Tε |eε(τ)| = O(ε), it suffices to show that
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
= O(1),
0≤τ≤T∧Tε
∫ τ/ε
Wε(s)δϕ1,ε(s)=1/2 −
Wε(s)s1,ε(s)
2Qε(s)
= O(1),
as well as two analogous claims about terms involving ϕ2,ε. Thus we have effec-
tively separated the effects of the different gas particles, so that we can deal with
each particle separately. We will only show that
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
= O(1).
The other three terms can be handled similarly.
Step 3: A sequence of times adapted for ergodization. Ergodization refers
to the convergence along an orbit of a function’s time average to its space average.
For example, because of the splitting according to particles above, one can easily
check that 1
H(z0(s))ds = H̄(h0) + O(1/t), even when z0(·) restricted to the
invariant tori Mh0 is not ergodic. In this step, for each initial condition zε(0) in
our phase space, we define a sequence of times tk,ε inductively as follows: t0,ε =
inf{t ≥ 0 : ϕ1,ε(t) = 0}, tk+1,ε = inf{t > tk,ε : ϕ1,ε(t) = 0}. This sequence is
chosen because δϕ1,0(s)=1/2 is “ergodizd” as time passes from tk,0 to tk+1,0. If ε is
sufficiently small and tk+1,ε ≤ (T ∧ Tε)/ε, then the spacings between these times
are uniformly of order 1, i.e. 1/ω < tk+1,ε − tk,ε < ω. Thus,
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
≤ O(1) +
tk+1,ε≤T∧Tεε
∫ tk+1,ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
(3.10)
Step 4: Control of individual terms by comparison with solutions along
fibers. The sum in Equation (3.10) has no more than O(1/ε) terms, and so it
suffices to show that each term is no larger than O(ε). We can accomplish this by
comparing the motions of zε(t) for tk,ε ≤ t ≤ tk+1,ε with the solution of the ε = 0
version of Equation (3.7) that, at time tk,ε, is located at zε(tk,ε). Since each term
in the sum has the same form, without loss of generality we will only examine the
first term and suppose that t0,ε = 0, i.e. that ϕ1,ε(0) = 0.
Lemma 3.3.2. If t1,ε ≤ T∧Tεε , then sup0≤t≤t1,ε |z0(t)− zε(t)| = O(ε).
Proof. To check that sup0≤t≤t1,ε |h0(t)− hε(t)| = O(ε), first note that h0(t) =
h0(0) = hε(0). Then dQε/dt = O(ε), so that Q0(t)−Qε(t) = O(εt). Furthermore,
the other slow variables change by O(ε) at collisions, while the number of collisions
in the time interval [0, t1,ε] is O(1).
It remains to show that sup0≤t≤t1,ε |ϕi,0(t)− ϕi,ε(t)| = O(ε). Using what we
know about the divergence of the slow variables,
ϕ1,0(t)− ϕ1,ε(t) =
s1,0(s)
2Q0(s)
− s1,ε(s)
2Qε(s)
+O(ε)ds =
O(ε)ds = O(ε)
for 0 ≤ t ≤ t1,ε. Showing that sup0≤t≤t1,ε |ϕ2,0(t)− ϕ2,ε(t)| = O(ε) is similar.
From Lemma 3.3.2, t1,ε = t1,0 +O(ε) = 2Q0/s1,0 +O(ε). We conclude that
∫ t1,ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
ds = O(ε) +
∫ t1,ε
s1,0(s)δϕ1,ε(s)=1/2 −
s1,0(s)
2Q0(s)
= O(ε) + s1,0 − t1,ε
s21,0
= O(ε).
It follows that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε), independent of the initial
condition in h−1V.
3.3.2 Extension to multiple gas particles
When n1, n2 > 1, only minor modifications are necessary to generalize the proof
above. We start by extending the slow variables h to a full set of coordinates on
phase space by defining the angle variables ϕi,j ∈ [0, 1]/ 0 ∼ 1 = S1 for 1 ≤ i ≤ 2,
1 ≤ j ≤ ni:
ϕ1,j = ϕ1,j(q1,j , v1,j, Q) =
if v1,j > 0
1− q1,j
if v1,j < 0
ϕ2,j = ϕ2,j(q2,j , v2,j, Q) =
1−q2,j
2(1−Q) if v2,j < 0
1− 1−q2,j
2(1−Q) if v2,j > 0
Then dϕ1,j/dt = s1,j(2Q)
−1 + O(ε), dϕ2,j/dt = s2,j(2(1 − Q))−1 + O(ε), and
z = (h, ϕ1,j , ϕ2,j) represents a choice of coordinates on our phase space, which
is homeomorphic to (a subset of Rn1+n2+2) × Tn1+n2 . In these coordinates, the
dynamical system yields a discontinuous flow zε(t) on phase space. The flow
preserves Liouville measure, which in our coordinates has a density proportional
to Qn1(1 − Q)n2 . As is Section 3.3.1, one can show that the measure of initial
conditions leading to multiple particle collisions is zero.
Next, define H(z) by
H(z) =
j=1 2m1,js1,jδϕ1,j=1/2 −
j=1 2m2s2,jδϕ2,j=1/2
−2Wδϕ1,j=1/2
2Wδϕ2,j=1/2
For 0 ≤ t ≤ T∧Tε
, hε(t) − hε(0) = ε
H(zε(s))ds + O(ε). From here, the rest of
the proof follows the same arguments made in Section 3.3.1.
3.4 Proof of the main result for the soft core
piston
For the remainder of this chapter, we consider the family of Hamiltonian systems
introduced in Section 3.1.2, which are parameterized by ε, δ ≥ 0. For simplicity,
we specialize to n1 = n2 = 1. As in Section 3.3, the generalization to n1, n2 > 1
is not difficult. The Hamiltonian dynamics are given by the following ordinary
differential equation:
= εW,
= ε (−κ′δ(Q− x1) + κ′δ(x2 −Q)) ,
= v1,
−κ′δ(x1) + κ′δ(Q− x1)
= v2,
−κ′δ(x2 −Q) + κ′δ(1− x2)
(3.11)
Recalling the particle energies defined by Equation (3.3), we find that
= εWκ′δ(Q− x1),
= −εWκ′δ(x2 −Q).
For the compact set V introduced in Section 3.1.2, fix a small positive number
E and an open set U ⊂ R4 such that V ⊂ U and h ∈ U ⇒ Q ∈ (E , 1−E),W ⊂⊂ R,
and E < E1, E2 < κ(0)− E . We only consider the dynamics for 0 < δ < E/2 and
h ∈ U .
Define
U1(q1) = U1(q1, Q, δ) := κδ(q1) + κδ(Q− q1),
U2(q2) = U2(q2, Q, δ) := κδ(q2 −Q) + κδ(1− q2).
Then the energies Ei satisfy Ei = miv
i /2 + Ui(xi).
Let T1 = T1(Q,E1, δ) and T2 = T2(Q,E2, δ) denote the periods of the motions
of the left and right gas particles, respectively, when ε = 0.
Lemma 3.4.1. For i = 1, 2,
Ti ∈ C1{(Q,Ei, δ) : Q ∈ (E , 1− E), Ei ∈ (E , κ(0)− E), 0 ≤ δ < E/2}.
Furthermore,
T1(Q,E1, δ) =
Q +O(δ),
T2(Q,E2, δ) =
(1−Q) +O(δ).
The proof of this lemma is mostly computational, and so we delay it until
Section 3.5. Note especially that the periods can be suitably defined such that
their regularity extends to δ = 0.
In this section, and in Section 3.5 below, we adopt the following convention
on the use of the O notation. All use of the O notation will explicitly contain the
dependence on ε and δ as ε, δ → 0. For example, if a function f(h, ε, δ) = O(ε),
then there exists δ′, ε′ > 0 such that sup0<ε≤ε′, 0<δ≤δ′, h∈V |f(h, ε, δ)/ε| <∞.
When ε = 0,
(Ei − Ui(xi)).
Define a = a(Ei, δ) by
κδ(a) = κ(a/δ) = Ei,
so that a(E1, δ) is a turning point for the left gas particle. Then a = δκ
−1(Ei),
where κ−1 is defined as follows: κ : [0, 1] → [0, κ(0)] takes 0 to κ(0) and 1 to 0.
Furthermore, κ ∈ C2([0, 1]), κ′ ≤ 0, and κ′(x) < 0 if x < 1. By monotonicity,
κ−1 : [0, κ(0)] → [0, 1] exists and takes 0 to 1 and κ(0) to 0. Also, by the Implicit
Function Theorem, κ−1 ∈ C2((0, κ(0)]), (κ−1)′(y) < 0 for y > 0, and (κ−1)′(y) →
−∞ as y → 0+. Because we only consider energies Ei ∈ (E , κ(0)− E), it follows
that a(Ei, δ) is a C2 function for the domains of interest.
3.4.1 Derivation of the averaged equation
As we previously pointed out, for each fixed δ > 0, Anosov’s theorem 2.1.1 and
Theorem 2.2.5 apply directly to the family of ordinary differential equations in
Equation (3.11), provided that δ is sufficiently small. The invariant fibers Mh of
the ε = 0 flow are tori described by a fixed value of the four slow variables and
{(Q,W, q1, v1, q2, v2) : E1 = m1v21/2 + U1(q1, Q, δ), E2 = m2v22/2 + U2(q2, Q, δ)}. If
we use (q1, q2) as local coordinates on Mh, which is valid except when v1 or v2 = 0,
the invariant measure µh of the unperturbed flow has the density
dq1dq2
(E1 − U1(q1)) T2
(E2 − U2(q2))
The restricted flow is ergodic for almost every h. See Corollary 3.5.1 in Section
−κ′δ(Q− q1) + κ′δ(q2 −Q)
Wκ′δ(Q− q1)
−Wκ′δ(q2 −Q)
κ′δ(Q− q1)dµh =
∫ Q−a
κ′δ(Q− q1)
(E1 − U1(q1))
∫ Q−a
κ′δ(Q− q1)
E1 − κδ(Q− q1)
E1 − u
8m1E1
Similarly,
κ′δ(q2 −Q)dµh = −
8m2E2
It follows that the averaged vector field is
H̄δ(h) =
8m1E1
8m2E2
8m1E1
8m2E2
where from Lemma 3.4.1 we see that H̄ ·(·) ∈ C1({(δ, h) : 0 ≤ δ < E/2, h ∈ V}).
H̄0(h) agrees with the averaged vector field for the hard core system from Equation
(3.1), once we account for the change of coordinates Ei = mis
i /2.
Remark 3.4.1. An argument due to Neishtadt and Sinai [NS04] shows that the
solutions to the averaged equation (3.4) are periodic. This argument also shows
that, as in the case δ = 0, the limiting dynamics of (Q,W ) are effectively Hamil-
tonian, with the shape of the Hamiltonian depending on δ, Q(0), and the initial
energies of the gas particles. The argument depends heavily on the observation
that the phase integrals
Ii(Q,Ei, δ) =
miv2+Ui(x,Q,δ)≤Ei
are adiabatic invariants, i.e. they are integrals of the solutions to the averaged
equation. Thus the four-dimensional phase space of the averaged equation is foli-
ated by invariant two-dimensional submanifolds, and one can think of the effective
Hamiltonians for the piston as living on these submanifolds.
3.4.2 Proof of Theorem 3.1.2
The following arguments are motivated by our proof in Section 3.3, although the
details are more involved as we show that the rate of convergence is independent
of all small δ.
A choice of coordinates on phase space
We wish to describe the dynamics in a coordinate system inspired by the one used
in Section 3.3.1. For each fixed δ ∈ (0, δ0], this change of coordinates will be C1
in all variables on the domain of interest. However, it is an exercise in analysis to
show this, and so we delay the proofs of the following two lemmas until Section
We introduce the angular coordinates ϕi ∈ [0, 1]/ 0 ∼ 1 = S1 defined by
ϕ1 = ϕ1(q1, v1, Q) =
0 if q1 = a
E1−U1(s)ds if v1 > 0
1/2 if q1 = Q− a
E1−U1(s)ds if v1 < 0
ϕ2 = ϕ2(q2, v2, Q) =
0 if q2 = 1− a
∫ 1−a
E2−U2(s)ds if v2 < 0
1/2 if q2 = Q+ a
∫ 1−a
E2−U2(s)ds if v2 > 0
. (3.12)
Then z = (h, ϕ1, ϕ2) is a choice of coordinates on h
−1U . As before, we will abuse
notation and let h(z) denote the projection onto the first four coordinates of z.
There is a fixed value of δ0 in the statement of Theorem 3.1.2. However, for
the purposes of our proof, it will be convenient to progressively choose δ0 smaller
when needed. At the end of the proof, we will have only shrunk δ0 a finite number
of times, and this final value will satisfies the requirements of the theorem. Our
first requirement on δ0 is that it is smaller than E/2.
Lemma 3.4.2. If δ0 > 0 is sufficiently small, then for each δ ∈ (0, δ0] the ordinary
differential equation (3.11) in the coordinates z takes the form
= Zδ(z, ε), (3.13)
where Zδ ∈ C1(h−1U × [0,∞)). When z ∈ h−1U ,
Zδ(z, ε) =
−κ′δ(Q− q1(z)) + κ′δ(q2(z)−Q)
εWκ′δ(Q− q1(z))
−εWκ′δ(q2(z)−Q)
+O(ε)
+O(ε)
. (3.14)
Recall that, by our conventions, the O(ε) terms in Equation (3.14) have a
size that can be bounded independent of all δ sufficiently small. Denote the flow
determined by Zδ(·, ε) by zδε(t), and its components by Qδε(t), W δε (t), Eδ1,ε(t), etc.
Also, set hδε(t) = h(z
ε(t)). From Equation (3.14),
Hδ(z, ε) :=
−κ′δ(Q− q1(z)) + κ′δ(q2(z)−Q)
Wκ′δ(Q− q1(z))
−Wκ′δ(q2(z)−Q)
. (3.15)
In particular, Hδ(z, ε) = Hδ(z, 0).
Before proceeding, we need one final technical lemma.
Lemma 3.4.3. If δ0 > 0 is chosen sufficiently small, there exists a constant K
such that for all δ ∈ (0, δ0], κ′δ(|Q− xi(z)|) = 0 unless ϕi ∈ [1/2−Kδ, 1/2 +Kδ].
Argument for uniform convergence
We start by proving the following lemma, which essentially says that an orbit
zδε(t) only spends a fraction O(δ) of its time in a region of phase space where
∣Hδ(zδε(t), ε)
∣Hδ(zδε(t), 0)
∣ is of size O(δ−1)
Lemma 3.4.4. For 0 ≤ T ′ ≤ T ≤ T∧T
∣Hδ(zδε(s), 0)
∣ ds = O(1 ∨ (T − T ′)).
Proof. Without loss of generality, T ′ = 0. From Lemmas 3.4.1 and 3.4.2 it follows
that if we choose δ0 sufficiently small, then there exists ω > 0 such that for all
sufficiently small ε and all δ ∈ (0, δ0], h ∈ V ⇒ 1/ω <
< ω. Define the set
B = [1/2 − Kδ, 1/2 + Kδ], where K comes from Lemma 3.4.3. Then we find a
crude bound on
Qδε(s)− q1(zδε(s))
∣ ds using that
dϕδ1,ε
≥ 1/ω if ϕδ1,ε ∈ B
≤ ω if ϕδ1,ε ∈ Bc.
This yields
Qδε(s)− q1(zδε(s))
∣ ds ≤ const
1ϕδ1,ε(s)∈Bds
≤ const
2Kωδ + 1−2Kδ
T + 2Kωδ
= O(1 ∨ T ).
Similarly,
∣κ′δ(q2(z
ε(s))−Qδε(s))
∣ ds = O(1∨ T ), and so
∣Hδ(zδε(s), 0)
∣ ds =
O(1 ∨ T ).
We now follow steps one through four from Section 3.3.1, making modifications
where necessary.
Step 1: Reduction using Gronwall’s Inequality. Now hδε(τ/ε) satisfies
hδε(τ/ε)− hδε(0) = ε
∫ τ/ε
Hδ(zδε(s), 0)ds.
Define
eδε(τ) = ε
∫ τ/ε
Hδ(zδε(s), 0)− H̄δ(hδε(s))ds.
It follows from Gronwall’s Inequality and the fact that H̄ ·(·) ∈ C1({(δ, h) : 0 ≤
δ ≤ δ0, h ∈ V}) that
0≤τ≤T∧T δε
∣hδε(τ/ε)− h̄δ(τ)
0≤τ≤T∧T δε
∣eδε(τ)
eLip(H̄
δ|V)T
0≤τ≤T∧T δε
∣eδε(τ)
(3.16)
Step 2: A splitting according to particles. Next,
Hδ(z, 0)− H̄δ(h)
−κ′δ(Q− q1(z))−
8m1E1
Wκ′δ(Q− q1(z)) +W
8m1E1
κ′δ(q2(z)−Q) +
8m2E2
−Wκ′δ(q2(z)−Q)−W
8m2E2
and so, in order to show that sup0≤τ≤T∧T δε
∣eδε(τ)
∣ = O(ε), it suffices to show that
for i = 1, 2,
0≤τ≤T∧T δε
∫ τ/ε
∣Qδε(s)− xi(zδε(s))
i,ε(s)
Ti(Qδε(s), E
i,ε(s), δ)
= O(1),
0≤τ≤T∧T δε
∫ τ/ε
Wε(s)κ
∣Qδε(s)− xi(zδε(s))
+Wε(s)
i,ε(s)
Ti(Qδε(s), E
i,ε(s), δ)
= O(1).
We only demonstrate that
0≤τ≤T∧T δε
∫ τ/ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
= O(1).
The other three terms are handled similarly.
Step 3: A sequence of times adapted for ergodization. Define the se-
quence of times tδk,ε inductively by t
0,ε = inf{t ≥ 0 : ϕδ1,ε(t) = 0}, tδk+1,ε = inf{t >
tδk,ε : ϕ
1,ε(t) = 0}. If ε and δ are sufficiently small and tδk+1,ε ≤ (T ∧ T δε )/ε, then
it follows from Lemma 3.4.2 and the discussion in the proof of Lemma 3.4.4 that
1/ω < tδk+1,ε − tδk,ε < ω. From Lemmas 3.4.2 and 3.4.4 it follows that
0≤τ≤T∧T δε
∫ τ/ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
≤ O(1) +
k+1,ε
k+1,ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
(3.17)
Step 4: Control of individual terms by comparison with solutions along
fibers. As before, it suffices to show that each term in the sum in Equation
(3.17) is no larger than O(ε). Without loss of generality we will only examine the
first term and suppose that tδ0,ε = 0, i.e. that ϕ
1,ε(0) = 0.
Lemma 3.4.5. If tδ1,ε ≤
T∧T δε
, then sup0≤t≤tδ1,ε
∣zδ0(t)− zδε(t)
∣ = O(ε).
Proof. By Lemma 3.4.4, hδ0(t) − hδε(t) = hδε(0) − hδε(t) = −ε
Hδ(zδε(s), 0)ds =
O(ε(1 ∨ t)) for t ≥ 0.
Using what we know about the divergence of the slow variables, we find that
ϕδ1,0(t)− ϕδ1,ε(t) =
0(s), E
0(s), δ)
T1(Qδε(s), E
ε(s), δ)
+O(ε)ds
O(ε)ds
= O(ε)
for 0 ≤ t ≤ tδ1,ε. Lemmas 3.4.1 and 3.4.2 ensure the desired uniformity in the sizes
of the orders of magnitudes. Showing that sup0≤t≤tδ1,ε
∣ϕδ2,0(t)− ϕδ2,ε(t)
∣ = O(ε) is
similar.
From Lemma 3.4.5 we find that t1,ε = t1,0+O(ε) = T1(Qδ0, Eδ0 , δ)+O(ε). Hence
∫ tδ1,ε
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
ds = O(ε) +
∫ tδ1,0
1,0, δ)
= O(ε) +
But when q1(z
ε) < Q
ε − a,
Eδ1,ε(s)− κδ
Qδε(s)− q1(zδε(s))
ε(s))
Qδε(s)− q1(zδε(s))
and so
∫ tδ1,ε
Qδε(s)− q1(zδε(s))
ds = −
1,ε(0)−
1,ε(t
= O(ε)−
Hence,
∫ tδ1,ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
ds = O(ε),
as desired.
3.5 Appendix to Section 3.4
Proof of Lemma 3.4.1:
Proof. For 0 < δ < E/2,
T1 = T1(Q,E1, δ) = 2
∫ Q−a
E1 − U1(s)
T2 = T2(Q,E2, δ) = 2
∫ 1−a
E2 − U2(s)
We only consider the claims about T1, and for convenience we take m1 = 2. Then
T1(Q,E1, δ) = 2
∫ Q−a
E1 − U1(s)
∫ Q/2
E1 − κδ(s)
Q/2− δ√
E1 − κδ(s)
2Q− 4δ√
κ−1(E1)
E1 − κ(s)
Define
F (E) :=
κ−1(E)
E − κ(s)
−(κ−1)′(u)√
E − u
Notice that (κ−1)′(u) diverges as u → 0+, while (E − u)−1/2 diverges as u → E−,
but both functions are still integrable on [0, E]. It follows that F (E) is well defined.
Then it suffices to show that F : [E , κ(0)− E ] → R is C1.
Write
F (E) =
∫ E/2
−(κ−1)′(u)√
E − u
−(κ−1)′(u)√
E − u
:= F1(E) + F2(E).
A standard application of the Dominated Convergence Theorem allows us to dif-
ferentiate inside the integral and conclude that F1 ∈ C∞([E , κ(0)− E ]), with
F ′1(E) =
∫ E/2
(κ−1)′(u)
2(E − u)3/2du.
To examine F2, we make the substitution v = E − u to find that
F2(E) =
∫ E−E/2
−(κ−1)′(E − v)√
Using the fact that (κ−1)′ ∈ C1([E/2, κ(0)]) and the Dominated Convergence The-
orem, we find that F2 is differentiable, with
F ′2(E) =
−(κ−1)′(E/2)
E − E/2
∫ E−E/2
−(κ−1)′′(E − v)√
Another application of the Dominated Convergence Theorem shows that F ′2 is
continuous, and so F2 ∈ C1([E , κ(0)− E ]).
T1(Q,E1, δ) =
−E−1/21 + F1(E1) + F2(E1)
has the desired regularity. For future reference, we note that
+O(δ). (3.18)
Corollary 3.5.1. For all δ sufficiently small, the flow zδ0(t) restricted to the in-
variant tori Mc = {h = c} is ergodic (with respect to the invariant Lebesgue
measure) for almost every c ∈ U .
Proof. The flow is ergodic whenever the periods T1 and T2 are irrationally related.
Fix δ sufficiently small such that ∂T1
= −Q/E3/21 + O(δ) < 0. Next, consider
Q, W , and E2 fixed, so that T2 is constant. Because T1 ∈ C1, it follows that,
as we let E1 vary,
/∈ Q for almost every E1. The result follows from Fubini’s
Theorem.
Proof of Lemma 3.4.2:
Proof. For the duration of this proof, we consider the dynamics for a small, fixed
value of δ > 0, which we generally suppress in our notation. For convenience, we
take m1 = 2.
Let ψ denote the map taking (Q,W, q1, v1, q2, v2) to (Q,W,E1, E2, ϕ1, ϕ2). We
claim that ψ is a C1 change of coordinates on the domain of interest. Since E1 =
v21 + κδ(q1) + κδ(Q− q1), E1 is a C2 function of q1, v1, and Q. A similar statement
holds for E2.
The angular coordinates ϕi(xi, vi, Q) are defined by Equation (3.12). We only
consider ϕ1, as the statements for ϕ2 are similar. Then ϕ1(q1, v1, Q) is clearly C1
whenever q1 6= a,Q−a. The apparent difficulties in regularity at the turning points
are only a result of how the definition of ϕ1 is presented in Equation (3.12). Recall
that the angle variables are actually defined by integrating the elapsed time along
orbits, and our previous definition expressed ϕ1 in a manner which emphasized
the dependence on q1. In fact, whenever |v1| <
ϕ1(q1, v1, Q) =
(κ−1δ )
′(E1 − v2)dv if q1 < δ
(κ−1δ )
′(E1 − v2)dv if q1 > Q− δ.
(3.19)
Here E1 is implicitly considered to be a function of q1, v1, and Q. One can verify
that Dψ is non-degenerate on the domain of interest, and so ψ is indeed a C1
change of coordinates.
Next observe that dϕ1,0/dt = 1/T1, so Hadamard’s Lemma implies that
dϕ1,ε
+O(εf(δ)).
It remains to show that, in fact, we may take f(δ) = 1. It is easy to verify this
whenever q1 ≤ Q−δ because dE1/dt = 0 there. We only perform the more difficult
verification when q1 > Q− δ.
When q1 > Q−δ, |v1| <
E1 and E1 = v
1+κδ(Q−q1). From Equation (3.19)
we find that
T1(Q,E1, δ)
(κ−1)′(E1 − v2)dv. (3.20)
To find dϕ1/dt, we consider ϕ1 as a function of v1, Q, and E1, so that
Then, using Equations (3.18) and (3.20), we compute
(κ−1δ )
′(E1 − v21)
κ′δ(Q− q1)
1/2− ϕ1
(εW ) = εW
1/2− ϕ1
1/2− ϕ1
(κ−1)′′(E1 − v2)dv
(εWκ′δ(Q− q1)).
Using that κ′δ(Q− q1) = κ′(κ−1(E1 − v21))/δ = (δ(κ−1)′(E1 − v21))−1, we find that
1/2− ϕ1
(κ−1)′(E1 − v21)
(κ−1)′′(E1 − v2)dv
But here 1/2− ϕ1 is O(δ). See the proof of Lemma 3.4.3 below. Thus the claims
about dϕ1/dt will be proven, provided we can uniformly bound
(κ−1)′(E1 − v21)
(κ−1)′′(E1 − v2)dv.
Note that the apparent divergence of the integral as |v1| →
E1 is entirely due to
the fact that our expression for ϕ1 from Equation (3.20) requires |v1| <
E1. If
we make the substitution u = E1− v2 and let e = E1− v21 , then it suffices to show
E≤E1≤κ(0)−E
0<e≤E1
(κ−1)′(e)
(κ−1)′′(u)√
E1 − u
< +∞.
The only difficulties occur when e is close to 0. Thus it suffices to show that
E≤E1≤κ(0)−E
0<e≤E/2
(κ−1)′(e)
∫ E/2
(κ−1)′′(u)√
E1 − u
is finite. But this is bounded by
0<e≤E/2
(κ−1)′(e)
∫ E/2
(κ−1)′′(u)
= sup
0<e≤E/2
(κ−1)′(e)
(κ−1)′(E/2)− (κ−1)′(e)
which is finite because (κ−1)′(e) → −∞ as e→ 0+. The claims about dϕ2/dt can
be proven similarly.
Proof of Lemma 3.4.3:
Proof. We continue in the notation of the proofs of Lemmas 3.4.1 and 3.4.2 above,
and we set m1 = 2. Then from Equation (3.20), we see that κ
δ(Q− q1) = 0 unless
|ϕ1 − 1/2| ≤
(κ−1)′(E1 − v2)dv
= δF (E1)/T1 = O(δ). Dealing with ϕ2
is similar.
Chapter 4
The periodic oscillation of an
adiabatic piston in two or three
dimensions
In this chapter, we present our results for the piston system in two or three di-
mensions. These results may also be found in [Wri07].
4.1 Statement of the main result
4.1.1 Description of the model
Consider a massive, insulating piston of mass M that separates a gas container
D in Rd, d = 2 or 3. See Figure 4.1. Denote the location of the piston by Q,
its velocity by dQ/dt = V , and its cross-sectional length (when d = 2, or area,
when d = 3) by ℓ. If Q is fixed, then the piston divides D into two subdomains,
D1(Q) = D1 on the left and D2(Q) = D2 on the right. By Ei we denote the
total energy of the gas inside Di, and by |Di| we denote the area (when d = 2, or
volume, when d = 3) of Di.
We are interested in the dynamics of the piston when the system’s total energy
is bounded and M → ∞. When M = ∞, the piston remains fixed in place, and
each energy Ei remains constant. When M is large but finite, MV
2/2 is bounded,
and so V = O(M−1/2). It is natural to define
ε =M−1/2,
so that W is of order 1 as ε → 0. This is equivalent to scaling time by ε.
D1(Q) D2(Q)
ℓ ✲ V = εW
M = ε−2 ≫ 1
❆❯s✟✟✙
s❍❍❍❥ s✘✘✿
Figure 4.1: A gas container D ⊂ R2 separated by a piston.
Next we precisely describe the gas container. It is a compact, connected billiard
domain D ⊂ Rd with a piecewise C3 boundary, i.e. ∂D consists of a finite number of
C3 embedded hypersurfaces, possibly with boundary and a finite number of corner
points. The container consists of a “tube,” whose perpendicular cross-section P is
the shape of the piston, connecting two disjoint regions. P ⊂ Rd−1 is a compact,
connected domain whose boundary is piecewise C3. Then the “tube” is the region
[0, 1] × P ⊂ D swept out by the piston for 0 ≤ Q ≤ 1, and [0, 1] × ∂P ⊂ ∂D. If
d = 2, P is just a closed line segment, and the “tube” is a rectangle. If d = 3, P
could be a circle, a square, a pentagon, etc.
Our fundamental assumption is as follows:
Main Assumption. For almost every Q ∈ [0, 1] the billiard flow of a single
particle on an energy surface in either of the two subdomains Di(Q) is ergodic
(with respect to the invariant Liouville measure).
If d = 2, the domain could be the Bunimovich stadium [Bun79]. Another possible
domain is indicated in Figure 4.1. The ergodicity of billiards in such domains,
which produce hyperbolic flows, goes back to the pioneering work of Sinai [Sin70],
although a number of individuals have contributed to the theory. A full accounting
of this history can be found in [CM06a]. Polygonal domains satisfying our assump-
tions can also be constructed [Vor97]. Suitable domains in d = 3 dimensions can be
constructed using a rectangular box with shallow spherical caps adjoined [BR98].
Note that we make no assumptions regarding the hyperbolicity of the billiard flow
in the domain.
The Hamiltonian system we consider consists of the massive piston of mass M
located at positionQ, as well as n1+n2 gas particles, n1 inD1 and n2 inD2. Here n1
and n2 are fixed positive integers. For convenience, the gas particles all have unit
mass, though all that is important is that each gas particle has a fixed mass. We
denote the positions of the gas particles in Di by qi,j, 1 ≤ j ≤ ni. The gas particles
are ideal point particles that interact with ∂D and the piston by hard core, elastic
collisions. Although it has no effect on the dynamics we consider, for convenience
we complete our description of the Hamiltonian dynamics by specifying that the
piston makes elastic collisions with walls located at Q = 0, 1 that are only visible
to the piston. We denote velocities by dQ/dt = V = εW and dqi,j/dt = vi,j, and
we set
Ei,j = v
i,j/2, Ei =
Ei,j.
Our system has d(n1 + n2) + 1 degrees of freedom, and so its phase space is
(2d(n1 + n2) + 2)-dimensional.
We let
h(z) = h = (Q,W,E1,1, E1,2, · · · , E1,n1 , E2,1, E2,2, · · · , E2,n2),
so that h is a function from our phase space to Rn1+n2+2. We often abbreviate
h = (Q,W,E1,j , E2,j), and we refer to h as consisting of the slow variables because
these quantities are conserved when ε = 0. We let hε(t, z) = hε(t) denote the
actual motions of these variables in time for a fixed value of ε. Here z represents
the initial condition in phase space, which we usually suppress in our notation.
One should think of hε(·) as being a random variable that takes initial conditions
in phase space to paths (depending on the parameter t) in Rn1+n2+2.
4.1.2 The averaged equation
From the work of Neishtadt and Sinai [NS04], one can derive
= H̄(h) :=
d |D1(Q)|
− 2E2ℓ
d |D2(Q)|
− 2WE1,jℓ
d |D1(Q)|
2WE2,jℓ
d |D2(Q)|
(4.1)
as the averaged equation (with respect to the slow time τ = εt) for the slow
variables. Later, in Section 4.2.3, we will give another heuristic derivation of the
averaged equation that is more suggestive of our proof.
Neishtadt and Sinai [Sin99, NS04] pointed out that the solutions of Equa-
tion (1.3) have (Q,W ) behaving as if they were the coordinates of a Hamiltonian
system describing a particle undergoing motion inside a potential well. As in
Section 1.2, the effective Hamiltonian is given by
W 2 +
E1(0) |D1(Q(0))|2/d
|D1(Q)|2/d
E2(0) |D2(Q(0))|2/d
|D2(Q)|2/d
This can be seen as follows. Since
∂ |D1(Q)|
= ℓ = −∂ |D2(Q)|
d ln(Ei,j)/dτ = −(2/d)d ln(|Di(Q)|)/dτ , and so
Ei,j(τ) = Ei,j(0)
|Di(Q(0))|
|Di(Q(τ))|
By summing over j, we find that
Ei(τ) = Ei(0)
|Di(Q(0))|
|Di(Q(τ))|
and so
d2Q(τ)
E1(0) |D1(Q(0))|2/d
|D1(Q(τ))|1+2/d
E2(0) |D2(Q(0))|2/d
|D2(Q(τ))|1+2/d
Let h̄(τ, z) = h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
Again, think of h̄(·) as being a random variable.
4.1.3 The main result
The solutions of the averaged equation approximate the motions of the slow vari-
ables, hε(t), on a time scale O(1/ε) as ε → 0. Precisely, fix a compact set
V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂ R, and Ei,j ⊂⊂ (0,∞)
for each i and j.1 We will be mostly concerned with the dynamics when h ∈ V.
Define
Qmin = inf
Q, Qmax = sup
Emin = inf
W 2 + E1 + E2, Emax = sup
W 2 + E1 + E2.
1 We have introduced this notation for convenience. For example, h ∈ V ⇒ Q ⊂⊂ (0, 1)
means that there exists a compact set A ⊂ (0, 1) such that h ∈ V ⇒ Q ∈ A, and similarly for
the other variables.
For a fixed value of ε > 0, we only consider the dynamics on the invariant subset
of phase space defined by
Mε = {(Q, V, qi,j, vi,j) ∈ R2d(n1+n2)+2 : Q ∈ [0, 1], qi,j ∈ Di(Q),
Emin ≤
V 2 + E1 + E2 ≤ Emax}.
Let Pε denote the probability measure obtained by restricting the invariant Liou-
ville measure to Mε. Define the stopping time
Tε(z) = Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 4.1.1. If D is a gas container in d = 2 or 3 dimensions satisfying the
assumptions in Subsection 4.1.1 above, then for each T > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣→ 0 in probability as ε =M−1/2 → 0,
i.e. for each fixed δ > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
→ 0 as ε =M−1/2 → 0.
Remark 4.1.1. It should be noted that the stopping time in the above result is not
unduly restrictive. If the initial pressures of the two gasses are not too mismatched,
then the solution to the averaged equation is a periodic orbit, with the effective
potential well keeping the piston away from the walls. Thus, if the actual motions
follow the averaged solution closely for 0 ≤ τ ≤ T ∧ Tε, and the averaged solution
stays in V, it follows that Tε > T .
Remark 4.1.2. The techniques of this work should immediately generalize to prove
the analogue of Theorem 4.1.1 above in the nonphysical dimensions d > 3, although
we do not pursue this here.
Remark 4.1.3. As in Subsection 3.1.3, Theorem 4.1.1 can be easily generalized
to cover a system of N − 1 pistons that divide N gas containers, so long as, for
almost every fixed location of the pistons, the billiard flow of a single gas particle
on an energy surface in any of the N subcontainers is ergodic (with respect to the
invariant Liouville measure). The effective Hamiltonian for the pistons has them
moving like an (N − 1)-dimensional particle inside a potential well.
4.2 Preparatory material concerning a
two-dimensional gas container with only one
gas particle on each side
Our results and techniques of proof are essentially independent of the dimension
and the fixed number of gas particles on either side of the piston. Thus, we focus
D1 D2
ℓ ✲V = εW
M = ε−2 ≫ 1
Figure 4.2: A choice of coordinates on phase space.
on the case when d = 2 and there is only one gas particle on either side. Later,
in Section 4.4, we will indicate the simple modifications that generalize our proof
to the general situation. For clarity, in this section and next, we denote q1,1 by q1,
v2,1 by v2, etc. We decompose the gas particle coordinates according to whether
they are perpendicular to or parallel to the piston’s face, for example q1 = (q
1 , q
See Figure 4.2.
The Hamiltonian dynamics define a flow on our phase space. We denote this
flow by zε(t, z) = zε(t), where z = zε(0, z). One should think of zε(·) as being
a random variable that takes initial conditions in phase space to paths in phase
space. Then hε(t) = h(zε(t)). By the change of coordinates W = V/ε, we may
identify all of the Mε defined in Section 4.1 with the space
M = {(Q,W, q1, v1, q2, v2) ∈ R10 : Q ∈ [0, 1], q1 ∈ D1(Q), q2 ∈ D2(Q),
Emin ≤
W 2 + E1 + E2 ≤ Emax}.
and all of the Pε with the probability measure P on M, which has the density
dP = const dQdWdq⊥1 dq
(Throughout this work we will use const to represent generic constants that are
independent of ε.) We will assume that these identifications have been made, so
that we may consider zε(·) as a family of measure preserving flows on the same
space that all preserve the same probability measure. We denote the components
of zε(t) by Qε(t), q
1,ε(t), etc.
The set {z ∈ M : q1 = Q = q2} has co-dimension two, and so
zε(t){q1 =
Q = q2} has co-dimension one, which shows that only a measure zero set of initial
conditions will give rise to three particle collisions. We ignore this and other
measures zero events, such as gas particles hitting singularities of the billiard flow,
in what follows.
Now we present some background material, as well as some lemmas that will
assist us in our proof of Theorem 4.1.1. We begin by studying the billiard flow
of a gas particle when the piston is infinitely massive. Next we examine collisions
between the gas particles and the piston when the piston has a large, but finite,
mass. Then we present a heuristic derivation of the averaged equation that is
suggestive of our proof. Finally we prove a lemma that allows us to disregard
the possibility that a gas particle will move nearly parallel to the piston’s face
– a situation that is clearly bad for having the motions of the piston follow the
solutions of the averaged equation.
4.2.1 Billiard flows and maps in two dimensions
In this section, we study the billiard flows of the gas particles when M = ∞ and
the slow variables are held fixed at a specific value h ∈ V. We will only study
the motions of the left gas particle, as similar definitions and results hold for the
motions of the right gas particle. Thus we wish to study the billiard flow of a point
particle moving inside the domain D1 at a constant speed
2E1. The results of
this section that are stated without proof can be found in [CM06a].
Let T D1 denote the tangent bundle to D1. The billiard flow takes place in
the three-dimensional space M1h = M1 = {(q1, v1) ∈ T D1 : q1 ∈ D1, |v1| =√
2E1}/ ∼. Here the quotient means that when q1 ∈ ∂D1, we identify velocity
vectors pointing outside of D1 with those pointing inside D1 by reflecting through
the tangent line to ∂D1 at q1, so that the angle of incidence with the unit normal
vector to ∂D1 equals the angle of reflection. Note that most of the quantities
defined in this subsection depend on the fixed value of h. We will usually suppress
this dependence, although, when necessary, we will indicate it by a subscript h.
We denote the resulting flow by y(t, y) = y(t), where y(0, y) = y. As the billiard
flow comes from a Hamiltonian system, it preserves Liouville measure restricted
to the energy surface. We denote the resulting probability measure by µ. This
measure has the density dµ = dq1dv1/(2π
2E1 |D1|). Here dq1 represents area on
R2, and dv1 represents length on S
v1 ∈ R2 : |v1| =
There is a standard cross-section to the billiard flow, the collision cross-section
Ω = {(q1, v1) ∈ T D1 : q1 ∈ ∂D1, |v1| =
2E1}/ ∼. It is customary to parameter-
ize Ω by {x = (r, ϕ) : r ∈ ∂D1, ϕ ∈ [−π/2,+π/2]}, where r is arc length and ϕ
represents the angle between the outgoing velocity vector and the inward pointing
normal vector to ∂D1. It follows that Ω may be realized as the disjoint union
of a finite number of rectangles and cylinders. The cylinders correspond to fixed
scatterers with smooth boundary placed inside the gas container. If F : Ω 	 is the
collision map, i.e. the return map to the collision cross-section, then F preserves the
projected probability measure ν, which has the density dν = cosϕdϕ dr/(2 |∂D1|).
Here |∂D1| is the length of ∂D1.
We suppose that the flow is ergodic, and so F is an invertible, ergodic mea-
sure preserving transformation. Because ∂D1 is piecewise C3, F is piecewise C2,
although it does have discontinuities and unbounded derivatives near discontinu-
ities corresponding to grazing collisions. Because of our assumptions on D1, the
free flight times and the curvature of ∂D1 are uniformly bounded. It follows that
if x /∈ ∂Ω ∪ F−1(∂Ω), then F is differentiable at x, and
‖DF (x)‖ ≤ const
cosϕ(Fx)
, (4.2)
where ϕ(Fx) is the value of the ϕ coordinate at the image of x.
Following the ideas in Section 4.5, we induce F on the subspace Ω̂ of Ω cor-
responding to collisions with the (immobile) piston. We denote the induced map
by F̂ and the induced measure by ν̂. We parameterize Ω̂ by {(r, ϕ) : 0 ≤ r ≤
ℓ, ϕ ∈ [−π/2,+π/2]}. As νΩ̂ = ℓ/ |∂D1|, it follows that ν̂ has the density
dν̂ = cosϕdϕ dr/(2ℓ).
For x ∈ Ω, define ζx to be the free flight time, i.e. the time it takes the billiard
particle traveling at speed
2E1 to travel from x to Fx. If x /∈ ∂Ω ∪ F−1(∂Ω),
‖Dζ(x)‖ ≤ const
cosϕ(Fx)
. (4.3)
Santaló’s formula [San76, Che97] tells us that
Eνζ =
π |D1|
|v1| |∂D1|
. (4.4)
If ζ̂ : Ω̂ → R is the free flight time between collisions with the piston, then it
follows from Proposition 4.5.1 that
Eν̂ ζ̂ =
π |D1|
|v1| ℓ
. (4.5)
The expected value of
∣ when the left gas particle collides with the (immo-
bile) piston is given by
∣ = Eν̂
2E1 cosϕ =
∫ +π/2
cos2 ϕdϕ =
. (4.6)
We wish to compute limt→∞ t
∣2v⊥1 (s)
∣ δq⊥1 (s)=Qds, the time average of the
change in momentum of the left gas particle when it collides with the piston. If this
limit exists and is equal for almost every initial condition of the left gas particle,
then it makes sense to define the pressure inside D1 to be this quantity divided
by ℓ. Because the collisions are hard-core, we cannot directly apply Birkhoff’s
Ergodic Theorem to compute this limit. However, we can compute this limit by
using the map F̂ .
Lemma 4.2.1. If the billiard flow y(t) is ergodic, then for µ− a.e. y ∈ M1,
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds =
2 |D1(Q)|
Proof. Because the billiard flow may be viewed as a suspension flow over the
collision cross-section with ζ as the height function, it suffices to show that the
convergence takes place for ν̂ − a.e. x ∈ Ω̂. For an initial condition x ∈ Ω̂, define
N̂t(x) = N̂t = #
s ∈ (0, t] : y(s, x) ∈ Ω̂
. By the Poincaré Recurrence Theorem,
N̂t → ∞ as t→ ∞, ν̂ − a.e.
n=0 ζ̂(F̂
∣ (F̂ nx) ≤ 1
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds
≤ N̂t
∑N̂t−1
n=0 ζ̂(F̂
∣ (F̂ nx),
and so the result follows from Birkhoff’s Ergodic Theorem and Equations (4.5)
and (4.6).
Corollary 4.2.2. If the billiard flow y(t) is ergodic, then for each δ > 0,
y ∈ M1 :
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds−
2 |D1(Q)|
→ 0 as t→ ∞.
4.2.2 Analysis of collisions
In this section, we return to studying our piston system when ε > 0. We will
examine what happens when a particle collides with the piston. For convenience,
we will only examine in detail collisions between the piston and the left gas particle.
Collisions with the right gas particle can be handled similarly.
When the left gas particle collides with the piston, v⊥1 and V instantaneously
change according to the laws of elastic collisions:
1−M 2M
2 M − 1
In our coordinates, this becomes
1 + ε2
ε2 − 1 2ε
2ε 1− ε2
. (4.7)
Recalling that v1,W = O(1), we find that to first order in ε,
v⊥+1 = −v⊥−1 +O(ε), W+ =W− +O(ε). (4.8)
Observe that a collision can only take place if v⊥−1 > εW
−. In particular, v⊥−1 >
2Emax. Thus, either v
1 > 0 or v
1 = O(ε). By expanding Equation (4.7)
to second order in ε, it follows that
E+1 − E−1 = −2εW
∣+O(ε2),
W+ −W− = +2ε
∣+O(ε2).
(4.9)
Note that it is immaterial whether we use the pre-collision or post-collision values
of W and
∣ on the right hand side of Equation (4.9), because any ambiguity
can be absorbed into the O(ε2) term.
It is convenient for us to define a “clean collision” between the piston and the
left gas particle:
Definition 4.2.1. The left gas particle experiences a clean collision with the
piston if and only if v⊥−1 > 0 and v
1 < −ε
2Emax.
In particular, after a clean collision, the left gas particle will escape from the
piston, i.e. the left gas particle will have to move into the region q⊥1 ≤ 0 before
it can experience another collision with the piston. It follows that there exists
a constant C1 > 0, which depends on the set V, such that for all ε sufficiently
small, so long as Q ≥ Qmin and
∣ > εC1 when q
1 ∈ [Qmin, Q], then the
left gas particle will experience only clean collisions with the piston, and the time
between these collisions will be greater than 2Qmin/(
2Emax). (Note that when we
write expressions such as q⊥1 ∈ [Qmin, Q], we implicitly mean that q1 is positioned
inside the “tube” discussed at the beginning of Section 4.1.) One can verify that
C1 = 5
2Emax would work.
Similarly, we can define clean collisions between the right gas particle and
the piston. We assume that C1 was chosen sufficiently large such that for all ε
sufficiently small, so long as Q ≤ Qmax and
∣ > εC1 when q
2 ∈ [Q,Qmax], then
the right gas particle will experience only clean collisions with the piston.
Now we define three more stopping times, which are functions of the initial
conditions in phase space.
T ′ε = inf{τ ≥ 0 : Qmin ≤ q⊥1,ε(τ/ε) ≤ Qε(τ/ε) ≤ Qmax and
∣v⊥1,ε(τ/ε)
∣ ≤ C1ε},
T ′′ε = inf{τ ≥ 0 : Qmin ≤ Qε(τ/ε) ≤ q⊥2,ε(τ/ε) ≤ Qmax and
∣v⊥2,ε(τ/ε)
∣ ≤ C1ε},
T̃ε =T ∧ Tε ∧ T ′ε ∧ T ′′ε
Define H(z) by
H(z) =
∣ δq⊥1 =Q − 2
∣ δq⊥2 =Q
∣ δq⊥1 =Q
∣ δq⊥2 =Q
Here we make use of Dirac delta functions. All integrals involving these delta
functions may be replaced by sums.
The following lemma is an immediate consequence of Equation (4.9) and the
above discussion:
Lemma 4.2.3. If 0 ≤ t1 ≤ t2 ≤ T̃ε/ε, the piston experiences O((t2 − t1) ∨ 1)
collisions with gas particles in the time interval [t1, t2], all of which are clean
collisions. Furthermore,
hε(t2)− hε(t1) = O(ε) + ε
H(zε(s))ds.
Here any ambiguities arising from collisions occurring at the limits of integration
can be absorbed into the O(ε) term.
4.2.3 Another heuristic derivation of the averaged equa-
The following heuristic derivation of Equation (4.1) when d = 2 was suggested
in [Dol05]. Let ∆t be a length of time long enough such that the piston experiences
many collisions with the gas particles, but short enough such that the slow variables
change very little, in this time interval. From each collision with the left gas
particle, Equation (4.9) states that W changes by an amount +2ε
∣ + O(ε2),
and from Equation (4.6) the average change in W at these collisions should be
approximately επ
2E1/2 + O(ε2). From Equation (4.5) the frequency of these
collisions is approximately
2E1 ℓ/(π |D1|). Arguing similarly for collisions with
the other particle, we guess that
|D1(Q)|
− ε E2ℓ|D2(Q)|
+O(ε2).
With τ = εt as the slow time, a reasonable guess for the averaged equation for W
|D1(Q)|
− E2ℓ|D2(Q)|
Similar arguments for the other slow variables lead to the averaged equation (4.1),
and this explains why we used Pi = Ei/ |Di| for the pressure of a 2-dimensional
gas in Section 1.2.
There is a similar heuristic derivation of the averaged equation in d > 2 dimen-
sions. Compare the analogues of Equations (4.5) and (4.6) in Subsection 4.4.2.
4.2.4 A priori estimate on the size of a set of bad initial
conditions
In this section, we give an a priori estimate on the size of a set of initial conditions
that should not give rise to orbits for which sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ is small.
In particular, when proving Theorem 4.1.1, it is convenient to focus on orbits that
only contain clean collisions with the piston. Thus, we show that P{T̃ε < T ∧ Tε}
vanishes as ε → 0. At first, this result may seem surprising, since P{T ′ε ∧ T ′′ε =
0} = O(ε), and one would expect ∪T/εt=0zε(−t){T ′ε ∧ T ′′ε = 0} to have a size of order
1. However, the rate at which orbits escape from {T ′ε ∧ T ′′ε = 0} is very small, and
so we can prove the following:
Lemma 4.2.4.
P{T̃ε < T ∧ Tε} = O(ε).
In some sense, this lemma states that the probability of having a gas particle
move nearly parallel to the piston’s face within the time interval [0, T/ε], when one
would expect the other gas particle to force the piston to move on a macroscopic
scale, vanishes as ε → 0. Thus, one can hope to control the occurrence of the
“nondiffusive fluctuations” of the piston described in [CD06a] on a time scale
O(ε−1).
Proof. As the left and the right gas particles can be handled similarly, it suffices
to show that P{T ′ε < T} = O(ε). Define
Bε = {z ∈ M : Qmin ≤ q⊥1 ≤ Q ≤ Qmax and
∣ ≤ C1ε}.
Then {T ′ε < T} ⊂ ∪
t=0zε(−t)Bε, and if γ = Qmin/
8Emax,
zε(−t)Bε
 = P
zε(t)Bε
 = P
Bε ∪
((zε(t)Bε)\Bε)
≤ PBε + P
T/(εγ)
zε(kγ)
(zε(t)Bε)\Bε
≤ PBε +
(zε(t)Bε)\Bε
Now PBε = O(ε), so if we can show that P (
t=0(zε(t)Bε)\Bε) = O(ε2), then it
will follow that P{T ′ε < T} = O(ε).
If z ∈
t=0(zε(t)Bε)\Bε, it is still true that
∣ = O(ε). This is because
changes by at most O(ε) at the collisions, and if a collision forces
∣ > C1ε, then
the gas particle must escape to the region q⊥1 ≤ 0 before v⊥1 can change again, and
this will take time greater than γ. Furthermore, if z ∈
t=0(zε(t)Bε)\Bε, then at
least one of the following four possibilities must hold:
∣q⊥1 −Qmin
∣ ≤ O(ε),
• |Q−Qmin| ≤ O(ε),
• |Q−Qmax| ≤ O(ε),
∣Q− q⊥1
∣ ≤ O(ε).
It follows that P (
t=0(zε(t)Bε)\Bε) = O(ε2). For example,
1{|v⊥1 |≤O(ε), |q⊥1 −Qmin|≤O(ε)}dP
= const
{Emin≤W 2/2+v21/2+v22/2≤Emax}
1{|v⊥1 |≤O(ε)}dWdv
{Q∈[0,1], q1∈D1, q2∈D2}
1{|q⊥1 −Qmin|≤O(ε)}dQdq
= O(ε2).
4.3 Proof of the main result for two-dimensional
gas containers with only one gas particle on
each side
As in Section 4.2, we continue with the case when d = 2 and there is only one gas
particle on either side of the piston.
4.3.1 Main steps in the proof of convergence in probability
By Lemma 4.2.4, it suffices to show that sup0≤τ≤T̃ε
∣hε(τ/ε)− h̄(τ)
∣→ 0 in prob-
ability as ε =M−1/2 → 0. Several of the ideas in the steps below were inspired by
a recent proof of Anosov’s averaging theorem for smooth systems that is due to
Dolgopyat [Dol05].
Step 1: Reduction using Gronwall’s Inequality. Observe that h̄(τ) satisfies
the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while from Lemma 4.2.3,
hε(τ/ε)− hε(0) = O(ε) + ε
∫ τ/ε
H(zε(s))ds
= O(ε) + ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T̃ε. Define
eε(τ) = ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T̃ε
∣hε(τ/ε)− h̄(τ)
O(ε) + sup
0≤τ≤T̃ε
|eε(τ)|
eLip(H̄|V)T . (4.10)
Gronwall’s Inequality is usually stated for continuous paths, but the standard
proof (found in [SV85]) still works for paths that are merely integrable, and
∣hε(τ/ε)− h̄(τ)
∣ is piecewise smooth.
Step 2: Introduction of a time scale for ergodization. Let L(ε) be a
real valued function such that L(ε) → ∞, but L(ε) ≪ log ε−1, as ε → 0. In
Section 4.3.2 we will place precise restrictions on the growth rate of L(ε). Think
of L(ε) as being a time scale that grows as ε → 0 so that ergodization, i.e. the
convergence along an orbit of a function’s time average to a space average, can
take place. However, L(ε) doesn’t grow too fast, so that on this time scale zε(t)
essentially stays on the submanifold {h = hε(0)}, where we have our ergodicity
assumption. Set tk,ε = kL(ε), so that
0≤τ≤T̃ε
|eε(τ)| ≤ O(εL(ε)) + ε
εL(ε)
∫ tk+1,ε
H(zε(s))− H̄(hε(s))ds
. (4.11)
Step 3: A splitting according to particles. Now H(z) − H̄(h(z)) divides
into two pieces, each of which depends on only one gas particle when the piston
is held fixed:
H(z)− H̄(h(z)) =
∣ δq⊥1 =Q −
|D1(Q)|
∣ δq⊥1 =Q +
|D1(Q)|
|D2(Q)| − 2
∣ δq⊥2 =Q
− WE2ℓ|D2(Q)| + 2W
∣ δq⊥2 =Q
We will only deal with the piece depending on the left gas particle, as the right
particle can be handled similarly. Define
G(z) =
∣ δq⊥1 =Q, Ḡ(h) =
2 |D1(Q)|
. (4.12)
Returning to Equation (4.11), we see that in order to prove Theorem 4.1.1, it
suffices to show that both
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
εL(ε)
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
converge to 0 in probability as ε→ 0.
Step 4: A splitting for using the triangle inequality. Now we let zk,ε(s)
be the orbit of the ε = 0 Hamiltonian vector field satisfying zk,ε(tk,ε) = zε(tk,ε).
Set hk,ε(t) = h(zk,ε(t)). Observe that hk,ε(t) is independent of t.
We emphasize that so long as 0 ≤ t ≤ T̃ε/ε, the times between collisions of a
specific gas particle and piston are uniformly bounded greater than 0, as explained
before Lemma 4.2.3. It follows that, so long as tk+1,ε ≤ T̃ε/ε,
tk,ε≤t≤tk+1,ε
|hk,ε(t)− hε(t)| = O(εL(ε)). (4.13)
This is because the slow variables change by at most O(ε) at collisions, and
dQε/dt = O(ε).
Also,
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
= O(εL(ε)2) +Wk,ε(tk,ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds,
and so
εL(ε)
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
≤ O(εL(ε)) + ε const
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
Thus, in order to prove Theorem 4.1.1, it suffices to show that
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
εL(ε)
|Ik,ε|+ |IIk,ε|+ |IIIk,ε|
converges to 0 in probability as ε → 0, where
Ik,ε =
∫ tk+1,ε
G(zε(s))−G(zk,ε(s))ds,
IIk,ε =
∫ tk+1,ε
G(zk,ε(s))− Ḡ(hk,ε(s))ds,
IIIk,ε =
∫ tk+1,ε
Ḡ(hk,ε(s))− Ḡ(hε(s))ds.
The term IIk,ε represents an “ergodicity term” that can be controlled by our
assumptions on the ergodicity of the flow z0(t), while the terms Ik,ε and IIIk,ε
represent “continuity terms” that can be controlled by controlling the drift of
zε(t) from zk,ε(t) for tk,ε ≤ t ≤ tk+1,ε.
Step 5: Control of drift from the ε = 0 orbits. Now Ḡ is uniformly
Lipschitz on the compact set V, and so it follows from Equation (4.13) that
IIIk,ε = O(εL(ε)2). Thus, ε
εL(ε)
k=0 |IIIk,ε| = O(εL(ε)) → 0 as ε→ 0.
Next, we show that for fixed δ > 0, P
εL(ε)
k=0 |Ik,ε| ≥ δ
→ 0 as ε → 0.
For initial conditions z ∈ M and for integers k ∈ [0, T/(εL(ε))− 1] define
Ak,ε =
|Ik,ε| >
and k ≤ T̃ε
εL(ε)
Az,ε = {k : z ∈ Ak,ε} .
Think of these sets as describing “poor continuity” between solutions of the ε = 0
and the ε > 0 Hamiltonian vector fields. For example, roughly speaking, z ∈ Ak,ε
if the orbit zε(t) starting at z does not closely follow zk,ε(t) for tk,ε ≤ t ≤ tk+1,ε.
One can easily check that |Ik,ε| ≤ O(L(ε)) for k ≤ T̃ε/(εL(ε))− 1, and so it
follows that
εL(ε)
|Ik,ε| ≤
+O(εL(ε)#(Az,ε)).
Therefore it suffices to show that P (#(Az,ε) ≥ δ(const εL(ε))−1) → 0 as ε → 0.
By Chebyshev’s Inequality, we need only show that
EP (εL(ε)#(Az,ε)) = εL(ε)
εL(ε)
P (Ak,ε)
tends to 0 with ε.
Observe that zε(tk,ε)Ak,ε ⊂ A0,ε. In words, the initial conditions giving rise to
orbits that are “bad” on the time interval [tk,ε, tk+1,ε], moved forward by time tk,ε,
are initial conditions giving rise to orbits which are “bad” on the time interval
[t0,ε, t1,ε]. Because the flow zε(·) preserves the measure, we find that
εL(ε)
εL(ε)
P (Ak,ε) ≤ constP (A0,ε).
To estimate P (A0,ε), it is convenient to use a different probability measure,
which is uniformly equivalent to P on the set {z ∈ M : h(z) ∈ V} ⊃ {T̃ε ≥ εL(ε)}.
We denote this new probability measure by P f , where the f stands for “factor.”
If we choose coordinates on M by using h and the billiard coordinates on the two
gas particles, then P f is defined on M by dP f = dh dµ1h dµ2h, where dh represents
the uniform measure on V ⊂ R4, and the factor measure dµih represents the in-
variant billiard measure of the ith gas particle coordinates for a fixed value of the
slow variables. One can verify that 1{h(z)∈V}dP ≤ const dP f , but that P f is not
invariant under the flow zε(·) when ε > 0.
We abuse notation, and consider µ1h to be a measure on the left particle’s initial
billiard coordinates once h and the initial coordinates of the right gas particle are
fixed. In this context, µ1h is simply the measure µ from Subsection 4.2.1. Then
P f(A0,ε)
dh dµ2h · µ1h
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
and we must show that the last term tends to 0 with ε. By the Bounded Con-
vergence Theorem, it suffices to show that for almost every h ∈ V and initial
condition for the right gas particle,
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
→ 0 as ε→ 0.
(4.14)
Note that if G were a smooth function and zε(·) were the flow of a smooth
family of vector fields Z(z, ε) that depended smoothly on ε, then from Gronwall’s
Inequality, it would follow that sup0≤t≤L(ε) |zε(t)− z0(t)| ≤ O(εL(ε)eLip(Z)L(ε)). If
this were the case, then
L(ε)−1
∫ L(ε)
G(zε(s))−G(z0(s))ds
= O(εL(ε)eLip(Z)L(ε)),
which would tend to 0 with ε. Thus, we need a Gronwall-type inequality for billiard
flows. We obtain the appropriate estimates in Section 4.3.2.
Step 6: Use of ergodicity along fibers to control IIk,ε. All that remains
to be shown is that for fixed δ > 0, P
εL(ε)
k=0 |IIk,ε| ≥ δ
→ 0 as ε→ 0.
For initial conditions z ∈ M and for integers k ∈ [0, T/(εL(ε))− 1] define
Bk,ε =
|IIk,ε| >
and k ≤ T̃ε
εL(ε)
Bz,ε = {k : z ∈ Bk,ε} .
Think of these sets as describing “bad ergodization.” For example, roughly speak-
ing, z ∈ Bk,ε if the orbit zε(t) starting at z spends the time between tk,ε and tk+1,ε
in a region of phase space where the function G(·) is “poorly ergodized” on the
time scale L(ε) by the flow z0(t) (as measured by the parameter δ/2T ). Note that
G(z) =
∣ δq⊥1 =Q is not really a function, but that we may still speak of the
convergence of t−1
G(z0(s))ds as t → ∞. As we showed in Lemma 4.2.1, the
limit is Ḡ(h0) for almost every initial condition.
Proceeding as in Step 5 above, we find that it suffices to show that for almost
every h ∈ V,
G(z0(s))ds− Ḡ(h0(0))
→ 0 as t→ ∞.
But this is simply a question of examining billiard flows, and it follows immediately
from Corollary 4.2.2 and our Main Assumption.
4.3.2 A Gronwall-type inequality for billiards
We begin by presenting a general version of Gronwall’s Inequality for billiard
maps. Then we will show how these results imply the convergence required in
Equation (4.14).
Some inequalities for the collision map
In this section, we consider the value of the slow variables to be fixed at h0 ∈ V.
We will use the notation and results presented in Section 4.2.1, but because the
value of the slow variables is fixed, we will omit it in our notation.
Let ρ, γ, and λ satisfy 0 < ρ≪ γ ≪ 1 ≪ λ <∞. Eventually, these quantities
will be chosen to depend explicitly on ε, but for now they are fixed.
Recall that the phase space Ω for the collision map F is a finite union of
disjoint rectangles and cylinders. Let d(·, ·) be the Euclidean metric on connected
components of Ω. If x and x′ belong to different components, then we set d(x, x′) =
∞. The invariant measure ν satisfies ν < const · (Lebesgue measure). For A ⊂ Ω
and a > 0, let Na(A) = {x ∈ Ω : d(x,A) < a} be the a-neighborhood of A.
For x ∈ Ω let xk(x) = xk = F kx, k ≥ 0, be its forward orbit. Suppose x /∈ Cγ,λ,
where
Cγ,λ =
∪λk=0F−kNγ(∂Ω)
∪λk=0F−kNγ(F−1Nγ(∂Ω))
Thus for 0 ≤ k ≤ λ, xk is well defined, and from Equation (4.2) it satisfies
d(x′, xk) ≤ γ ⇒ d(Fx′, xk+1) ≤
const
d(x′, xk). (4.15)
Next, we consider any ρ-pseudo-orbit x′k obtained from x by adding on an
error of size ≤ ρ at each application of the map, i.e. d(x′0, x0) ≤ ρ, and for k ≥ 1,
d(x′k, Fx
k−1) ≤ ρ. Provided d(xj , x′j) < γ for each j < k, it follows that
d(xk, x
k) ≤ ρ
const
≤ const ρ
const
. (4.16)
In particular, if ρ, γ, and λ were chosen such that
const ρ
const
< γ, (4.17)
then Equation (4.16) will hold for each k ≤ λ. We assume that Equation (4.17)
is true. Then we can also control the differences in elapsed flight times using
Equation (4.3):
|ζxk − ζx′k| ≤
const ρ
const
. (4.18)
It remains to estimate the size νCγ,λ of the set of x for which the above estimates
do not hold. Using Lemma 4.3.1 below,
νCγ,λ ≤ (λ+ 1)
νNγ(∂Ω) + νNγ(F−1Nγ(∂Ω))
≤ O(λ(γ + γ1/3)) = O(λγ1/3).
(4.19)
Lemma 4.3.1. As γ → 0,
νNγ(F−1Nγ(∂Ω)) = O(γ1/3).
This estimate is not necessarily the best possible. For example, for dispersing
billiard tables, where the curvature of the boundary is positive, one can show that
νNγ(F−1Nγ(∂Ω)) = O(γ). However, the estimate in Lemma 4.3.1 is general and
sufficient for our needs.
Proof. First, we note that it is equivalent to estimate νNγ(FNγ(∂Ω)), as F has
the measure-preserving involution I(r, ϕ) = (r,−ϕ), i.e. F−1 = I ◦F ◦I [CM06b].
Fix α ∈ (0, 1/2), and cover Nγ(∂Ω) with O(γ−1) starlike sets, each of diameter
no greater than O(γ). For example, these sets could be squares of side length γ.
Enumerate the sets as {Ai}. Set G = {i : FAi ∩ Nγα(∂Ω) = ∅}.
If i ∈ G, F |Ai is a diffeomorphism satisfying ‖DF |Ai‖ ≤ O(γ−α). See Equa-
tion (4.2). Thus diameter (FAi) ≤ O(γ1−α), and so
diameter (Nγ(FAi)) ≤ O(γ1−α).
Hence νNγ(FAi) ≤ O(γ2(1−α)), and νNγ(∪i∈GFAi) ≤ O(γ1−2α).
If i /∈ G, Ai ∩ F−1(Nγα(∂Ω)) 6= ∅. Thus Ai might be cut into many pieces
by F−1(∂Ω), but each of these pieces must be mapped near ∂Ω. In fact, FAi ⊂
NO(γα)(∂Ω). This is because outside F−1(Nγα(∂Ω)), ‖DF‖ ≤ O(γ−α), and so
points in FAi are no more than a distance O(γ/γα) away from Nγα(∂Ω), and γ <
γ1−α < γα. It follows that Nγ(FAi) ⊂ NO(γα)(∂Ω), and νNO(γα)(∂Ω) = O(γα).
Thus νNγ(F−1Nγ(∂Ω)) = O(γ1−2α + γα), and we obtain the lemma by taking
α = 1/3.
Application to a perturbed billiard flow
Returning to the end of Step 5 in Section 4.3.1, let the initial conditions of the slow
variables be fixed at h0 = (Q0,W0, E1,0, E2,0) ∈ V throughout the remainder of this
section. We can assume that the billiard dynamics of the left gas particle in D1(Q0)
are ergodic. Also, fix a particular value of the initial conditions for the right gas
particle for the remainder of this section. Then zε(t) and T̃ε may be thought of as
random variables depending on the left gas particle’s initial conditions y ∈ M1.
Now if hε(t) = (Qε(t),Wε(t), E1,ε(t), E2,ε(t)) denotes the actual motions of the slow
variables when ε > 0, it follows from Equation (4.13) that, provided εL(ε) ≤ T̃ε,
0≤t≤L(ε)
|h0 − hε(t)| = O(εL(ε)). (4.20)
Furthermore, we only need to show that
y ∈ M1 :
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
(4.21)
as ε→ 0, where G is defined in Equation (4.12).
For definiteness, we take the following quantities from Subsection 4.3.2 to de-
pend on ε as follows:
L(ε) = L = log log
γ(ε) = γ = e−L,
λ(ε) = λ =
ρ(ε) = ρ = const
(4.22)
The constant in the choice of ρ and ρ’s dependence on ε will be explained in the
proof of Lemma 4.3.3, which is at the end of this subsection. The other choices may
be explained as follows. We wish to use continuity estimates for the billiard map
to produce continuity estimates for the flow on the time scale L. As the divergence
of orbits should be exponentially fast, we choose L to grow sublogarithmically in
ε−1. Since from Equation (4.4) the expected flight time between collisions with
∂D1(Q0) when ε = 0 is Eνζ = π |D1(Q0)| /(
2E1,0 |∂D1(Q0)|), we expect to see
roughly λ/2 collisions on this time scale. Considering λ collisions gives us some
margin for error. Furthermore, we will want orbits to keep a certain distance,
γ, away from the billiard discontinuities. γ → 0 as ε → 0, but γ is very large
compared to the possible drift O(εL) of the slow variables on the time scale L. In
fact, for each C,m, n > 0,
= O(ε econstL2) → 0 as ε→ 0. (4.23)
Let X : M1 → Ω be the map taking y ∈ M1 to x = X(y) ∈ Ω, the location
of the billiard orbit of y in the collision cross-section that corresponds to the most
recent time in the past that the orbit was in the collision cross-section. We consider
the set of initial conditions
Eε = X−1(Ω\Cγ,λ)
x ∈ Ω :
ζ(F kx) > L
Now from Equations (4.19) and (4.22), νCγ,λ → 0 as ε → 0. Furthermore, by the
ergodicity of F ,
x ∈ Ω :
ζ(F kx) ≤ L
x ∈ Ω : λ−1
ζ(F kx) ≤ Eνζ/2
as ε → 0. But because the free flight time is bounded above, µX−1 ≤ const ·ν, and
so µEε → 1 as ε→ 0. Hence, the convergence in Equation (4.21) and the conclusion
of the proof in Section 4.3.1 follow from the lemma below and Equation (4.23).
Lemma 4.3.2 (Analysis of deviations along good orbits). As ε→ 0,
y∈Eε∩{εL≤T̃ε}
G(zε(s))−G(z0(s))ds
const
+O(L−1) → 0.
Proof. Fix a particular value of y ∈ Eε∩
εL ≤ T̃ε
. For convenience, suppose that
y = X(y) = x ∈ Ω. Let y0(t) denote the time evolution of the billiard coordinates
for the left gas particle when ε = 0. Then there is some N ≤ λ such that the
orbit xk = F
kx = (rk, ϕk) for 0 ≤ k ≤ N corresponds to all of the instances
(in order) when y0(t) enters the collision cross-section Ω = Ωh0 corresponding to
collisions with ∂D1(Q0) for 0 ≤ t ≤ L. We write Ωh0 to emphasize that in this
subsection we are only considering the collision cross-section corresponding to the
billiard dynamics in the domain D1(Q0) at the energy level E1,0. In particular, F
will always refer to the return map on Ωh0 .
Also, define an increasing sequence of times tk corresponding to the actual
times y0(t) enters the collision cross-section, i.e.
t0 = 0,
tk = tk−1 + ζxk−1 for k > 0.
Then xk = y0(tk). Furthermore, define inductively
N1 = inf {k > 0 : tk corresponds to a collision with the piston} ,
Nj = inf {k > Nj−1 : tk corresponds to a collision with the piston} .
Next, let yε(t) denote the time evolution of the billiard coordinates for the left
gas particle when ε > 0. We will construct a pseudo-orbit x′k,ε = (r
k,ε, ϕ
k,ε) of
points in Ωh0 that essentially track the collisions (in order) of the left gas particle
with the boundary under the dynamics of yε(t) for 0 ≤ t ≤ L.
First, define an increasing sequence of times t′k,ε corresponding to the actual
times yε(t) experiences a collision with the boundary of the gas container or the
moving piston. Define
N ′ε = sup
k ≥ 0 : t′k,ε ≤ L
N ′1,ε = inf
k > 0 : t′k,ε corresponds to a collision with the piston
N ′j,ε = inf
k > N ′j−1,ε : t
k,ε corresponds to a collision with the piston
Because L ≤ T̃ε(y)/ε, we know that as long as N ′j+1,ε ≤ N ′ε, then N ′j+1,ε−N ′j,ε ≥ 2.
See the discussion in Subsection 4.2.2. Then we define x′k,ε ∈ Ωh0 by
x′k,ε =
k,ε) if k /∈
N ′j,ε
F−1x′k+1,ε if k ∈
N ′j,ε
Lemma 4.3.3. Provided ε is sufficiently small, the following hold for each k ∈
[0, N∧N ′ε). Furthermore, the requisite smallness of ε and the sizes of the constants
in these estimates may be chosen independent of the initial condition y ∈ Eε ∩
εL ≤ T̃ε
and of k:
(a) x′k,ε is well defined. In particular, if k /∈
N ′j,ε
, yε(t
k,ε) corresponds to a
collision point on ∂D1(Q0), and not to a collision point on a piece of ∂D to
the right of Q0.
(b) If k > 0 and k /∈
N ′j,ε
, then x′k,ε = Fx
k−1,ε.
(c) If k > 0 and k ∈
N ′j,ε
, then d(x′k,ε, Fx
k−1,ε) ≤ ρ and the ϕ coordinate of
k,ε) satisfies ϕ(yε(t
k,ε)) = ϕ
k,ε +O(ε).
(d) d(xk, x
k,ε) ≤ const ρ(const/γ)k .
(e) k = N ′j,ε if and only if k = Nj.
(f) If k > 0, t′k,ε − t′k−1,ε = tk − tk−1 +O(ρ (const/γ)
We defer the proof of Lemma 4.3.3 until the end of this subsection. Assuming
that ε is sufficiently small for the conclusions of Lemma 4.3.3 to be valid, we
continue with the proof of Lemma 4.3.2.
Set M = N ∧ N ′ε − 1. Note that M ≤ λ ∼ L. From (f) in Lemma 4.3.3 and
Equations (4.22) and (4.23), we see that
∣tM − t′M,ε
∣t′k,ε − t′k−1,ε − (tk − tk−1)
∣ = O
constλ
→ 0 as ε → 0.
Because the flight times t′k,ε − t′k−1,ε and tk − tk−1 are uniformly bounded above,
it follows from the definitions of N and N ′ε that tM , t
M,ε ≥ L − const. But from
Subsection 4.2.2, the time between the collisions of the left gas particle with the
piston are uniformly bounded away from zero. Using (c) and Equation (4.20), it
follows that
G(zε(s))−G(z0(s))ds
= O(L−1) +
k∈{Nj :Nj≤M}
2E1,0 cosϕk −
2E1,ε(t
k,ε) cos(ϕ
k,ε +O(ε))
= O(L−1) +
k∈{Nj :Nj≤M}
2E1,0 cosϕk −
2E1,0 cosϕ
k,ε +O(εL)
= O(L−1) +O(εL2) +
2E1,0
k∈{Nj :Nj≤M}
∣cosϕk − cosϕ′k,ε
But using (d),
k∈{Nj :Nj≤M}
∣cosϕk − cosϕ′k,ε
O(ρ(const/γ)k) = O(ρ(const/γ)λ).
Since εL2 = O(ρ(const/γ)λ), this finishes the proof of Lemma 4.3.2.
Proof of Lemma 4.3.3. The proof is by induction. We take ε to be so small that
Equation (4.17) is satisfied. This is possible by Equation (4.23).
It is trivial to verify (a)-(f) for k = 0. So let 0 < l < N ∧N ′ε, and suppose that
(a)-(f) have been verified for all k < l. We have three cases to consider:
Case 1: l − 1 and l /∈
N ′j,ε
In this case, verifying (a)-(f) for k = l is a relatively straightforward application
of the machinery developed in Subsection 4.3.2, because for t′l−1,ε ≤ t ≤ t′l,ε, yε(t)
traces out the billiard orbit between x′l−1,ε and x
l,ε corresponding to free flight in
the domain D1(Q0). We make only two remarks.
First, as long as ε is sufficiently small, it really is true that x′l,ε = yε(t
corresponds to a true collision point on ∂D1(Q0). Indeed, if this were not the
case, then it must be that Qε(t
l,ε) > Q0, and yε(t
l,ε) would have to correspond
to a collision with the side of the “tube” to the right of Q0. But then x
l,ε =
Fx′l−1,ε ∈ Ωh0 would correspond to a collision with an immobile piston at Q0 and
would satisfy d(xk, x
k,ε) ≤ const ρ(const/γ)k ≤ const ρ(const/γ)λ = o(γ), using
Equations (4.16) and (4.23). But xk /∈ Nγ(∂Ωh0), and so it follows that when the
trajectory of yε(t) crosses the plane {Q = Q0}, it is at least a distance ∼ γ away
from the boundary of the face of the piston, and its velocity vector is pointed
no closer than ∼ γ to being parallel to the piston’s face. As Qε(t′l,ε) − Q0 =
O(εL) = o(γ), and it is geometrically impossible (for small ε) to construct a right
triangle whose sides s1, s2 satisfy |s1| ≥∼ γ, |s2| ≤ O(εL), with the measure of
the acute angle adjacent to s1 being greater than ∼ γ, we have a contradiction.
After crossing the plane {Q = Q0}, yε(t) must experience its next collision with
the face of the piston, which violates the fact that l /∈
N ′j,ε
Second, t′l,ε − t′l−1,ε = ζx′l−1,ε +O(εL), because v1,ε = v1,0 +O(εL). See Equa-
tion (4.20). From Equation 4.18,
∣ζxl−1 − ζx′l−1,ε
∣ ≤ O((ρ/γ) (const/γ)l−1). As
tl − tl−1 = ζxl−1 and εL = O((ρ/γ) (const/γ)l−1), we obtain (f).
Case 2: There exists i such that l = N ′i,ε:
For definiteness, we suppose that Qε(t
l,ε) ≥ Q0, so that the left gas particle collides
with the piston to the right of Q0. The case when Qε(t
l,ε) ≤ Q0 can be handled
similarly.
We know that xl−1, xl, xl+1 /∈ Nγ(∂Ωh0)∪Nγ(F−1Nγ(∂Ωh0)). Using the induc-
tive hypothesis and Equation (4.16), we can define
x′′l,ε = Fx
l−1,ε, x
l+1,ε = F
2x′l−1,ε,
and d(xl, x
l,ε) ≤ const ρ(const/γ)l, d(xl+1, x′′l+1,ε) ≤ const ρ(const/γ)l+1. In partic-
ular, x′′l,ε and x
l+1,ε are both a distance ∼ γ away from ∂Ωh0 . Furthermore, when
the left gas particle collides with the moving piston, it follows from Equation (4.8)
that the difference between its angle of incidence and its angle of reflection is
O(ε). Referring to Figure 4.3, this means that ϕ′l,ε = ϕ′′l,ε + O(ε). Geometric
arguments similar to the one given in Case 1 above show that the yε-trajectory
of the left gas particle has precisely one collision with the piston and no other
collisions with the sides of the gas container when the gas particle traverses the
region Q0 ≤ Q ≤ Qε(t′l,ε). Note that x′l,ε was defined to be the point in the
collision cross-section Ωh0 corresponding to the return of the yε-trajectory into
the region Q ≤ Q0. See Figure 4.3. From this figure, it is also evident that
d(r′l,ε, r
l,ε) ≤ O(εL/γ). Thus d(x′′l,ε, x′l,ε) = O(εL/γ), and this explains the choice
of ρ(ε) in Equation (4.22).
From the above discussion and the machinery of Subsection 4.3.2, (a)-(e) now
follow readily for both k = l and k = l + 1. Furthermore, property (f) follows in
much the same manner as it did in Case 1 above. However, one should note that
t′l,ε− t′l−1,ε = ζx′l−1,ε+O(εL)+O(εL/γ) and t′l+1,ε− t′l,ε = ζx′l,ε+O(εL)+O(εL/γ),
because of the extra distance O(εL/γ) that the gas particle travels to the right of
Q0. But εL/γ = O((ρ/γ) (const/γ)l−1), and so property (f) follows.
Case 3: There exists i such that l − 1 = N ′i,ε:
As mentioned above, the inductive step in this case follows immediately from our
analysis in Case 2.
r−coordinate Q0 Qε(t′l,ε)
D1(Q0)
γ/2 γ/2
O(εL)
O(εL/γ)
r′′l,ε
ϕ′′l,ε
r′l,ε
ϕ′l,ε
r′l−1,ε
ϕ′l−1,ε
r′l+1,ε
r′′l+1,ε
Figure 4.3: An analysis of the divergences of orbits when ε > 0 and the left
gas particle collides with the moving piston to the right of Q0. Note that the
dimensions are distorted for visual clarity, but that εL and εL/γ are both o(γ) as
ε→ 0.
Furthermore, ϕ′′l,ε ∈ (−π/2 + γ/2, π/2 − γ/2) and ϕ′l,ε = ϕ′′l,ε + O(ε), and so
r′l,ε = r
l,ε + O(εL/γ). In particular, the yε-trajectory of the left gas particle has
precisely one collision with the piston and no other collisions with the sides of the
gas container when the gas particle traverses the region Q0 ≤ Q ≤ Qε(t′l,ε)
4.4 Generalization to a full proof
of Theorem 4.1.1
It remains to generalize the proof in Sections 4.2 and 4.3 to the cases when n1, n2 ≥
1 and d = 3.
4.4.1 Multiple gas particles on each side of the piston
When d = 2, but n1, n2 ≥ 1, only minor modifications are necessary to generalize
the proof above. As in Subsection 4.2.2, one defines a stopping time T̃ε satisfying
T̃ε < T ∧ Tε
= O(ε) such that for 0 ≤ t ≤ T̃ε/ε, gas particles will only
experience clean collisions with the piston.
Next, define H(z) by
H(z) =
∣v⊥1,j
∣ δq⊥1,j=Q
∣v⊥2,j
∣ δq⊥2,j=Q
∣v⊥1,j
∣ δq⊥1,j=Q
∣v⊥2,j
∣ δq⊥2,j=Q
It follows that for 0 ≤ t ≤ T̃ε/ε, hε(t)−hε(0) = O(ε)+ε
H(zε(s))ds. From here,
the rest of the proof follows the same steps made in Subsection 4.3.1. We note
that at Step 3, we find that H(z) − H̄(h(z)) divides into n1 + n2 pieces, each of
which depends on only one gas particle when the piston is held fixed.
4.4.2 Three dimensions
The proof of Theorem 4.1.1 in d = 3 dimensions is essentially the same as the proof
in two dimensions given above. The principal differences are due to differences in
the geometry of billiards. We indicate the necessary modifications.
In analogy with Section 4.2.1, we briefly summarize the necessary facts for the
billiard flows of the gas particles when M = ∞ and the slow variables are held
fixed at a specific value h ∈ V. As before, we will only consider the motions of one
gas particle moving in D1. Thus we consider the billiard flow of a point particle
moving inside the domain D1 at a constant speed
2E1. Unless otherwise noted,
we use the notation from Section 4.2.1.
The billiard flow takes place in the five-dimensional space M1 = {(q1, v1) ∈
T D1 : q1 ∈ D1, |v1| =
2E1}/ ∼. Here the quotient means that when q1 ∈ ∂D1,
we identify velocity vectors pointing outside of D1 with those pointing inside D1
by reflecting orthogonally through the tangent plane to ∂D1 at q1. The billiard
flow preserves Liouville measure restricted to the energy surface. This measure
has the density dµ = dq1dv1/(8πE1 |D1|). Here dq1 represents volume on R3, and
dv1 represents area on S
v1 ∈ R3 : |v1| =
The collision cross-section Ω = {(q1, v1) ∈ T D1 : q1 ∈ ∂D1, |v1| =
2E1}/ ∼ is
properly thought of as a fiber bundle, whose base consists of the smooth pieces of
∂D1 and whose fibers are the set of outgoing velocity vectors at q1 ∈ ∂D1. This and
other facts about higher-dimensional billiards, with emphasis on the dispersing
case, can be found in [BCST03]. For our purposes, Ω can be parameterized as
follows. We decompose ∂D1 into a finite union ∪jΓj of pieces, each of which
is diffeomorphic via coordinates r to a compact, connected subset of R2 with
a piecewise C3 boundary. The Γj are nonoverlapping, except possibly on their
boundaries. Next, if (q1, v1) ∈ Ω and v1 is the outward going velocity vector, let
v̂ = v1/ |v1|. Then Ω can be parameterized by {x = (r, v̂)}. It follows that Ω it is
diffeomorphic to ∪jΓj × S2+, where S2+ is the upper unit hemisphere, and by ∂Ω
we mean the subset diffeomorphic to (∪j∂Γj × S2+)
(∪jΓj × ∂S2+). If x ∈ Ω,
we let ϕ ∈ [0, π/2] represent the angle between the outgoing velocity vector and
the inward pointing normal vector n to ∂D1, i.e. cosϕ = 〈v̂, n〉. Note that we no
longer allow ϕ to take on negative values. The return map F : Ω 	 preserves the
projected probability measure ν, which has the density dν = cosϕdv̂ dr/(π |∂D1|).
Here |∂D1| is the area of ∂D1.
F is an invertible, measure preserving transformation that is piecewise C2.
Because of our assumptions on D1, the free flight times and the curvature of ∂D1
are uniformly bounded. The bound on ‖DF (x)‖ given in Equation (4.2) is still
true. A proof of this fact for general three-dimensional billiard tables with finite
horizon does not seem to have made it into the literature, although see [BCST03]
for the case of dispersing billiards. For completeness, we provide a sketch of a
proof for general billiard tables in Section 4.6.
We suppose that the billiard flow is ergodic, so that F is ergodic. Again, we
induce F on the subspace Ω̂ of Ω corresponding to collisions with the (immobile)
piston to obtain the induced map F̂ : Ω̂ 	 that preserves the induced measure ν̂.
The free flight time ζ : Ω → R again satisfies the derivative bound given in
Equation (4.3). The generalized Santaló’s formula [Che97] yields
Eνζ =
4 |D1|
|v1| |∂D1|
If ζ̂ : Ω̂ → R is the free flight time between collisions with the piston, then it
follows from Proposition 4.5.1 that
Eν̂ ζ̂ =
4 |D1|
|v1| ℓ
The expected value of
∣ when the left gas particle collides with the (immo-
bile) piston is given by
∣ = Eν̂
2E1 cosϕ =
cos2 ϕdv̂1 =
As a consequence, we obtain
Lemma 4.4.1. For µ− a.e. y ∈ M1,
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds =
3 |D1(Q)|
Compare the proof of Lemma 4.2.1.
With these differences in mind, the rest of the proof of Theorem 4.1.1 when
d = 3 proceeds in the same manner as indicated in Sections 4.2, 4.3 and 4.4.1
above. The only notable difference occurs in the proof of the Gronwall-type in-
equality for billiards. Due to dimensional considerations, if one follows the proof
of Lemma 4.3.1 for a three-dimensional billiard table, one finds that
νNγ(F−1Nγ(∂Ω)) = O(γ1−4α + γα).
The optimal value of α is 1/5, and so νNγ(F−1Nγ(∂Ω)) = O(γ1/5) as γ → 0.
Hence νCγ,λ = O(λγ1/5), which is a slightly worse estimate than the one in Equa-
tion (4.19). However, it is still sufficient for all of the arguments in Section 4.3.2,
and this finishes the proof.
4.5 Inducing maps on subspaces
Here we present some well-known facts on inducing measure preserving transfor-
mations on subspaces. Let F : (Ω,B, ν) 	 be an invertible, ergodic, measure
preserving transformation of the probability space Ω endowed with the σ-algebra
B and the probability measure ν. Let Ω̂ ∈ B satisfy 0 < νΩ̂ < 1. Define
R : Ω̂ → N to be the first return time to Ω̂, i.e. Rω = inf{n ∈ N : F nω ∈ Ω̂}.
Then if ν̂ := ν(· ∩ Ω̂)/νΩ̂ and B̂ := {B ∩ Ω̂ : B ∈ B}, F̂ : (Ω̂, B̂, ν̂) 	 defined
by F̂ω = FRωω is also an invertible, ergodic, measure preserving transforma-
tion [Pet83]. Furthermore Eν̂R =
Rdν̂ = (νΩ̂)−1.
This last fact is a consequence of the following proposition:
Proposition 4.5.1. If ζ : Ω → R≥0 is in L1(ν), then ζ̂ =
n=0 ζ ◦F n is in L1(ν̂),
Eν̂ ζ̂ =
Proof.
ζ ◦ F n dν̂ =
ζ ◦ F n dν =
Ω̂∩{R=k}
ζ ◦ F n dν
Fn(Ω̂∩{R=k})
ζ dν =
ζ dν,
because {F n(Ω̂ ∩ {R = k}) : 0 ≤ n < k <∞} is a partition of Ω.
4.6 Derivative bounds for the billiard map
in three dimensions
Returning to Section 4.4.2, we need to show that for a billiard table D1 ⊂ R3 with
a piecewise C3 boundary and the free flight time uniformly bounded above, the
billiard map F satisfies the following: If x0 /∈ ∂Ω ∪ F−1(∂Ω), then
‖DF (x0)‖ ≤
const
cosϕ(Fx0)
Fix x0 = (r0, v̂0) ∈ Ω, and let x1 = (r1, v̂1) = Fx0. Let Σ be the plane that
perpendicularly bisects the straight line between r0 and r1, and let r1/2 denote
the point of intersection. We consider Σ as a “transparent” wall, so that in a
neighborhood of x0, we can write F = F2 ◦ F1. Here, F1 is like a billiard map in
that it takes points (i.e. directed velocity vectors with a base) near x0 to points
with a base on Σ and a direction pointing near r1. (F1 would be a billiard map if
we reflected the image velocity vectors orthogonally through Σ.) F2 is a billiard
map that takes points in the image of F1 and maps them near x1. Let x1/2 =
F1x0 = F
2 x1. Then ‖DF (x0)‖ ≤ ‖DF1(x0)‖
∥DF2(x1/2)
It is easy to verify that ‖DF1(x0)‖ ≤ const, with the constant depending
only on the curvature of ∂D1 at r0. In other words, the constant may be chosen
independent of x0. Similarly,
∥DF−12 (x1)
∥ ≤ const. Because billiard maps pre-
serve a probability measure with a density proportional to cosϕ, detDF−12 (x1) =
cosϕ1/ cosϕ1/2 = cosϕ1. As Ω is 4-dimensional, it follows from Cramer’s Rule for
the inversion of linear transformations that
∥DF2(x1/2)
const
∥DF−12 (x1)
detDF−12 (x1)
≤ const
cosϕ1
and we are done.
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http://www.cims.nyu.edu/~paulrite
	Dedication
	Acknowledgements
	Abstract
	List of Figures
	Introduction
	The adiabatic piston
	Physical motivation for the results
	Background Averaging Material
	The averaging framework
	Some classical averaging results
	A proof of Anosov's theorem
	Moral
	Results for piston systems in one dimension
	Statement of results
	Heuristic derivation of the averaged equation for the hard core piston
	Proof of the main result for the hard core piston
	Proof of the main result for the soft core piston
	Appendix to Section 3.4
	The periodic oscillation of an adiabatic piston in two or three dimensions
	Statement of the main result
	Preparatory material concerning a two-dimensional gas container with only one gas particle on each side
	Proof of the main result for two-dimensional gas containers with only one gas particle on each side
	Generalization to a full proof of Theorem 4.1.1
	Inducing maps on subspaces
	Derivative bounds for the billiard map in three dimensions
	Bibliography

  We study a heavy piston of mass $M$ that moves in one dimension. The piston
separates two gas chambers, each of which contains finitely many ideal, unit
mass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averaging
techniques, we prove that the actual motions of the piston converge in
probability to the predicted averaged behavior on the time scale $M^ {1/2} $
when $M$ tends to infinity while the total energy of the system is bounded and
the number of gas particles is fixed. Neishtadt and Sinai previously pointed
out that an averaging theorem due to Anosov should extend to this situation.
  When $ d=1$, the gas particles move in just one dimension, and we prove that
the rate of convergence of the actual motions of the piston to its averaged
behavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. The
convergence is uniform over all initial conditions in a compact set. We also
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.
  In addition, we prove generalizations of our results to $N$ pistons
separating $N+1$ gas chambers. We also provide a general discussion of
averaging theory and the proofs of a number of previously known averaging
results. In particular, we include a new proof of Anosov's averaging theorem
for smooth systems that is primarily due to Dolgopyat.

Introduction 1
1.1 The adiabatic piston . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Physical motivation for the results . . . . . . . . . . . . . . . . . . . 4
2 Background Averaging Material 8
2.1 The averaging framework . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Some classical averaging results . . . . . . . . . . . . . . . . . . . . 12
2.3 A proof of Anosov’s theorem . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Moral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Results for piston systems in one dimension 27
3.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Heuristic derivation of the averaged equation for the hard core piston 34
3.3 Proof of the main result for the hard core piston . . . . . . . . . . . 35
3.4 Proof of the main result for the soft core piston . . . . . . . . . . . 41
3.5 Appendix to Section 3.4 . . . . . . . . . . . . . . . . . . . . . . . . 50
4 The periodic oscillation of an adiabatic piston in two or three
dimensions 54
4.1 Statement of the main result . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Preparatory material concerning a
two-dimensional gas container with only one gas particle on each side 58
4.3 Proof of the main result for two-dimensional gas containers with
only one gas particle on each side . . . . . . . . . . . . . . . . . . . 66
4.4 Generalization to a full proof
of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Inducing maps on subspaces . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Derivative bounds for the billiard map
in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography 84
List of Figures
1.1 A gas container D in d = 2 dimensions separated by an adiabatic
piston. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 An effective potential. . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 A schematic of the phase space M. Note that although the level
set Mc = {h = c} is depicted as a torus, it need not be a torus. It
could be any compact, co-dimension m submanifold. . . . . . . . . . 11
3.1 The piston system with n1 = 3 and n2 = 4. Note that the gas
particles do not interact with each other, but only with the piston
and the walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 A gas container D ⊂ R2 separated by a piston. . . . . . . . . . . . . 55
4.2 A choice of coordinates on phase space. . . . . . . . . . . . . . . . . 59
4.3 An analysis of the divergences of orbits when ε > 0 and the left gas
particle collides with the moving piston to the right of Q0. Note
that the dimensions are distorted for visual clarity, but that εL and
εL/γ are both o(γ) as ε→ 0. . . . . . . . . . . . . . . . . . . . . . 79
Chapter 1
Introduction
What can be rigorously understood about the nonequilibrium dynamics of chaotic,
many particle systems? Although much progress has been made in understanding
the infinite time behavior of such systems, our understanding on finite time scales
is still far from complete. Systems of many particles contain a large number of
degrees of freedom, and it is often impractical or impossible to keep track of their
full dynamics. However, if one is only interested in the evolution of macroscopic
quantities, then these variables form a small subset of all of the variables. The evo-
lution of these quantities does not itself form a closed dynamical system, because
it depends on events happening in all of the (very large) phase space. We must
therefore develop techniques for describing the evolution of just a few variables in
phase space. Such descriptions are valid on limited time scales because a large
amount of information about the dynamics of the full system is lost. However, the
time scales of validity can often be long enough to enable a good prediction of the
observable dynamics.
Averaging techniques help to describe the evolution of certain variables in
some physical systems, especially when the system has components that move on
different time scales. The primary results of this thesis involve applying averaging
techniques to chaotic microscopic models of gas particles separated by an adiabatic
piston for the purposes of justifying and understanding macroscopic laws.
This thesis is organized as follows. In Section 1.1 we briefly introduce the the
adiabatic piston problem and our results. In Section 1.2 we review the physical
motivations for our results. The following three chapters may each be read inde-
pendently. Chapter 2 presents an introduction to averaging theory and the proofs
of a number of averaging theorems for smooth systems that motivate our later
proofs for the piston problem. Chapter 3 contains our results for piston systems
in one dimension, and Chapter 4 contains our results for the piston system in
dimensions two and three.
1.1 The adiabatic piston
Consider the following simple model of an adiabatic piston separating two gas
containers: A massive piston of mass M ≫ 1 divides a container in Rd, d =
1, 2, or 3, into two halves. The piston has no internal degrees of freedom and can
only move along one axis of the container. On either side of the piston there are a
finite number of ideal, unit mass, point gas particles that interact with the walls
of the container and with the piston via elastic collisions. When M = ∞, the
piston remains fixed in place, and each gas particle performs billiard motion at a
constant energy in its sub-container. We make an ergodicity assumption on the
behavior of the gas particles when the piston is fixed. Then we study the motions
of the piston when the number of gas particles is fixed, the total energy of the
system is bounded, but M is very large.
Heuristically, after some time, one expects the system to approach a steady
state, where the energy of the system is equidistributed amongst the particles
and the piston. However, even if we could show that the full system is ergodic,
an abstract ergodic theorem says nothing about the time scale required to reach
such a steady state. Because the piston will move much slower than a typical gas
particle, it is natural to try to determine the intermediate behavior of the piston
by averaging techniques. By averaging over the motion of the gas particles on a
time scale chosen short enough that the piston is nearly fixed, but long enough
that the ergodic behavior of individual gas particles is observable, we will show
that the system does not approach the expected steady state on the time scale
M1/2. Instead, the piston oscillates periodically, and there is no net energy transfer
between the gas particles.
The results of this thesis follow earlier work by Neishtadt and Sinai [Sin99,
NS04]. They determined that for a wide variety of Hamiltonians for the gas par-
ticles, the averaged behavior of the piston is periodic oscillation, with the piston
moving inside an effective potential well whose shape depends on the initial po-
sition of the piston and the gas particles’ Hamiltonians. They pointed out that
an averaging theorem due to Anosov [Ano60, LM88], proved for smooth systems,
should extend to this case. The main result of the present work, Theorem 4.1.1,
is that Anosov’s theorem does extend to the particular gas particle Hamiltonian
described above. Thus, if we examine the actual motions of the piston with respect
to the slow time τ = t/M1/2, then, as M → ∞, in probability (with respect to
Liouville measure) most initial conditions give rise to orbits whose actual motion is
accurately described by the averaged behavior for 0 ≤ τ ≤ 1, i.e. for 0 ≤ t ≤M1/2.
A recent study involving some similar ideas by Chernov and Dolgopyat [CD06a]
considered the motion inside a two-dimensional domain of a single heavy, large gas
particle (a disk) of mass M ≫ 1 and a single unit mass point particle. They as-
sumed that for each fixed location of the heavy particle, the light particle moves
inside a dispersing (Sinai) billiard domain. By averaging over the strongly hy-
perbolic motions of the light particle, they showed that under an appropriate
scaling of space and time the limiting process of the heavy particle’s velocity is a
(time-inhomogeneous) Brownian motion on a time scale O(M1/2). It is not clear
whether a similar result holds for the piston problem, even for gas containers with
good hyperbolic properties, such as the Bunimovich stadium. In such a container
the motion of a gas particle when the piston is fixed is only nonuniformly hyper-
bolic because it can experience many collisions with the flat walls of the container
immediately preceding and following a collision with the piston.
The present work provides a weak law of large numbers, and it is an open
problem to describe the sizes of the deviations for the piston problem [CD06b].
Although our result does not yield concrete information on the sizes of the devia-
tions, it is general in that it imposes very few conditions on the shape of the gas
container. Most studies of billiard systems impose strict conditions on the shape
of the boundary, generally involving the sign of the curvature and how the corners
are put together. The proofs in this work require no such restrictions. In particu-
lar, the gas container can have cusps as corners and need satisfy no hyperbolicity
conditions.
If the piston divides a container in R2 or R3 with axial symmetry, such as a
rectangle or a cylinder, then our ergodicity assumption on the behavior of the gas
particles when the piston is fixed does not hold. In this case, the interactions of the
gas particles with the piston and the ends of the container are completely specified
by their motions along the normal axis of the container. Thus, this system projects
onto a system inside an interval consisting of a massive point particle, the piston,
which interacts with the gas particles on either side of it. These gas particles make
elastic collisions with the walls at the ends of the container and with the piston,
but they do not interact with each other. For such one-dimensional containers,
the effects of the gas particles are quasi-periodic and can be essentially decoupled,
and we recover a strong law of large numbers with a uniform rate, reminiscent
of classical averaging over just one fast variable in S1: The convergence of the
actual motions to the averaged behavior is uniform over all initial conditions, with
the size of the deviations being no larger than O(M−1/2) on the time scale M+1/2.
See Theorem 3.1.1. Gorelyshev and Neishtadt [GN06] independently obtained this
result.
For systems in d = 1 dimension, we also investigate the behavior of the system
when the interactions of the gas particles with the walls and the piston have been
smoothed, so that Anosov’s theorem applies directly. Let δ ≥ 0 be a parameter
of smoothing, so that δ = 0 corresponds to the hard core setting above. Then
the averaged behavior of the piston is still a periodic oscillation, which depends
smoothly on δ. We show that the deviations of the actual motions of the piston
from the averaged behavior are again not more than O(M−1/2) on the time scale
M1/2. The size of the deviations is bounded uniformly, both over initial conditions
and over the amount of smoothing, Theorem 3.1.2.
Our results for a single heavy piston separating two gas containers generalize
to the case of N heavy pistons separating N +1 gas containers. Here the averaged
behavior of the pistons has them moving like an N -dimensional particle inside an
effective potential well. Compare Section 3.1.3.
The systems under consideration in this work are simple models of an adiabatic
piston. The general adiabatic piston problem [Cal63], well-known from physics,
consists of the following: An insulating piston separates two gas containers, and
initially the piston is fixed in place, and the gas in each container is in a separate
thermal equilibrium. At some time, the piston is no longer externally constrained
and is free to move. One hopes to show that eventually the system will come to a
full thermal equilibrium, where each gas has the same pressure and temperature.
Whether the system will evolve to thermal equilibrium and the interim behavior of
the piston are mechanical problems, not adequately described by thermodynam-
ics [Gru99], that have recently generated much interest within the physics and
mathematics communities following Lieb’s address [Lie99]. One expects that the
system will evolve in at least two stages. First, the system relaxes deterministically
toward a mechanical equilibrium, where the pressures on either side of the piston
are equal. In the second, much longer, stage, the piston drifts stochastically in the
direction of the hotter gas, and the temperatures of the gases equilibrate. See for
example [GPL03, CL02, Che04] and the references therein. Previously, rigorous
results have been limited mainly to models where the effects of gas particles rec-
olliding with the piston can be neglected, either by restricting to extremely short
time scales [LSC02, CLS02] or to infinite gas containers [Che04].
1.2 Physical motivation for the results
In this section, we briefly review the physical motivations for our results on the
adiabatic piston.
Consider a massive, insulating piston of massM that separates a gas container
D in Rd, d = 1, 2, or 3. See Figure 1.1. Denote the location of the piston by Q
and its velocity by dQ/dt = V . If Q is fixed, then the piston divides D into two
subdomains, D1(Q) = D1 on the left and D2(Q) = D2 on the right. By |Di| we
denote the area (when d = 2, or length, when d = 1, or volume, when d = 3) of
Di. Define
∂ |D1(Q)|
= −∂ |D2(Q)|
so that ℓ is the piston’s cross-sectional length (when d = 2, or area, when d = 3).
If d = 1, then ℓ = 1. By Ei we denote the total energy of the gas inside Di.
D1(Q) D2(Q)
E1 E2
D = D1(Q) ⊔ D2(Q)
✲ V = εW
M = ε−2 ≫ 1
s✟✟✙ℓ
Figure 1.1: A gas container D in d = 2 dimensions separated by an adiabatic
piston.
We are interested in the dynamics of the piston when the system’s total energy
is bounded and M → ∞. When M = ∞, the piston remains fixed in place, and
each energy Ei remains constant. When M is large but finite, MV
2/2 is bounded,
and so V = O(M−1/2). It is natural to define
ε =M−1/2,
so that W is of order 1 as ε → 0. This is equivalent to scaling time by ε, and so
we introduce the slow time
τ = εt.
If we let Pi denote the pressure of the gas inside Di, then heuristically the
dynamics of the piston should be governed by the following differential equation:
= V, M
= P1ℓ− P2ℓ,
= P1ℓ− P2ℓ.
(1.1)
To find differential equations for the energies of the gases, note that in a short
amount of time dt, the change in energy should come entirely from the work done
on a gas, i.e. the force applied to the gas times the distance the piston has moved,
because the piston is adiabatic. Thus, one expects that
= −V P1ℓ,
= +V P2ℓ,
= −WP1ℓ,
= +WP2ℓ.
(1.2)
To obtain a closed system of differential equations, it is necessary to insert an
expression for the pressures. Piℓ should be the average force from the gas particles
in Di experienced by the piston when it is held fixed in place. Whether such an
expression, depending only on Ei and Di(Q), exists and is the same for (almost)
every initial condition of the gas particles depends strongly on the microscopic
model of the gas particle dynamics. Sinai and Neishtadt [Sin99, NS04] pointed
out that for many microscopic models where the pressures are well defined, the
solutions of Equations (1.1) and (1.2) have the piston moving according to a model-
dependent effective Hamiltonian.
Because the pressure of an ideal gas in d dimensions is proportional to the
energy density, with the constant of proportionality 2/d, we choose to insert
d |Di|
Later, we will make assumptions on the microscopic gas particle dynamics to justify
this substitution. However, if we accept this definition of the pressure, we obtain
the following ordinary differential equations for the four macroscopic variables of
the system:
d |D1(Q)|
− 2E2ℓ
d |D2(Q)|
− 2WE1ℓ
d |D1(Q)|
2WE2ℓ
d |D2(Q)|
. (1.3)
For these equations, one can see the effective Hamiltonian as follows. Since
d ln(Ei)
d ln(|Di(Q)|)
Ei(τ) = Ei(0)
|Di(Q(0))|
|Di(Q(τ))|
Hence
d2Q(τ)
E1(0) |D1(Q(0))|2/d
|D1(Q(τ))|1+2/d
E2(0) |D2(Q(0))|2/d
|D2(Q(τ))|1+2/d
effective potential
P1 = P2
Figure 1.2: An effective potential.
and so (Q,W ) behave as if they were the coordinates of a Hamiltonian system
describing a particle undergoing motion inside a potential well. The effective
Hamiltonian may be expressed as
W 2 +
E1(0) |D1(Q(0))|2/d
|D1(Q)|2/d
E2(0) |D2(Q(0))|2/d
|D2(Q)|2/d
. (1.4)
The question is, do the solutions of Equation (1.3) give an accurate description
of the actual motions of the macroscopic variables when M tends to infinity? The
main result of this thesis, Theorem 4.1.1, is that, for an appropriately defined
system, the answer to this question is affirmative for 0 ≤ t ≤ M1/2, at least for
most initial conditions of the microscopic variables. Observe that one should not
expect the description to be accurate on time scales much longer than O(M1/2) =
O(ε−1). The reason for this is that, presumably, there are corrections of size O(ε)
in Equation (1.3) that we are neglecting. For τ = εt > O(1), these corrections
should become significant. Such higher order corrections for the adiabatic piston
were studied by Crosignani et al. [CDPS96].
Chapter 2
Background Averaging Material
In this chapter, we present a number of well-known classical averaging results
for smooth systems, as well as a proof of Anosov’s averaging theorem, which is
the first general multi-phase averaging result. All of these theorems are at least
45 years old. However, we present them here because our proofs of the classical
results are at least slightly novel, and the ideas in them lend themselves well to
certain higher-dimensional generalizations. In particular, they are fairly close to
the ideas in the proof we give for our piston results in one dimension. The proof of
Anosov’s theorem is a new and unpublished proof due mainly to Dolgopyat, with
some further simplifications made. The ideas in this proof underly the ideas we
will use to prove the weak law of large numbers for our piston system in dimensions
two and three.
We begin by giving a discussion of a framework for general averaging theory
and some averaging results. A number of classical averaging theorems are then
proved, followed by the proof of Anosov’s theorem.
2.1 The averaging framework
In this section, consider a family of ordinary differential equations
= Z(z, ε) (2.1)
on a smooth, finite-dimensional Riemannian manifold M, which is indexed by the
real parameter ε ∈ [0, ε0]. Assume
• Regularity: the functions Z and ∂Z/∂ε are both C1 on M× [0, ε0].
We denote the flow generated by Z(·, ε) by zε(t, z) = zε(t). We will usually
suppress the dependence on the initial condition z = zε(0, z). Think of zε(·)
as being a random variable whose domain is the space of initial conditions for
the differential equation (2.1) and whose range is the space of continuous paths
(depending on the parameter t) in M.
• Existence of smooth integrals: z0(t) has m independent C2 first integrals
h = (h1, . . . , hm) : M → Rm.
Then h is conserved by z0(t), and at every point the linear operator ∂h/∂z has
full rank. It follows from the implicit function theorem that each level set
Mc := {h = c}
is a smooth submanifold of co-dimension m, which is invariant under z0(t). Fur-
ther, assume that there exists an open ball U ⊂ Rm satisfying:
• Compactness: ∀c ∈ U , Mc is compact.
• Preservation of smooth measures: ∀c ∈ U , z0(t)|Mc preserves a smooth
measure µc that varies smoothly with c, i.e. there exists a C1 function
g : M → R>0 such that g|Mc is the density of µc with respect to the
restriction of Riemannian volume.
hε(t, z) = hε(t) := h(zε(t)).
Again, think of hε(·) as being a random variable that takes initial conditions z ∈ M
to continuous paths (depending on the parameter t) in U . Since dh0/dt ≡ 0,
Hadamard’s Lemma allows us to write
= εH(zε, ε)
for some C1 function H : M× [0, ε0] → U . Observe that
(t) = Dh(zε(t))Z(zε(t), ε) = Dh(zε(t))
Z(zε(t), ε)− Z(zε(t), 0)
so that
H(z, 0) = L ∂Z
|ε=0h.
Here L denotes the Lie derivative.
Define the averaged vector field H̄ by
H̄(h) =
H(z, 0)dµh(z). (2.2)
Then H̄ is C1. Fix a compact set V ⊂ U , and introduce the slow time
τ = εt.
Let h̄(τ, z) = h̄(τ) be the random variable that is the solution of
= H̄(h̄), h̄(0) = hε(0).
We only consider the dynamics in a compact subset of phase space, so for initial
conditions z ∈ h−1U , define the stopping time
Tε(z) = Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Heuristically, think of the phase space M as being a fiber bundle whose base
is the open set U and whose fibers are the compact sets Mh. See Figure 2.1.
Then the vector field Z(·, 0) is perpendicular to the base, so its orbits z0(t) flow
only along the fibers. Now when 0 < ε ≪ 1, the vector field Z(·, ε) acquires a
component of size O(ε) along the base, and so its orbits zε(t) have a small drift
along the base, which we can follow by observing the evolution of hε(t). Because
of this, we refer to h as consisting of the slow variables. Other variables, used
to complete h to a parameterization of (a piece of) phase space, are called fast
variables. Note that hε(t) depends on all the dimensions of phase space, and so it
is not the flow of a vector field on the m-dimensional space U . However, because
the motion along each fiber is relatively fast compared to the motion across fibers,
we hope to be able to average over the fast motions and obtain a vector field
on U that gives a good description of hε(t) over a relatively long time interval,
independent of where the solution zε(t) started on Mhε(0). Because our averaged
vector field, as defined by Equation (2.2), only accounts for deviations of size O(ε),
we cannot expect this time interval to be longer than size O(1/ε). In terms of
the slow time τ = εt, this length becomes O(1). In other words, the goal of the
first-order averaging method described above should be to show that, in some
sense, sup0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣→ 0 as ε → 0. This is often referred to as the
averaging principle.
Note that the assumptions of regularity, existence of smooth integrals, com-
pactness, and preservation of smooth measures above are not sufficient for the
averaging principle to hold in any form. As an example of just one possible
obstruction, the level sets Mc could separate into two completely disjoint sets,
Mc = M+c ⊔ M−c . If this were the case, then it would be implausible that the
solutions of the averaged vector field defined by averaging over all of Mc would
accurately describe hε(t, z), independent of whether z ∈ M+c or z ∈ M−c .
Some averaging results
So far, we are in a general averaging setting. Frequently, one also assumes that the
invariant submanifolds, Mh, are tori, and that there exists a choice of coordinates
z = (h, ϕ)
M = {(h, ϕ)}
✲ h ∈ U ⊂ Rm
“slow variables”
ϕ =“fast variables”
✄✗Z(·, 0) Z(·, ε)
Figure 2.1: A schematic of the phase space M. Note that although the level set
Mc = {h = c} is depicted as a torus, it need not be a torus. It could be any
compact, co-dimension m submanifold.
on M in which the differential equation (2.1) takes the form
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε).
Then if ϕ ∈ S1 and the differential equation for the fast variable is regular,
i.e. Φ(h, ϕ, 0) is bounded away from zero for h ∈ U ,
initial conditions
s.t. hε(0)∈V
0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε → 0.
See for example Chapter 5 in [SV85], Chapter 3 in [LM88], or Theorem 2.2.3 in
the following section.
When the differential equation for the fast variable is not regular, or when
there is more than one fast variable, the typical averaging result becomes much
weaker than the uniform convergence above. For example, consider the case when
ϕ ∈ Tn, n > 1, and the unperturbed motion is quasi-periodic, i.e. Φ(h, ϕ, 0) =
Ω(h). Also assume that H ∈ Cn+2 and that Ω is nonvanishing and satisfies a
nondegeneracy condition on U (for example, Ω : U → Tn is a submersion). Let P
denote Riemannian volume on M. Neishtadt [LM88, Nei76] showed that in this
situation, for each fixed δ > 0,
0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
ε/δ),
and that this result is optimal. Thus, the averaged equation only describes the
actual motions of the slow variables in probability on the time scale 1/ε as ε→ 0.
Neishtadt’s result was motivated by a general averaging theorem for smooth
systems due to Anosov. This theorem requires none of the additional assumptions
in the averaging results above. Under the conditions of regularity, existence of
smooth integrals, compactness, and preservation of smooth measures, as well as
• Ergodicity: for Lebesgue almost every c ∈ U , (z0(·), µc) is ergodic,
Anosov showed that sup0≤τ≤1∧Tε
∣hε(τ/ε)− h̄(τ)
∣ → 0 in probability (w.r.t. Rie-
mannian volume on initial conditions) as ε→ 0, i.e.
Theorem 2.1.1 (Anosov’s averaging theorem [Ano60]). For each T > 0 and for
each fixed δ > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
as ε→ 0.
We present a recent proof of this theorem in Section 2.3 below.
If we consider hε(·) and h̄(·) to be random variables, Anosov’s theorem is a
version of the weak law of large numbers. In general, we can do no better: There
is no general strong law in this setting. There exists a simple example due to
Neishtadt (which comes from the equations for the motion of a pendulum with
linear drag being driven by a constant torque) where for no initial condition in a
positive measure set do we have convergence of hε(t) to h̄(εt) on the time scale 1/ε
as ε → 0 [Kif04b]. Here, the phase space is R× S1, and the unperturbed motion
is (uniquely) ergodic on all but one fiber.
2.2 Some classical averaging results
In this section we present some simple, well-known averaging results. See for
example Chapter 5 in [SV85] or Chapter 3 in [LM88].
2.2.1 Averaging for time-periodic vector fields
Consider a family of time dependent ordinary differential equations
= εH(h, t, ε), (2.3)
indexed by the real parameter ε ≥ 0, where h ∈ Rm. Fix V ⊂⊂ U ⊂ Rm, and
suppose
• Regularity: H ∈ C1(U × R× [0,∞)).
• Periodicity: There exists T > 0 such that for each h ∈ U , H(h, t, 0) is
T -periodic in time.
= εH(h, t, 0) +O(ε2).
Let hε(t) denote the solution of Equation (2.3). We seek a time independent
vector field whose solutions approximate hε(t), at least for a long length of time.
It is natural to define the averaged vector field H̄ by
H̄(h) =
H(h, s, 0)ds.
Then H̄ ∈ C1(U). Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
It is reasonable to hope that h̄(εt) and hε(t) are close together for 0 ≤ t ≤ ε−1.
We only consider the dynamics in a compact subset of phase space, so for initial
conditions in U , we define the stopping time
Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.1 (Time-periodic averaging). For each T > 0,
hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Proof. We divide our proof into three essential steps.
Step 1: Reduction using Gronwall’s Inequality. Now, h̄(τ) satisfies the
integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(hε(s), s, ε)ds
= O(ε) + ε
∫ τ/ε
H(hε(s), s, 0)ds
= O(ε) + ε
∫ τ/ε
H(hε(s), s, 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(hε(s), s, 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄|V)T .
Step 2: A sequence of times adapted for ergodization. Ergodization refers
to the convergence along an orbit of a function’s time average to its space average.
We define a sequence of times tk for k ≥ 0 by tk = kT . This sequence of times is
motivated by the fact that
tk+1 − tk
∫ tk+1
H(h0(s), s, 0)ds = H̄(h0).
Note that h0(t) is independent of time. Thus,
0≤τ≤T∧Tε
|eε(τ)| ≤ O(ε) + ε
tk+1≤T∧Tεε
∫ tk+1
H(hε(s), s, 0)− H̄(hε(s))ds
. (2.4)
Step 3: Control of individual terms by comparison with solutions of the
ε = 0 equation. The sum in Equation (2.4) has no more than O(1/ε) terms,
and so it suffices to show that each term
∫ tk+1
H(hε(s), s, 0) − H̄(hε(s))ds is no
larger than O(ε). We can accomplish this by comparing the motions of hε(t) for
tk ≤ t ≤ tk+1 with hk,ε(t), which is defined to be the solution of the ε = 0 ordinary
differential equation satisfying hk,ε(tk) = hε(tk), i.e. hk,ε(t) ≡ hε(tk).
Lemma 2.2.2. If tk+1 ≤ T∧Tεε , then suptk≤t≤tk+1 |hk,ε(t)− hε(t)| = O(ε).
Proof. dhε/dt = O(ε).
Using that H and H̄ are Lipschitz continuous, we conclude that
∫ tk+1
H(hε(s), s, 0)− H̄(hε(s))ds
∫ tk+1
H(hε(s), s, 0)−H(hk,ε(s), s, 0)ds
∫ tk+1
H(hk,ε(s), s, 0)− H̄(hk,ε(s))ds
∫ tk+1
H̄(hk,ε(s))− H̄(hε(s))ds
=O(ε) + 0 +O(ε)
=O(ε).
Thus we see that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≤ O(ε), independent of the
initial condition hε(0) ∈ V.
Remark 2.2.1. Note that the O(ε) control in Theorem 2.2.1 on a time scale t =
O(ε−1) is generally optimal. For example, take H(h, t, ε) = cos(t) + ε.
2.2.2 Averaging for vector fields with one regular fast vari-
For h ∈ Rm and ϕ ∈ S1 = [0, 1]/0 ∼ 1, consider the family of ordinary differential
equations
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε), (2.5)
indexed by the real parameter ε ≥ 0. With z = (h, ϕ), we write this family of
differential equations as dz/dt = Z(z, ε).
Fix V ⊂⊂ U ⊂ Rm, and suppose
• Regularity: Z ∈ C1(U × S1 × [0,∞)).
• Regular fast variable: Φ(h, ϕ, 0) is bounded away from 0 for h ∈ U , i.e.
(h,ϕ)∈U×S1
|Φ(h, ϕ, 0)| > 0.
Without loss of generality, we assume that Φ(h, ϕ, 0) > 0.
Let zε(t) = (hε(t), ϕε(t)) denote the solution of Equation (2.5). Then z0(t)
leaves invariant the circles Mc = {h = c} in phase space. In fact, z0(t) preserves
an uniquely ergodic invariant probability measure on Mc, whose density is given
dµc =
Φ(c, ϕ, 0)
where Kc =
Φ(c,ϕ,0)
is a normalization constant.
The averaged vector field H̄ is defined by averaging H(h, ϕ, 0) over ϕ:
H̄(h) =
H(h, ϕ, 0)dµh(ϕ) =
H(h, ϕ, 0)
Φ(h, ϕ, 0)
Then H̄ ∈ C1(U). Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
For initial conditions in U × S1, we have the usual stopping time Tε = inf{τ ≥ 0 :
h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.3 (Averaging over one regular fast variable). For each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Remark 2.2.2. This result encompasses Theorem 2.2.1 for time-periodic averaging.
For example, if T = 1, simply take ϕ = t mod 1 and Φ(h, ϕ, ε) = 1.
Remark 2.2.3. Many of the proofs of the above theorem of which we are aware
hinge on considering ϕ as a time-like variable. For example, one could write
H(h, ϕ, 0)
Φ(h, ϕ, 0)
+O(ε2),
and this looks very similar to the time-periodic situation considered previously.
However, it does take some work to justify such arguments rigorously, and the
traditional proofs do not easily generalize to averaging over multiple fast variables.
Our proof essentially uses ϕ to mark off time, and it will immediately generalize
to a specific instance of multiphase averaging.
Proof. Again, we have three steps.
Step 1: Reduction using Gronwall’s Inequality. Now
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(zε(s), ε)ds = O(ε) + ε
∫ τ/ε
H(zε(s), 0)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄|V)T .
Step 2: A sequence of times adapted for ergodization. Now for each
initial condition in our phase space and for each fixed ε, we define a sequence of
times tk,ε and a sequence of solutions zk,ε(t) inductively as follows: t0,ε = 0 and
z0,ε(t) = z0(t). For k > 0, tk,ε = inf{t > tk−1,ε : ϕk−1,ε(t) = ϕε(0)}, and zk,ε(t) is
defined as the solution of
dzk,ε
= Z(zk,ε, 0) = (0,Φ(zk,ε, 0)), zk,ε(tk,ε) = zε(tk,ε).
This sequence of times is motivated by the fact that
tk+1,ε − tk,ε
∫ tk+1,ε
tk ,ε
H(zk,ε(s), 0)ds = H̄(hk,ε).
Recall that hk,ε(t) is independent of time. The elements of this sequence of times
are approximately uniformly spaced, i.e. if we fix ω > 0 such that z ∈ V × S1 ⇒
1/ω < Φ(z, 0) < ω, then if tk+1,ε ≤ (T ∧ Tε)/ε, 1/ω < tk+1,ε − tk,ε < ω.
Thus,
0≤τ≤T∧Tε
|eε(τ)| ≤ O(ε) + ε
tk+1,ε≤T∧Tεε
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
where the sum in in this equation has no more than O(1/ε) terms.
Step 3: Control of individual terms by comparison with solutions along
fibers. It suffices to show that each term
∫ tk+1,ε
H(zε(s), 0) − H̄(hε(s))ds is no
larger than O(ε). We can accomplish this by comparing the motions of zε(t) for
tk,ε ≤ t ≤ tk+1,ε with zk,ε(t).
Lemma 2.2.4. If tk+1,ε ≤ T∧Tεε , then suptk,ε≤t≤tk+1,ε |zk,ε(t)− zε(t)| = O(ε).
Proof. Without loss of generality, we take k = 0, so that zk,ε(t) = z0(t). Since
h0(t) = hε(0) and dhε/dt = O(ε), supt0,ε≤t≤t1,ε |h0(t)− hε(t)| = O(ε).
Now ϕε(t) − ϕε(0) =
Φ(hε(s), ϕε(s), ε)ds, and because Φ is Lipschitz, we
find that
|ϕε(t)− ϕ0(t)| ≤ O(ε) + Lip (Φ)
|ϕε(s)− ϕ0(s)| ds
for 0 ≤ t ≤ ω. The result follows from Gronwall’s Inequality.
Using that H and H̄ are Lipschitz continuous, we conclude that
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
∫ tk+1,ε
H(zε(s), 0)−H(zk,ε(s), 0)ds
∫ tk+1,ε
H(zk,ε(s), 0)− H̄(hk,ε(s))ds
∫ tk+1,ε
H̄(hk,ε(s))− H̄(hε(s))ds
=O(ε) + 0 +O(ε)
=O(ε).
Thus we see that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε), independent of the
initial condition (hε(0), ϕε(0)) ∈ V × S1.
2.2.3 Multiphase averaging for vector fields with separa-
ble, regular fast variables
As explained in Section 2.1, when the differential equation for the fast variable is
not regular, or when there is more than one fast variable, the typical averaging
result becomes much weaker than the uniform convergence in Theorems 2.2.1
and 2.2.3 above. Nonetheless, if the differential equations under consideration
satisfy some very specific hypotheses, the proof in the previous section immediately
generalizes to yield uniform convergence.
For h ∈ Rm and ϕ = (ϕ1, · · · , ϕn) ∈ Tn = ([0, 1]/0 ∼ 1)n, consider the family
of ordinary differential equations
= εH(h, ϕ, ε),
= Φ(h, ϕ, ε), (2.6)
indexed by the real parameter ε ≥ 0. We also write z = (h, ϕ) and dz/dt = Z(z, ε).
Fix V ⊂⊂ U ⊂ Rm, and suppose
• Regularity: Z ∈ C1(U × Tn × [0,∞)).
• Separable fast variables: H(h, ϕ, 0) and Φ(h, ϕ, 0) have the following specific
forms:
– There exist C1 functionsHj(h, ϕj) such thatH(h, ϕ, 0) =
j=1Hj(h, ϕ
This can be thought of as saying that, to first order in ε, each fast vari-
able affects the slow variables independently of the other fast variables.
– The components Φj of Φ satisfy Φj(h, ϕ, 0) = Φj(h, ϕj, 0), i.e. the un-
perturbed motion has each fast variable moving independently of the
other fast variables. Note that this assumption is satisfied if the unper-
turbed motion is quasi-periodic, i.e. Φ(h, ϕ, 0) = Ω(h).
• Regular fast variables: For each j,
(h,ϕj)∈U×S1
∣Φj(h, ϕj, 0)
∣ > 0.
Let zε(t) = (hε(t), ϕε(t)) denote the solution of Equation (2.6). Then z0(t)
leaves invariant the tori Mc = {h = c} in phase space. In fact, z0(t) preserves a
(not necessarily ergodic) invariant probability measure on Mc, whose density is
given by
dµc =
|Φj(c, ϕj, 0)| ,
where Kjc =
|Φj(c,ϕj ,0)| .
The averaged vector field H̄ is defined by
H̄(h) =
H(h, ϕ, 0)dµh(ϕ) =
Hj(h, ϕ
j)dµh(ϕ)
Hj(h, ϕ
|Φj(h, ϕj, 0)|dϕ
H̄j(h).
Let h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0),
and the stopping time Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 2.2.5 (Averaging over multiple separable, regular fast variables). For
each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε→ 0.
Proof. The proof is essentially the same as the proof of Theorem 2.2.3. As before,
we need only show that sup0≤τ≤T∧Tε |eε(τ)| = O(ε), where
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
But by our separability assumptions, it suffices to show that for each j,
0≤τ≤T∧Tε
|ej,ε(τ)| = O(ε),
where ej,ε(τ) is defined by
ej,ε(τ) = ε
∫ τ/ε
Hj(hε(s), ϕ
ε(s))− H̄j(hε(s))ds.
Thus, we have effectively separated the effects of each fast variable, and now the
proof can be completed by essentially following steps 2 and 3 in the proof of
Theorem 2.2.3.
2.3 A proof of Anosov’s theorem
Anosov’s original proof of Theorem 2.1.1 from 1960 may be found in [Ano60]. An
exposition of the theorem and Anosov’s proof in English may be found in [LM88].
Recently, Kifer [Kif04a] proved necessary and sufficient conditions for the averaging
principle to hold in an averaged with respect to initial conditions sense. He also
showed explicitly that his conditions are met in the setting of Anosov’s theorem.
The proof of Anosov’s theorem given here is mainly due to Dolgopyat [Dol05],
although some further simplifications have been made.
Proof of Anosov’s theorem. We begin by showing that without loss of generality
we may take Tε = ∞. This is just for convenience, and not an essential part of
the proof. To accomplish this, let ψ(h) be a smooth bump function satisfying
• ψ(h) = 1 if h ∈ V,
• ψ(h) > 0 if h ∈ interior(Ṽ),
• ψ(h) = 0 if h /∈ Ṽ,
where Ṽ is a compact set chosen such that V ⊂⊂ interior(Ṽ) ⊂⊂ U . Next,
set Z̃(z, ε) = ψ(h(z))Z(z, ε). Because the bump function was chosen to depend
only on the slow variables, our assumption about preservation of measures is still
satisfied; on each fiber, Z̃(z, 0) is a scaler multiple of Z(z, 0). Furthermore, the
flow of Z̃(·, 0)|Mh is ergodic for almost every h ∈ Ṽ. Then it would suffice to prove
our theorem for the vector fields Z̃(z, ε) with the set Ṽ replacing V. We assume
that this reduction has been made, although we will not use it until Step 5 below.
Step 1: Reduction using Gronwall’s Inequality. Observe that h̄(τ) satisfies
the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = ε
∫ τ/ε
H(zε(s), ε)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)ds
= O(ε) + ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε. Here we have used the fact that h−1V × [0, ε0] is compact to
achieve uniformity over all initial conditions in the size of the O(ε) term above. We
use this fact repeatedly in what follows. In particular, H , H̄, and Z are uniformly
bounded and have uniform Lipschitz constants on the domains of interest.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s), 0)− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄ |V)T . (2.7)
Step 2: Introduction of a time scale for ergodization. Choose a real-valued
function L(ε) such that L(ε) → ∞, L(ε) = o(log ε−1) as ε → 0. Think of L(ε) as
being a time scale which grows as ε→ 0 so that ergodization, i.e. the convergence
along an orbit of a function’s time average to a space average, can take place.
However, L(ε) doesn’t grow too fast, so that on this time scale zε(t) essentially
stays on one fiber, where we have our ergodicity assumption. Set tk,ε = kL(ε), so
0≤τ≤T∧Tε
|eε(τ)| ≤ O(εL(ε)) + ε
εL(ε)
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
. (2.8)
Step 3: A splitting for using the triangle inequality. Now we let zk,ε(s)
be the solution of
dzk,ε
= Z(zk,ε, 0), zk,ε(tk,ε) = zε(tk,ε).
Set hk,ε(t) = h(zk,ε(t)). Observe that hk,ε(t) is independent of t. We break up the
integral
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds into three parts:
∫ tk+1,ε
H(zε(s), 0)− H̄(hε(s))ds
∫ tk+1,ε
H(zε(s), 0)−H(zk,ε(s), 0)ds
∫ tk+1,ε
H(zk,ε(s), 0)− H̄(hk,ε(s))ds
∫ tk+1,ε
H̄(hk,ε(s))ds− H̄(hε(s))ds
:=Ik,ε + IIk,ε + IIIk,ε.
The term IIk,ε represents an “ergodicity term” that can be controlled by our
assumptions on the ergodicity of the flow z0(t), while the terms Ik,ε and IIIk,ε
represent “continuity terms” that can be controlled using the following control on
the drift from solutions along fibers.
Step 4: Control of drift from solutions along fibers.
Lemma 2.3.1. If 0 < tk+1,ε ≤ T∧Tεε ,
tk,ε≤t≤tk+1,ε
|zk,ε(t)− zε(t)| ≤ O(εL(ε)eLip(Z)L(ε))
Proof. Without loss of generality we may set k = 0, so that zk,ε(t) = z0(t). Then
for 0 ≤ t ≤ L(ε),
|z0(t)− zε(t)| =
Z(z0(s), 0)− Z(zε(s), ε)ds
≤ Lip (Z)
|ε|+ |z0(s)− zε(s)| ds
= O(εL(ε)) + Lip (Z)
|z0(s)− zε(s)| ds.
The result follows from Gronwall’s Inequality.
From Lemma 2.3.1 we find that Ik,ε, IIIk,ε = O(εL(ε)2eLip(Z)L(ε)).
Step 5: Use of ergodicity along fibers to control IIk,ε. From Equations
(2.7) and (2.8) and the triangle inequality, we already know that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
≤ O(ε) +O(εL(ε)) + ε T
εL(ε)
O(εL(ε)2eLip(Z)L(ε)) +O
εL(ε)
|IIk,ε|
= O(εL(ε)eLip(Z)L(ε)) +O
εL(ε)
|IIk,ε|
(2.9)
Fix δ > 0. Recalling that Tε = ∞, it suffices to show that
εL(ε)
|IIk,ε| ≥ δ
as ε→ 0.
For initial conditions z ∈ M and for 0 ≤ k ≤ T
εL(ε)
define
Bk,ε =
|IIk,ε| >
Bz,ε = {k : z ∈ Bk,ε} .
Think of these sets as describing “bad ergodization.” For example, roughly speak-
ing, z ∈ Bk,ε if the orbit zε(t) starting at z spends the time between tk,ε and tk+1,ε
in a region of phase space where the function H(·, 0) is “poorly ergodized” on the
time scale L(ε) by the flow z0(t) (as measured by the parameter δ/2T ). As IIk,ε
is clearly never larger than O(L(ε)), it follows that
εL(ε)
|IIk,ε| ≤
+O(εL(ε)#(Bz,ε)).
Therefore it suffices to show that
#(Bz,ε) ≥
const εL(ε)
as ε→ 0. By Chebyshev’s Inequality, we need only show that
E(εL(ε)#(Bz,ε)) = εL(ε)
εL(ε)
P (Bk,ε)
tends to 0 with ε.
In order to estimate the size of P (Bk,ε), it is convenient to introduce a new
measure P f that is uniformly equivalent to the restriction of Riemannian volume
P to h−1V. Here the f stands for “factor,” and P f is defined by
dP f = dh · dµh,
where dh represents integration with respect to the uniform measure on V.
Observe that B0,ε = zε(tk,ε)Bk,ε. In words, the initial conditions giving rise
to orbits that are “bad” on the time interval [tk,ε, tk+1,ε], moved forward by time
tk,ε, are precisely the initial conditions giving rise to orbits that are “bad” on the
time interval [t0,ε, t1,ε]. Because the flow z0(·) preserves the measure P f , we expect
P f(B0,ε) and P f(Bk,ε) to have roughly the same size. This is made precise by the
following lemma.
Lemma 2.3.2. There exists a constant K such that for each Borel set B ⊂ M
and each t ∈ [−T/ε, T/ε], P f(zε(t)B) ≤ eKTP f(B).
Proof. Assume that P f(B) > 0, and set γ(t) = ln
P f(zε(t)B)/P
. Then
γ(0) = 0, and
(t) =
zε(t)B
f̃(z)dz
zε(t)B
f̃(z)dz
zε(t)B
divP fZ(z, ε)dz
zε(t)B
f̃(z)dz
where f̃ > 0 is the C1 density of P f with respect to Riemannian volume on
h−1V, dz represents integration with respect to that volume, and divP fZ(z, ε) =
divzf̃(z)Z(z, ε). Because z0(t) preserves P
f , divP fZ(z, 0) ≡ 0. By Hadamard’s
Lemma, it follows that divP f Z(z, ε) = O(ε) on the compact set h−1V. Hence
dγ(t)/dt = O(ε), and the result follows.
Returning to our proof of Anosov’s theorem, it suffices to show that
P f(B0,ε) =
dh · µh
∫ L(ε)
H(z0(s), 0)− H̄(h0(0))ds
tends to 0 with ε. By our ergodicity assumption, for almost every h,
∫ L(ε)
H(z0(s), 0)− H̄(h0(0))ds
→ 0 as ε→ 0.
Finally, an application of the Bounded Convergence Theorem finishes the proof.
2.4 Moral
From the proofs of the theorems in this chapter, it should be apparent that there
are at least two key steps necessary for proving a version of the averaging principle
in the setting presented in Section 2.1.
The first step is estimating the continuity between the ε = 0 and the ε > 0
solutions of
= Z(z, ε).
In particular, on some relatively long timescale L = L(ε) ≪ ε−1, we need to show
0≤t≤L
|z0(t)− zε(t)| → 0
as ε → 0. As long as L is sub-logarithmic in ε−1, such estimates for smooth
systems can be made using Gronwall’s Inequality.
The second step is estimating the rate of ergodization of H(·, 0) by z0(t), i.e. es-
timating how fast
H(z0(s), 0) ds→ H̄(h0)
(generally as L → ∞). Note that the estimates in this step compete with those
in the first step in that, if L is small we obtain better continuity, but if L is large
we usually obtain better ergodization. Also, we do not need the full force of the
assumption of ergodicity of (z0(t), µh) on the fibers Mh. We only need z0(t) to
ergodize the specific function H(·, 0). Compare the proof of Theorem 2.2.5.
Note that in the setting of Anosov’s theorem, uniform ergodization leads to
uniform convergence in the averaging principle. Returning to the proof of Theo-
rem 2.1.1 above, suppose that
∫ L(ε)
H(z0(s), 0)ds→ H̄(h0)
uniformly over all initial conditions as L(ε) → ∞. Then for all ε sufficiently small
and each k, Bk,ε = ∅, and hence for all ε sufficiently small and each z, #(Bz,ε) = 0.
From Equation (2.9), it follows that sup0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
∣ → 0 as ε → 0,
uniformly over all initial conditions z ∈ h−1V. However, uniform convergence in
Birkhoff’s Ergodic Theorem is extremely rare and usually comes about because of
unique ergodicity, so it is unreasonable to expect this sort of uniform convergence
in most situations where Anosov’s theorem applies.
Chapter 3
Results for piston systems in one
dimension
In this chapter, we present our results for piston systems in one dimension. These
results may also be found in [Wri06].
3.1 Statement of results
3.1.1 The hard core piston problem
Consider the system of n1 + n2 + 1 point particles moving inside the unit interval
indicated in Figure 3.1. One distinguished particle, the piston, has position Q and
mass M . To the left of the piston there are n1 > 0 particles with positions q1,j
and masses m1,j, 1 ≤ j ≤ n1, and to the right there are n2 > 0 particles with
positions q2,j and masses m2,j, 1 ≤ j ≤ n2. These gas particles do not interact
with each other, but they interact with the piston and with walls located at the
end points of the unit interval via elastic collisions. We denote the velocities by
dQ/dt = V and dxi,j/dt = vi,j. There is a standard method for transforming
this system into a billiard system consisting of a point particle moving inside an
(n1 + n2 + 1)-dimensional polytope [CM06a], but we will not use this in what
follows.
We are interested in the dynamics of this system when the numbers and masses
of the gas particles are fixed, the total energy is bounded, and the mass of the
piston tends to infinity. When M = ∞, the piston remains at rest, and each
gas particle performs periodic motion. More interesting are the motions of the
system when M is very large but finite. Because the total energy of the system is
bounded, MV 2/2 ≤ const, and so V = O(M−1/2). Set
ε =M−1/2,
t t t t t t t
Figure 3.1: The piston system with n1 = 3 and n2 = 4. Note that the gas particles
do not interact with each other, but only with the piston and the walls.
and let
so that
with W = O(1).
When ε = 0, the system has n1 + n2 + 2 independent first integrals (conserved
quantities), which we take to be Q, W , and si,j = |vi,j |, the speeds of the gas
particles. We refer to these variables as the slow variables because they should
change slowly with time when ε is small, and we denote them by
h = (Q,W, s1,1, s1,2, · · · , s1,n1, s2,1, s2,2, · · · , s2,n2) ∈ Rn1+n2+2.
We will often abbreviate by writing h = (Q,W, s1,j, s2,j). Let hε(t, z) = hε(t)
denote the dynamics of these variables in time for a fixed value of ε, where z
represents the dependence on the initial condition in phase space. We usually
suppress the initial condition in our notation. Think of hε(·) as a random variable
which, given an initial condition in the 2(n1 + n2 + 1)-dimensional phase space,
produces a piecewise continuous path in Rn1+n2+2. These paths are the projection
of the actual motions in our phase space onto a lower dimensional space. The goal
of averaging is to find a vector field on Rn1+n2+2 whose orbits approximate hε(t).
The averaged equation
Sinai [Sin99] derived
= H̄(h) :=
j=1 m1,js
j=1 m2,js
−s1,jW
s2,jW
(3.1)
as the averaged equation (with respect to the slow time τ = εt) for the slow
variables. We provide a heuristic derivation in Section 3.2. Sinai solved this
equation as follows: From
d ln(s1,j)
= −d ln(Q)
s1,j(τ) = s1,j(0)Q(0)/Q(τ). Similarly, s2,j(τ) = s2,j(0)(1−Q(0))/(1−Q(τ)). Hence
j=1m1,js1,j(0)
2Q(0)2
j=1m2,js2,j(0)
2(1−Q(0))2
(1−Q)3 ,
and so (Q,W ) behave as if they were the coordinates of a Hamiltonian system
describing a particle undergoing periodic motion inside a potential well. If we let
s2i,j
be the kinetic energy of the gas particles on one side of the piston, the effective
Hamiltonian may be expressed as
W 2 +
E1(0)Q(0)
E2(0)(1−Q(0))2
(1−Q)2 . (3.2)
Hence, the solutions to the averaged equation are periodic for all initial conditions
under consideration.
Main result in the hard core setting
The solutions of the averaged equation approximate the motions of the slow vari-
ables, hε(t), on a time scale O(1/ε) as ε → 0. Precisely, let h̄(τ, z) = h̄(τ) be the
solution of
= H̄(h̄), h̄(0) = hε(0).
Again, think of h̄(·) as being a random variable that takes an initial condition in
our phase space and produces a path in Rn1+n2+2.
Next, fix a compact set V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂
R, and si,j ⊂⊂ (0,∞) for each i and j.1 For the remainder of this discussion we
will restrict our attention to the dynamics of the system while the slow variables
remain in the set V. To this end, we define the stopping time
Tε(z) = Tε := inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 3.1.1. For each T > 0,
initial conditions
s.t. hε(0)∈V
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε) as ε =M−1/2 → 0.
This result was independently obtained by Gorelyshev and Neishtadt [GN06].
Note that the stopping time does not unduly restrict the result. Given any
c such that h = c ⇒ Q ∈ (0, 1), si,j ∈ (0,∞), then by an appropriate choice of
the compact set V we may ensure that, for all ε sufficiently small and all initial
conditions in our phase space with hε(0) = c, Tε ≥ T . We do this by choosing
V ∋ c such that the distance between ∂V and the periodic orbit h̄(τ) with h̄(0) = c
is positive. Call this distance d. Then Tε can only occur before T if hε(τ/ε) has
deviated by at least d from h̄(τ) for some τ ∈ [0, T ). Since the size of the deviations
tends to zero uniformly with ε, this is impossible for all small ε.
3.1.2 The soft core piston problem
In this section, we consider the same system of one piston and gas particles inside
the unit interval considered in Section 3.1.1, but now the interactions of the gas
particles with the walls and with the piston are smooth. Let κ : R → R be a C2
function satisfying
• κ(x) = 0 if x ≥ 1,
• κ′(x) < 0 if x < 1.
Let δ > 0 be a parameter of smoothing, and set
κδ(x) = κ(x/δ).
1 We have introduced this notation for convenience. For example, h ∈ V ⇒ Q ⊂⊂ (0, 1)
means that there exists a compact set A ⊂ (0, 1) such that h ∈ V ⇒ Q ∈ A, and similarly for
the other variables.
Then consider the Hamiltonian system obtained by having the gas particles inter-
act with the piston and the walls via the potential
κδ(q1,j) + κδ(Q− q1,j) +
κδ(q2,j −Q) + κδ(1− q2,j).
As before, we set ε =M−1/2 and W = V/ε. If we let
E1,j =
m1,jv
1,j + κδ(q1,j) + κδ(Q− q1,j), 1 ≤ j ≤ n1,
E2,j =
m2,jv
2,j + κδ(q2,j −Q) + κδ(1− q2,j), 1 ≤ j ≤ n2,
(3.3)
then Ei,j may be thought of as the energy associated with a gas particle, and
W 2/2 +
j=1E1,j +
j=1E2,j is the conserved energy.
When ε = 0, the Hamiltonian system admits n1 + n2 + 2 independent first
integrals, which we choose this time as h = (Q,W,E1,j , E2,j). While discussing
the soft core dynamics we use the energies Ei,j rather than the variables si,j =
2Ei,j/mi,j, which we used for the hard core dynamics, for convenience.
For comparison with the hard core results, we formally consider the dynamics
described by setting δ = 0 to be the hard core dynamics described in Section 3.1.1.
This is reasonable because we will only consider gas particle energies below the
barrier height κ(0). Then for any ε, δ ≥ 0, hδε(t) denotes the actual time evolution
of the slow variables. While discussing the soft core dynamics we often use δ as a
superscript to specify the dynamics for a certain value of δ. We usually suppress
the dependence on δ, unless it is needed for clarity.
Main result in the soft core setting
We have already seen that when δ = 0, there is an appropriate averaged vector
field H̄0 whose solutions approximate the actual motions of the slow variables,
h0ε(t). We will show that when δ > 0, there is also an appropriate averaged vector
field H̄δ whose solutions still approximate the actual motions of the slow variables,
hδε(t). We delay the derivation of H̄
δ until Section 3.4.1.
Fix a compact set V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂ R,
and Ei,j ⊂⊂ (0, κ(0)) for each i and j. For each ε, δ ≥ 0 we define the functions
h̄δ(·) and T δε on our phase space by letting h̄δ(τ) be the solution of
= H̄δ(h̄δ), h̄δ(0) = hδε(0), (3.4)
T δε = inf{τ ≥ 0 : h̄δ(τ) /∈ V or hδε(τ/ε) /∈ V}.
Theorem 3.1.2. There exists δ0 > 0 such that the averaged vector field H̄
δ(h) is
C1 on the domain {(δ, h) : 0 ≤ δ ≤ δ0, h ∈ V}. Furthermore, for each T > 0,
0≤δ≤δ0
initial conditions
s.t. hδε(0)∈V
0≤τ≤T∧T δε
∣hδε(τ/ε)− h̄δ(τ)
∣ = O(ε) as ε =M−1/2 → 0.
As in Section 3.1.1, for any fixed c there exists a suitable choice of the compact
set V such that for all sufficiently small ε and δ, T δε ≥ T whenever hδε(0) = c.
As we will see, for each fixed δ > 0, Anosov’s theorem 2.1.1 applies to the soft
core system and yields a weak law of large numbers, and Theorem 2.2.5 applies and
yields a strong law of large numbers with a uniform rate of convergence. However,
neither of these theorems yields the uniformity over δ in the result above.
3.1.3 Applications and generalizations
Relationship between the hard core and the soft core piston
It is not a priori clear that we can compare the motions of the slow variables on
the time scale 1/ε for δ > 0 versus δ = 0, i.e. compare the motions of the soft core
piston with the motions of the hard core piston on a relatively long time scale. It
is impossible to compare the motions of the fast-moving gas particles on this time
scale as ε → 0. As we see in Section 3.4, the frequency with which a gas particle
hits the piston changes by an amount O(δ) when we smooth the interaction. Thus,
on the time scale 1/ε, the number of collisions is altered by roughly O(δ/ε), and
this number diverges if δ is held fixed while ε → 0.
Similarly, one might expect that it is impossible to compare the motions of the
soft and hard core pistons as ε → 0 without letting δ → 0 with ε. However, from
Gronwall’s Inequality it follows that if h̄δ(0) = h̄0(0), then
0≤τ≤T∧T δε ∧T 0ε
∣h̄δ(τ)− h̄0(τ)
∣ = O(δ).
From the triangle inequality and Theorems 3.1.1 and 3.1.2 we obtain the following
corollary, which allows us to compare the motions of the hard core and the soft
core piston.
Corollary 3.1.3. As ε =M−1/2, δ → 0,
initial conditions
s.t. hδε(0)=c=h
0≤t≤(T∧T δε ∧T 0ε )/ε
∣hδε(t)− h0ε(t)
∣ = O(ε) +O(δ).
This shows that, provided the slow variables have the same initial conditions,
0≤t≤1/ε
∣hδε(t)− h0ε(t)
∣ = O(ε) +O(δ).
Thus the motions of the slow variables converge on the time scale 1/ε as ε, δ → 0,
and it is immaterial in which order we let these parameters tend to zero.
The adiabatic piston problem
We comment on what Theorem 3.1.1 says about the adiabatic piston problem. The
initial conditions of the adiabatic piston problem require thatW (0) = 0. Although
our system is so simple that a proper thermodynamical pressure is not defined,
we can define the pressure of a gas to be the average force received from the gas
particles by the piston when it is held fixed, i.e. P1 =
j=1 2m1,js1,j
= 2E1/Q
and P2 = 2E2/(1 − Q). Then if P1(0) > P2(0), the initial condition for our
averaged equation (3.1) has the motion of the piston starting at the left turning
point of a periodic orbit determined by the effective potential well. Up to errors not
much bigger than M−1/2, we see the piston oscillate periodically on the time scale
M1/2. If P1(0) < P2(0), the motion of the piston starts at a right turning point.
However, if P1(0) = P2(0), then the motion of the piston starts at the bottom of
the effective potential well. In this case of mechanical equilibrium, h̄(τ) = h̄(0),
and we conclude that, up to errors not much bigger thanM−1/2, we see no motion
of the piston on the time scale M1/2. A much longer time scale is required to see
if the temperatures equilibrate.
Generalizations
A simple generalization of Theorem 3.1.1, proved by similar techniques, follows.
The system consists of N − 1 pistons, that is, heavy point particles, located inside
the unit interval at positions Q1 < Q2 < . . . < QN−1. Walls are located at Q0 ≡ 0
and QN ≡ 1, and the piston at position Qi has mass Mi. Then the pistons divide
the unit interval into N chambers. Inside the ith chamber, there are ni ≥ 1 gas
particles whose locations and masses will be denoted by xi,j and mi,j , respectively,
where 1 ≤ j ≤ ni. All of the particles are point particles, and the gas particles
interact with the pistons and with the walls via elastic collisions. However, the
gas particles do not directly interact with each other. We scale the piston masses
as Mi = M̂i/ε
2 with M̂i constant, define Wi by dQi/dt = εWi, and let Ei be
the kinetic energy of the gas particles in the ith chamber. Then we can find an
appropriate averaged equation whose solutions have the pistons moving like an
(N − 1)-dimensional particle inside a potential well with an effective Hamiltonian
Ei(0)(Qi(0)−Qi−1(0))2
(Qi −Qi−1)2
If we write the slow variables as h = (Qi,Wi, |vi,j|) and fix a compact set V such
that h ∈ V ⇒ Qi+1 − Qi ⊂⊂ (0, 1),Wi ⊂⊂ R, and |vi,j| ⊂⊂ (0,∞), then the
convergence of the actual motions of the slow variables to the averaged solutions
is exactly the same as the convergence given in Theorem 3.1.1.
Remark 3.1.1. The inverse quadratic potential between adjacent pistons in the
effective Hamiltonian above is also referred to as the Calogero-Moser-Sutherland
potential. It has also been observed as the effective potential created between two
adjacent tagged particles in a one-dimensional Rayleigh gas by the insertion of one
very light particle inbetween the tagged particles [BTT07].
3.2 Heuristic derivation of the averaged equa-
tion for the hard core piston
We present here a heuristic derivation of Sinai’s averaged equation (3.1) that is
found in [Dol05].
First, we examine interparticle collisions when ε > 0. When a particle on the
left, say the one at position q1,j , collides with the piston, s1,j andW instantaneously
change according to the laws of elastic collisions:
v+1,j
m1,j +M
m1,j −M 2M
2m1,j M −m1,j
v−1,j
. (3.5)
If the speed of the left gas particle is bounded away from zero, and W = M1/2V
is also bounded, it follows that for all ε sufficiently small, any collision will have
v−1,j > 0 and v
1,j < 0. In this case, when we translate Equation (3.5) into our new
coordinates, we find that
s+1,j
1 + ε2m1,j
1− ε2m1,j −2ε
2εm1,j 1− ε2m1,j
s−1,j
, (3.6)
so that
∆s1,j = s
1,j − s−1,j = −2εW− +O(ε2),
∆W =W+ −W− = +2εm1,js−1,j +O(ε2).
The situation is analogous when particles on the right collide with the piston.
For all ε sufficiently small, s2,j and W instantaneously change by
∆W = W+ −W− = −2εm2,js−2,j +O(ε2),
∆s2,j = s
2,j − s−2,j = +2εW− +O(ε2).
We defer discussing the rare events in which multiple gas particles collide with the
piston simultaneously, although we will see that they can be handled appropriately.
Let ∆t be a length of time long enough such that the piston experiences many
collisions with the gas particles, but short enough such that the slow variables
change very little, in this time interval. From each collision with the particle at
position q1,j , W changes by an amount +2εm1,js1,j + O(ε2), and the frequency
of these collisions is approximately
. Arguing similarly for collisions with the
other particles, we guess that
2m1,js1,j
2m2,js2,j
2(1−Q) +O(ε
Note that not only does the position of the piston change slowly in time, but its
velocity also changes slowly, i.e. the piston has inertia. With τ = εt as the slow
time, a reasonable guess for the averaged equation for W is
m1,js
m2,js
1−Q .
Similar arguments for the other slow variables lead to the averaged equation (3.1).
3.3 Proof of the main result for the hard core
piston
3.3.1 Proof of Theorem 3.1.1 with only one gas particle on
each side
We specialize to the case when there is only one gas particle on either side of
the piston, i.e. we assume that n1 = n2 = 1. We then denote x1,1 by q1, m2,2
by m2, etc. This allows the proof’s major ideas to be clearly expressed, without
substantially limiting their applicability. At the end of this section, we outline the
simple generalizations needed to make the proof apply in the general case.
A choice of coordinates on the phase space for a three particle system
As part of our proof, we choose a set of coordinates on our six-dimensional phase
space such that, in these coordinates, the ε = 0 dynamics are smooth. Complete
the slow variables h = (Q,W, s1, s2) to a full set of coordinates by adding the
coordinates ϕi ∈ [0, 1]/ 0 ∼ 1 = S1, i = 1, 2, defined as follows:
ϕ1 = ϕ1(q1, v1, Q) =
if v1 > 0
1− q1
if v1 < 0
ϕ2 = ϕ2(q2, v2, Q) =
2(1−Q) if v2 < 0
1− 1−q2
2(1−Q) if v2 > 0
When ε = 0, these coordinates are simply the angle variable portion of action-
angle coordinates for an integrable Hamiltonian system. They are defined such
that collisions occur between the piston and the gas particles precisely when ϕ1
or ϕ2 = 1/2. Then z = (h, ϕ1, ϕ2) represents a choice of coordinates on our phase
space, which is homeomorphic to (a subset of R4) × T2. We abuse notation and
also let h(z) represent the projection onto the first four coordinates of z.
Now we describe the dynamics of our system in these coordinates. When
ϕ1, ϕ2 6= 1/2,
ϕ1 if 0 ≤ ϕ1 < 1/2
(1− ϕ1) if 1/2 < ϕ1 ≤ 1
2(1−Q) +
1−Qϕ2 if 0 ≤ ϕ2 < 1/2
2(1−Q) −
1−Q(1− ϕ2) if 1/2 < ϕ2 ≤ 1
Hence between interparticle collisions, the dynamics are smooth and are described
= εW,
+O(ε),
2(1−Q) +O(ε).
(3.7)
When ϕ1 reaches 1/2, while ϕ2 6= 1/2, the coordinates Q, s2, ϕ1, and ϕ2 are
instantaneously unchanged, while s1 andW instantaneously jump, as described by
Equation (3.6). As an aside, it is curious that s+1 +εW
+ = s−1 −εW−, so that dϕ1/dt
is continuous as ϕ1 crosses 1/2. However, the collision induces discontinuous jumps
of size O(ε2) in dQ/dt and dϕ2/dt. Denote the linear transformation in Equation
(3.6) with j = 1 by A1,ε. Then
A1,ε =
1 −2ε
2εm1 1
+O(ε2).
The situation is analogous when ϕ2 reaches 1/2, while ϕ1 6= 1/2. Then W and
s2 are instantaneously transformed by a linear transformation
A2,ε =
1 −2εm2
+O(ε2).
We also account for the possibility of all three particles colliding simultane-
ously. There is no completely satisfactory way to do this, as the dynamics have an
essential singularity near {ϕ1 = ϕ2 = 1/2}. Furthermore, such three particle colli-
sions occur with probability zero with respect to the invariant measure discussed
below. However, the two 3× 3 matrices
A1,ε 0
0 A2,ε
have a commutator of size O(ε2). We will see that this small of an error will make
no difference to us as ε → 0, and so when ϕ1 = ϕ2 = 1/2, we pretend that the
left particle collides with the piston instantaneously before the right particle does.
Precisely, we transform the variables s1, W, and s2 by
0 A2,ε
A1,ε 0
We find that
∆s1 = s
1 − s−1 = −2εW− +O(ε2),
∆W = W+ −W− = +2εm1s−1 − 2εm2s−2 +O(ε2),
∆s2 = s
2 − s−2 = +2εW− +O(ε2).
The above rules define a flow on the phase space, which we denote by zε(t).
We denote its components by Qε(t), Wε(t), s1,ε(t), etc. When ε > 0, the flow is
not continuous, and for definiteness we take zε(t) to be left continuous in t.
Because our system comes from a Hamiltonian system, it preserves Liouville
measure. In our coordinates, this measure has a density proportional to Q(1 −
Q). That this measure is preserved also follows from the fact that the ordinary
differential equation (3.7) preserves this measure, and the matrices A1,ε, A2,ε have
determinant 1. Also note that the set {ϕ1 = ϕ2 = 1/2} has co-dimension two,
and so
t zε(t){ϕ1 = ϕ2 = 1/2} has co-dimension one, which shows that only a
measure zero set of initial conditions will give rise to three particle collisions.
Argument for uniform convergence
Step 1: Reduction using Gronwall’s Inequality. Define H(z) by
H(z) =
2m1s1δϕ1=1/2 − 2m2s2δϕ2=1/2
−2Wδϕ1=1/2
2Wδϕ2=1/2
Here we make use of Dirac delta functions. All integrals involving these delta
functions may be replaced by sums. We explicitly deal with any ambiguities arising
from collisions occurring at the limits of integration.
Lemma 3.3.1. For 0 ≤ t ≤ T∧Tε
hε(t)− hε(0) = ε
H(zε(s))ds+O(ε),
where any ambiguity about changes due to collisions occurring precisely at times 0
and t is absorbed in the O(ε) term.
Proof. There are four components to verify. The first component requires that
Qε(t)−Qε(0) = ε
Wε(s)ds+O(ε). This is trivially true because Qε(t)−Qε(0) =
Wε(s)ds.
The second component states that
Wε(t)−Wε(0) = ε
2m1s1,ε(s)δϕ1,ε(s)=1/2−2m2s2,ε(s)δϕ2,ε(s)=1/2ds+O(ε). (3.8)
Let rk and qj be the times in (0, t) such that ϕ1,ε(rk) = 1/2 and ϕ2,ε(qj) = 1/2,
respectively. Then
Wε(t)−Wε(0) =
∆Wε(rk) +
∆Wε(qj) +O(ε).
Observe that there exists ω > 0 such that for all sufficiently small ε and all h ∈ V,
1/ω < dϕi
< ω. Thus the number of collisions in a time interval grows no faster
than linearly in the length of that time interval. Because t ≤ T/ε, it follows that
Wε(t)−Wε(0) = ε
2m1s1,ε(rk)− ε
2m2s2,ε(qj) +O(ε),
and Equation (3.8) is verified. Note that because V is compact, there is uniformity
over all initial conditions in the size of the O(ε) terms above. The third and fourth
components are handled similarly.
Next, h̄(τ) satisfies the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while hε(τ/ε) satisfies
hε(τ/ε)− hε(0) = O(ε) + ε
∫ τ/ε
H(zε(s))ds
= O(ε) + ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T ∧ Tε.
Define
eε(τ) = ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T∧Tε
∣h̄(τ)− hε(τ/ε)
O(ε) + sup
0≤τ≤T∧Tε
|eε(τ)|
eLip(H̄ |V)T . (3.9)
Gronwall’s Inequality is usually stated for continuous paths, but the standard
proof (found in [SV85]) still works for paths that are merely integrable, and
∣h̄(τ)− hε(τ/ε)
∣ is piecewise smooth.
Step 2: A splitting according to particles. Now
H(z)− H̄(h) =
2m1s1δϕ1=1/2 −m1s21/Q
−2Wδϕ1=1/2 + s1W/Q
−2m2s2δϕ2=1/2 +m2s22/(1−Q)
2Wδϕ2=1/2 − s2W/(1−Q)
and so, in order to show that sup0≤τ≤T∧Tε |eε(τ)| = O(ε), it suffices to show that
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
= O(1),
0≤τ≤T∧Tε
∫ τ/ε
Wε(s)δϕ1,ε(s)=1/2 −
Wε(s)s1,ε(s)
2Qε(s)
= O(1),
as well as two analogous claims about terms involving ϕ2,ε. Thus we have effec-
tively separated the effects of the different gas particles, so that we can deal with
each particle separately. We will only show that
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
= O(1).
The other three terms can be handled similarly.
Step 3: A sequence of times adapted for ergodization. Ergodization refers
to the convergence along an orbit of a function’s time average to its space average.
For example, because of the splitting according to particles above, one can easily
check that 1
H(z0(s))ds = H̄(h0) + O(1/t), even when z0(·) restricted to the
invariant tori Mh0 is not ergodic. In this step, for each initial condition zε(0) in
our phase space, we define a sequence of times tk,ε inductively as follows: t0,ε =
inf{t ≥ 0 : ϕ1,ε(t) = 0}, tk+1,ε = inf{t > tk,ε : ϕ1,ε(t) = 0}. This sequence is
chosen because δϕ1,0(s)=1/2 is “ergodizd” as time passes from tk,0 to tk+1,0. If ε is
sufficiently small and tk+1,ε ≤ (T ∧ Tε)/ε, then the spacings between these times
are uniformly of order 1, i.e. 1/ω < tk+1,ε − tk,ε < ω. Thus,
0≤τ≤T∧Tε
∫ τ/ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
≤ O(1) +
tk+1,ε≤T∧Tεε
∫ tk+1,ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
(3.10)
Step 4: Control of individual terms by comparison with solutions along
fibers. The sum in Equation (3.10) has no more than O(1/ε) terms, and so it
suffices to show that each term is no larger than O(ε). We can accomplish this by
comparing the motions of zε(t) for tk,ε ≤ t ≤ tk+1,ε with the solution of the ε = 0
version of Equation (3.7) that, at time tk,ε, is located at zε(tk,ε). Since each term
in the sum has the same form, without loss of generality we will only examine the
first term and suppose that t0,ε = 0, i.e. that ϕ1,ε(0) = 0.
Lemma 3.3.2. If t1,ε ≤ T∧Tεε , then sup0≤t≤t1,ε |z0(t)− zε(t)| = O(ε).
Proof. To check that sup0≤t≤t1,ε |h0(t)− hε(t)| = O(ε), first note that h0(t) =
h0(0) = hε(0). Then dQε/dt = O(ε), so that Q0(t)−Qε(t) = O(εt). Furthermore,
the other slow variables change by O(ε) at collisions, while the number of collisions
in the time interval [0, t1,ε] is O(1).
It remains to show that sup0≤t≤t1,ε |ϕi,0(t)− ϕi,ε(t)| = O(ε). Using what we
know about the divergence of the slow variables,
ϕ1,0(t)− ϕ1,ε(t) =
s1,0(s)
2Q0(s)
− s1,ε(s)
2Qε(s)
+O(ε)ds =
O(ε)ds = O(ε)
for 0 ≤ t ≤ t1,ε. Showing that sup0≤t≤t1,ε |ϕ2,0(t)− ϕ2,ε(t)| = O(ε) is similar.
From Lemma 3.3.2, t1,ε = t1,0 +O(ε) = 2Q0/s1,0 +O(ε). We conclude that
∫ t1,ε
s1,ε(s)δϕ1,ε(s)=1/2 −
s1,ε(s)
2Qε(s)
ds = O(ε) +
∫ t1,ε
s1,0(s)δϕ1,ε(s)=1/2 −
s1,0(s)
2Q0(s)
= O(ε) + s1,0 − t1,ε
s21,0
= O(ε).
It follows that sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ = O(ε), independent of the initial
condition in h−1V.
3.3.2 Extension to multiple gas particles
When n1, n2 > 1, only minor modifications are necessary to generalize the proof
above. We start by extending the slow variables h to a full set of coordinates on
phase space by defining the angle variables ϕi,j ∈ [0, 1]/ 0 ∼ 1 = S1 for 1 ≤ i ≤ 2,
1 ≤ j ≤ ni:
ϕ1,j = ϕ1,j(q1,j , v1,j, Q) =
if v1,j > 0
1− q1,j
if v1,j < 0
ϕ2,j = ϕ2,j(q2,j , v2,j, Q) =
1−q2,j
2(1−Q) if v2,j < 0
1− 1−q2,j
2(1−Q) if v2,j > 0
Then dϕ1,j/dt = s1,j(2Q)
−1 + O(ε), dϕ2,j/dt = s2,j(2(1 − Q))−1 + O(ε), and
z = (h, ϕ1,j , ϕ2,j) represents a choice of coordinates on our phase space, which
is homeomorphic to (a subset of Rn1+n2+2) × Tn1+n2 . In these coordinates, the
dynamical system yields a discontinuous flow zε(t) on phase space. The flow
preserves Liouville measure, which in our coordinates has a density proportional
to Qn1(1 − Q)n2 . As is Section 3.3.1, one can show that the measure of initial
conditions leading to multiple particle collisions is zero.
Next, define H(z) by
H(z) =
j=1 2m1,js1,jδϕ1,j=1/2 −
j=1 2m2s2,jδϕ2,j=1/2
−2Wδϕ1,j=1/2
2Wδϕ2,j=1/2
For 0 ≤ t ≤ T∧Tε
, hε(t) − hε(0) = ε
H(zε(s))ds + O(ε). From here, the rest of
the proof follows the same arguments made in Section 3.3.1.
3.4 Proof of the main result for the soft core
piston
For the remainder of this chapter, we consider the family of Hamiltonian systems
introduced in Section 3.1.2, which are parameterized by ε, δ ≥ 0. For simplicity,
we specialize to n1 = n2 = 1. As in Section 3.3, the generalization to n1, n2 > 1
is not difficult. The Hamiltonian dynamics are given by the following ordinary
differential equation:
= εW,
= ε (−κ′δ(Q− x1) + κ′δ(x2 −Q)) ,
= v1,
−κ′δ(x1) + κ′δ(Q− x1)
= v2,
−κ′δ(x2 −Q) + κ′δ(1− x2)
(3.11)
Recalling the particle energies defined by Equation (3.3), we find that
= εWκ′δ(Q− x1),
= −εWκ′δ(x2 −Q).
For the compact set V introduced in Section 3.1.2, fix a small positive number
E and an open set U ⊂ R4 such that V ⊂ U and h ∈ U ⇒ Q ∈ (E , 1−E),W ⊂⊂ R,
and E < E1, E2 < κ(0)− E . We only consider the dynamics for 0 < δ < E/2 and
h ∈ U .
Define
U1(q1) = U1(q1, Q, δ) := κδ(q1) + κδ(Q− q1),
U2(q2) = U2(q2, Q, δ) := κδ(q2 −Q) + κδ(1− q2).
Then the energies Ei satisfy Ei = miv
i /2 + Ui(xi).
Let T1 = T1(Q,E1, δ) and T2 = T2(Q,E2, δ) denote the periods of the motions
of the left and right gas particles, respectively, when ε = 0.
Lemma 3.4.1. For i = 1, 2,
Ti ∈ C1{(Q,Ei, δ) : Q ∈ (E , 1− E), Ei ∈ (E , κ(0)− E), 0 ≤ δ < E/2}.
Furthermore,
T1(Q,E1, δ) =
Q +O(δ),
T2(Q,E2, δ) =
(1−Q) +O(δ).
The proof of this lemma is mostly computational, and so we delay it until
Section 3.5. Note especially that the periods can be suitably defined such that
their regularity extends to δ = 0.
In this section, and in Section 3.5 below, we adopt the following convention
on the use of the O notation. All use of the O notation will explicitly contain the
dependence on ε and δ as ε, δ → 0. For example, if a function f(h, ε, δ) = O(ε),
then there exists δ′, ε′ > 0 such that sup0<ε≤ε′, 0<δ≤δ′, h∈V |f(h, ε, δ)/ε| <∞.
When ε = 0,
(Ei − Ui(xi)).
Define a = a(Ei, δ) by
κδ(a) = κ(a/δ) = Ei,
so that a(E1, δ) is a turning point for the left gas particle. Then a = δκ
−1(Ei),
where κ−1 is defined as follows: κ : [0, 1] → [0, κ(0)] takes 0 to κ(0) and 1 to 0.
Furthermore, κ ∈ C2([0, 1]), κ′ ≤ 0, and κ′(x) < 0 if x < 1. By monotonicity,
κ−1 : [0, κ(0)] → [0, 1] exists and takes 0 to 1 and κ(0) to 0. Also, by the Implicit
Function Theorem, κ−1 ∈ C2((0, κ(0)]), (κ−1)′(y) < 0 for y > 0, and (κ−1)′(y) →
−∞ as y → 0+. Because we only consider energies Ei ∈ (E , κ(0)− E), it follows
that a(Ei, δ) is a C2 function for the domains of interest.
3.4.1 Derivation of the averaged equation
As we previously pointed out, for each fixed δ > 0, Anosov’s theorem 2.1.1 and
Theorem 2.2.5 apply directly to the family of ordinary differential equations in
Equation (3.11), provided that δ is sufficiently small. The invariant fibers Mh of
the ε = 0 flow are tori described by a fixed value of the four slow variables and
{(Q,W, q1, v1, q2, v2) : E1 = m1v21/2 + U1(q1, Q, δ), E2 = m2v22/2 + U2(q2, Q, δ)}. If
we use (q1, q2) as local coordinates on Mh, which is valid except when v1 or v2 = 0,
the invariant measure µh of the unperturbed flow has the density
dq1dq2
(E1 − U1(q1)) T2
(E2 − U2(q2))
The restricted flow is ergodic for almost every h. See Corollary 3.5.1 in Section
−κ′δ(Q− q1) + κ′δ(q2 −Q)
Wκ′δ(Q− q1)
−Wκ′δ(q2 −Q)
κ′δ(Q− q1)dµh =
∫ Q−a
κ′δ(Q− q1)
(E1 − U1(q1))
∫ Q−a
κ′δ(Q− q1)
E1 − κδ(Q− q1)
E1 − u
8m1E1
Similarly,
κ′δ(q2 −Q)dµh = −
8m2E2
It follows that the averaged vector field is
H̄δ(h) =
8m1E1
8m2E2
8m1E1
8m2E2
where from Lemma 3.4.1 we see that H̄ ·(·) ∈ C1({(δ, h) : 0 ≤ δ < E/2, h ∈ V}).
H̄0(h) agrees with the averaged vector field for the hard core system from Equation
(3.1), once we account for the change of coordinates Ei = mis
i /2.
Remark 3.4.1. An argument due to Neishtadt and Sinai [NS04] shows that the
solutions to the averaged equation (3.4) are periodic. This argument also shows
that, as in the case δ = 0, the limiting dynamics of (Q,W ) are effectively Hamil-
tonian, with the shape of the Hamiltonian depending on δ, Q(0), and the initial
energies of the gas particles. The argument depends heavily on the observation
that the phase integrals
Ii(Q,Ei, δ) =
miv2+Ui(x,Q,δ)≤Ei
are adiabatic invariants, i.e. they are integrals of the solutions to the averaged
equation. Thus the four-dimensional phase space of the averaged equation is foli-
ated by invariant two-dimensional submanifolds, and one can think of the effective
Hamiltonians for the piston as living on these submanifolds.
3.4.2 Proof of Theorem 3.1.2
The following arguments are motivated by our proof in Section 3.3, although the
details are more involved as we show that the rate of convergence is independent
of all small δ.
A choice of coordinates on phase space
We wish to describe the dynamics in a coordinate system inspired by the one used
in Section 3.3.1. For each fixed δ ∈ (0, δ0], this change of coordinates will be C1
in all variables on the domain of interest. However, it is an exercise in analysis to
show this, and so we delay the proofs of the following two lemmas until Section
We introduce the angular coordinates ϕi ∈ [0, 1]/ 0 ∼ 1 = S1 defined by
ϕ1 = ϕ1(q1, v1, Q) =
0 if q1 = a
E1−U1(s)ds if v1 > 0
1/2 if q1 = Q− a
E1−U1(s)ds if v1 < 0
ϕ2 = ϕ2(q2, v2, Q) =
0 if q2 = 1− a
∫ 1−a
E2−U2(s)ds if v2 < 0
1/2 if q2 = Q+ a
∫ 1−a
E2−U2(s)ds if v2 > 0
. (3.12)
Then z = (h, ϕ1, ϕ2) is a choice of coordinates on h
−1U . As before, we will abuse
notation and let h(z) denote the projection onto the first four coordinates of z.
There is a fixed value of δ0 in the statement of Theorem 3.1.2. However, for
the purposes of our proof, it will be convenient to progressively choose δ0 smaller
when needed. At the end of the proof, we will have only shrunk δ0 a finite number
of times, and this final value will satisfies the requirements of the theorem. Our
first requirement on δ0 is that it is smaller than E/2.
Lemma 3.4.2. If δ0 > 0 is sufficiently small, then for each δ ∈ (0, δ0] the ordinary
differential equation (3.11) in the coordinates z takes the form
= Zδ(z, ε), (3.13)
where Zδ ∈ C1(h−1U × [0,∞)). When z ∈ h−1U ,
Zδ(z, ε) =
−κ′δ(Q− q1(z)) + κ′δ(q2(z)−Q)
εWκ′δ(Q− q1(z))
−εWκ′δ(q2(z)−Q)
+O(ε)
+O(ε)
. (3.14)
Recall that, by our conventions, the O(ε) terms in Equation (3.14) have a
size that can be bounded independent of all δ sufficiently small. Denote the flow
determined by Zδ(·, ε) by zδε(t), and its components by Qδε(t), W δε (t), Eδ1,ε(t), etc.
Also, set hδε(t) = h(z
ε(t)). From Equation (3.14),
Hδ(z, ε) :=
−κ′δ(Q− q1(z)) + κ′δ(q2(z)−Q)
Wκ′δ(Q− q1(z))
−Wκ′δ(q2(z)−Q)
. (3.15)
In particular, Hδ(z, ε) = Hδ(z, 0).
Before proceeding, we need one final technical lemma.
Lemma 3.4.3. If δ0 > 0 is chosen sufficiently small, there exists a constant K
such that for all δ ∈ (0, δ0], κ′δ(|Q− xi(z)|) = 0 unless ϕi ∈ [1/2−Kδ, 1/2 +Kδ].
Argument for uniform convergence
We start by proving the following lemma, which essentially says that an orbit
zδε(t) only spends a fraction O(δ) of its time in a region of phase space where
∣Hδ(zδε(t), ε)
∣Hδ(zδε(t), 0)
∣ is of size O(δ−1)
Lemma 3.4.4. For 0 ≤ T ′ ≤ T ≤ T∧T
∣Hδ(zδε(s), 0)
∣ ds = O(1 ∨ (T − T ′)).
Proof. Without loss of generality, T ′ = 0. From Lemmas 3.4.1 and 3.4.2 it follows
that if we choose δ0 sufficiently small, then there exists ω > 0 such that for all
sufficiently small ε and all δ ∈ (0, δ0], h ∈ V ⇒ 1/ω <
< ω. Define the set
B = [1/2 − Kδ, 1/2 + Kδ], where K comes from Lemma 3.4.3. Then we find a
crude bound on
Qδε(s)− q1(zδε(s))
∣ ds using that
dϕδ1,ε
≥ 1/ω if ϕδ1,ε ∈ B
≤ ω if ϕδ1,ε ∈ Bc.
This yields
Qδε(s)− q1(zδε(s))
∣ ds ≤ const
1ϕδ1,ε(s)∈Bds
≤ const
2Kωδ + 1−2Kδ
T + 2Kωδ
= O(1 ∨ T ).
Similarly,
∣κ′δ(q2(z
ε(s))−Qδε(s))
∣ ds = O(1∨ T ), and so
∣Hδ(zδε(s), 0)
∣ ds =
O(1 ∨ T ).
We now follow steps one through four from Section 3.3.1, making modifications
where necessary.
Step 1: Reduction using Gronwall’s Inequality. Now hδε(τ/ε) satisfies
hδε(τ/ε)− hδε(0) = ε
∫ τ/ε
Hδ(zδε(s), 0)ds.
Define
eδε(τ) = ε
∫ τ/ε
Hδ(zδε(s), 0)− H̄δ(hδε(s))ds.
It follows from Gronwall’s Inequality and the fact that H̄ ·(·) ∈ C1({(δ, h) : 0 ≤
δ ≤ δ0, h ∈ V}) that
0≤τ≤T∧T δε
∣hδε(τ/ε)− h̄δ(τ)
0≤τ≤T∧T δε
∣eδε(τ)
eLip(H̄
δ|V)T
0≤τ≤T∧T δε
∣eδε(τ)
(3.16)
Step 2: A splitting according to particles. Next,
Hδ(z, 0)− H̄δ(h)
−κ′δ(Q− q1(z))−
8m1E1
Wκ′δ(Q− q1(z)) +W
8m1E1
κ′δ(q2(z)−Q) +
8m2E2
−Wκ′δ(q2(z)−Q)−W
8m2E2
and so, in order to show that sup0≤τ≤T∧T δε
∣eδε(τ)
∣ = O(ε), it suffices to show that
for i = 1, 2,
0≤τ≤T∧T δε
∫ τ/ε
∣Qδε(s)− xi(zδε(s))
i,ε(s)
Ti(Qδε(s), E
i,ε(s), δ)
= O(1),
0≤τ≤T∧T δε
∫ τ/ε
Wε(s)κ
∣Qδε(s)− xi(zδε(s))
+Wε(s)
i,ε(s)
Ti(Qδε(s), E
i,ε(s), δ)
= O(1).
We only demonstrate that
0≤τ≤T∧T δε
∫ τ/ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
= O(1).
The other three terms are handled similarly.
Step 3: A sequence of times adapted for ergodization. Define the se-
quence of times tδk,ε inductively by t
0,ε = inf{t ≥ 0 : ϕδ1,ε(t) = 0}, tδk+1,ε = inf{t >
tδk,ε : ϕ
1,ε(t) = 0}. If ε and δ are sufficiently small and tδk+1,ε ≤ (T ∧ T δε )/ε, then
it follows from Lemma 3.4.2 and the discussion in the proof of Lemma 3.4.4 that
1/ω < tδk+1,ε − tδk,ε < ω. From Lemmas 3.4.2 and 3.4.4 it follows that
0≤τ≤T∧T δε
∫ τ/ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
≤ O(1) +
k+1,ε
k+1,ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
(3.17)
Step 4: Control of individual terms by comparison with solutions along
fibers. As before, it suffices to show that each term in the sum in Equation
(3.17) is no larger than O(ε). Without loss of generality we will only examine the
first term and suppose that tδ0,ε = 0, i.e. that ϕ
1,ε(0) = 0.
Lemma 3.4.5. If tδ1,ε ≤
T∧T δε
, then sup0≤t≤tδ1,ε
∣zδ0(t)− zδε(t)
∣ = O(ε).
Proof. By Lemma 3.4.4, hδ0(t) − hδε(t) = hδε(0) − hδε(t) = −ε
Hδ(zδε(s), 0)ds =
O(ε(1 ∨ t)) for t ≥ 0.
Using what we know about the divergence of the slow variables, we find that
ϕδ1,0(t)− ϕδ1,ε(t) =
0(s), E
0(s), δ)
T1(Qδε(s), E
ε(s), δ)
+O(ε)ds
O(ε)ds
= O(ε)
for 0 ≤ t ≤ tδ1,ε. Lemmas 3.4.1 and 3.4.2 ensure the desired uniformity in the sizes
of the orders of magnitudes. Showing that sup0≤t≤tδ1,ε
∣ϕδ2,0(t)− ϕδ2,ε(t)
∣ = O(ε) is
similar.
From Lemma 3.4.5 we find that t1,ε = t1,0+O(ε) = T1(Qδ0, Eδ0 , δ)+O(ε). Hence
∫ tδ1,ε
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
ds = O(ε) +
∫ tδ1,0
1,0, δ)
= O(ε) +
But when q1(z
ε) < Q
ε − a,
Eδ1,ε(s)− κδ
Qδε(s)− q1(zδε(s))
ε(s))
Qδε(s)− q1(zδε(s))
and so
∫ tδ1,ε
Qδε(s)− q1(zδε(s))
ds = −
1,ε(0)−
1,ε(t
= O(ε)−
Hence,
∫ tδ1,ε
Qδε(s)− q1(zδε(s))
1,ε(s)
T1(Qδε(s), E
1,ε(s), δ)
ds = O(ε),
as desired.
3.5 Appendix to Section 3.4
Proof of Lemma 3.4.1:
Proof. For 0 < δ < E/2,
T1 = T1(Q,E1, δ) = 2
∫ Q−a
E1 − U1(s)
T2 = T2(Q,E2, δ) = 2
∫ 1−a
E2 − U2(s)
We only consider the claims about T1, and for convenience we take m1 = 2. Then
T1(Q,E1, δ) = 2
∫ Q−a
E1 − U1(s)
∫ Q/2
E1 − κδ(s)
Q/2− δ√
E1 − κδ(s)
2Q− 4δ√
κ−1(E1)
E1 − κ(s)
Define
F (E) :=
κ−1(E)
E − κ(s)
−(κ−1)′(u)√
E − u
Notice that (κ−1)′(u) diverges as u → 0+, while (E − u)−1/2 diverges as u → E−,
but both functions are still integrable on [0, E]. It follows that F (E) is well defined.
Then it suffices to show that F : [E , κ(0)− E ] → R is C1.
Write
F (E) =
∫ E/2
−(κ−1)′(u)√
E − u
−(κ−1)′(u)√
E − u
:= F1(E) + F2(E).
A standard application of the Dominated Convergence Theorem allows us to dif-
ferentiate inside the integral and conclude that F1 ∈ C∞([E , κ(0)− E ]), with
F ′1(E) =
∫ E/2
(κ−1)′(u)
2(E − u)3/2du.
To examine F2, we make the substitution v = E − u to find that
F2(E) =
∫ E−E/2
−(κ−1)′(E − v)√
Using the fact that (κ−1)′ ∈ C1([E/2, κ(0)]) and the Dominated Convergence The-
orem, we find that F2 is differentiable, with
F ′2(E) =
−(κ−1)′(E/2)
E − E/2
∫ E−E/2
−(κ−1)′′(E − v)√
Another application of the Dominated Convergence Theorem shows that F ′2 is
continuous, and so F2 ∈ C1([E , κ(0)− E ]).
T1(Q,E1, δ) =
−E−1/21 + F1(E1) + F2(E1)
has the desired regularity. For future reference, we note that
+O(δ). (3.18)
Corollary 3.5.1. For all δ sufficiently small, the flow zδ0(t) restricted to the in-
variant tori Mc = {h = c} is ergodic (with respect to the invariant Lebesgue
measure) for almost every c ∈ U .
Proof. The flow is ergodic whenever the periods T1 and T2 are irrationally related.
Fix δ sufficiently small such that ∂T1
= −Q/E3/21 + O(δ) < 0. Next, consider
Q, W , and E2 fixed, so that T2 is constant. Because T1 ∈ C1, it follows that,
as we let E1 vary,
/∈ Q for almost every E1. The result follows from Fubini’s
Theorem.
Proof of Lemma 3.4.2:
Proof. For the duration of this proof, we consider the dynamics for a small, fixed
value of δ > 0, which we generally suppress in our notation. For convenience, we
take m1 = 2.
Let ψ denote the map taking (Q,W, q1, v1, q2, v2) to (Q,W,E1, E2, ϕ1, ϕ2). We
claim that ψ is a C1 change of coordinates on the domain of interest. Since E1 =
v21 + κδ(q1) + κδ(Q− q1), E1 is a C2 function of q1, v1, and Q. A similar statement
holds for E2.
The angular coordinates ϕi(xi, vi, Q) are defined by Equation (3.12). We only
consider ϕ1, as the statements for ϕ2 are similar. Then ϕ1(q1, v1, Q) is clearly C1
whenever q1 6= a,Q−a. The apparent difficulties in regularity at the turning points
are only a result of how the definition of ϕ1 is presented in Equation (3.12). Recall
that the angle variables are actually defined by integrating the elapsed time along
orbits, and our previous definition expressed ϕ1 in a manner which emphasized
the dependence on q1. In fact, whenever |v1| <
ϕ1(q1, v1, Q) =
(κ−1δ )
′(E1 − v2)dv if q1 < δ
(κ−1δ )
′(E1 − v2)dv if q1 > Q− δ.
(3.19)
Here E1 is implicitly considered to be a function of q1, v1, and Q. One can verify
that Dψ is non-degenerate on the domain of interest, and so ψ is indeed a C1
change of coordinates.
Next observe that dϕ1,0/dt = 1/T1, so Hadamard’s Lemma implies that
dϕ1,ε
+O(εf(δ)).
It remains to show that, in fact, we may take f(δ) = 1. It is easy to verify this
whenever q1 ≤ Q−δ because dE1/dt = 0 there. We only perform the more difficult
verification when q1 > Q− δ.
When q1 > Q−δ, |v1| <
E1 and E1 = v
1+κδ(Q−q1). From Equation (3.19)
we find that
T1(Q,E1, δ)
(κ−1)′(E1 − v2)dv. (3.20)
To find dϕ1/dt, we consider ϕ1 as a function of v1, Q, and E1, so that
Then, using Equations (3.18) and (3.20), we compute
(κ−1δ )
′(E1 − v21)
κ′δ(Q− q1)
1/2− ϕ1
(εW ) = εW
1/2− ϕ1
1/2− ϕ1
(κ−1)′′(E1 − v2)dv
(εWκ′δ(Q− q1)).
Using that κ′δ(Q− q1) = κ′(κ−1(E1 − v21))/δ = (δ(κ−1)′(E1 − v21))−1, we find that
1/2− ϕ1
(κ−1)′(E1 − v21)
(κ−1)′′(E1 − v2)dv
But here 1/2− ϕ1 is O(δ). See the proof of Lemma 3.4.3 below. Thus the claims
about dϕ1/dt will be proven, provided we can uniformly bound
(κ−1)′(E1 − v21)
(κ−1)′′(E1 − v2)dv.
Note that the apparent divergence of the integral as |v1| →
E1 is entirely due to
the fact that our expression for ϕ1 from Equation (3.20) requires |v1| <
E1. If
we make the substitution u = E1− v2 and let e = E1− v21 , then it suffices to show
E≤E1≤κ(0)−E
0<e≤E1
(κ−1)′(e)
(κ−1)′′(u)√
E1 − u
< +∞.
The only difficulties occur when e is close to 0. Thus it suffices to show that
E≤E1≤κ(0)−E
0<e≤E/2
(κ−1)′(e)
∫ E/2
(κ−1)′′(u)√
E1 − u
is finite. But this is bounded by
0<e≤E/2
(κ−1)′(e)
∫ E/2
(κ−1)′′(u)
= sup
0<e≤E/2
(κ−1)′(e)
(κ−1)′(E/2)− (κ−1)′(e)
which is finite because (κ−1)′(e) → −∞ as e→ 0+. The claims about dϕ2/dt can
be proven similarly.
Proof of Lemma 3.4.3:
Proof. We continue in the notation of the proofs of Lemmas 3.4.1 and 3.4.2 above,
and we set m1 = 2. Then from Equation (3.20), we see that κ
δ(Q− q1) = 0 unless
|ϕ1 − 1/2| ≤
(κ−1)′(E1 − v2)dv
= δF (E1)/T1 = O(δ). Dealing with ϕ2
is similar.
Chapter 4
The periodic oscillation of an
adiabatic piston in two or three
dimensions
In this chapter, we present our results for the piston system in two or three di-
mensions. These results may also be found in [Wri07].
4.1 Statement of the main result
4.1.1 Description of the model
Consider a massive, insulating piston of mass M that separates a gas container
D in Rd, d = 2 or 3. See Figure 4.1. Denote the location of the piston by Q,
its velocity by dQ/dt = V , and its cross-sectional length (when d = 2, or area,
when d = 3) by ℓ. If Q is fixed, then the piston divides D into two subdomains,
D1(Q) = D1 on the left and D2(Q) = D2 on the right. By Ei we denote the
total energy of the gas inside Di, and by |Di| we denote the area (when d = 2, or
volume, when d = 3) of Di.
We are interested in the dynamics of the piston when the system’s total energy
is bounded and M → ∞. When M = ∞, the piston remains fixed in place, and
each energy Ei remains constant. When M is large but finite, MV
2/2 is bounded,
and so V = O(M−1/2). It is natural to define
ε =M−1/2,
so that W is of order 1 as ε → 0. This is equivalent to scaling time by ε.
D1(Q) D2(Q)
ℓ ✲ V = εW
M = ε−2 ≫ 1
❆❯s✟✟✙
s❍❍❍❥ s✘✘✿
Figure 4.1: A gas container D ⊂ R2 separated by a piston.
Next we precisely describe the gas container. It is a compact, connected billiard
domain D ⊂ Rd with a piecewise C3 boundary, i.e. ∂D consists of a finite number of
C3 embedded hypersurfaces, possibly with boundary and a finite number of corner
points. The container consists of a “tube,” whose perpendicular cross-section P is
the shape of the piston, connecting two disjoint regions. P ⊂ Rd−1 is a compact,
connected domain whose boundary is piecewise C3. Then the “tube” is the region
[0, 1] × P ⊂ D swept out by the piston for 0 ≤ Q ≤ 1, and [0, 1] × ∂P ⊂ ∂D. If
d = 2, P is just a closed line segment, and the “tube” is a rectangle. If d = 3, P
could be a circle, a square, a pentagon, etc.
Our fundamental assumption is as follows:
Main Assumption. For almost every Q ∈ [0, 1] the billiard flow of a single
particle on an energy surface in either of the two subdomains Di(Q) is ergodic
(with respect to the invariant Liouville measure).
If d = 2, the domain could be the Bunimovich stadium [Bun79]. Another possible
domain is indicated in Figure 4.1. The ergodicity of billiards in such domains,
which produce hyperbolic flows, goes back to the pioneering work of Sinai [Sin70],
although a number of individuals have contributed to the theory. A full accounting
of this history can be found in [CM06a]. Polygonal domains satisfying our assump-
tions can also be constructed [Vor97]. Suitable domains in d = 3 dimensions can be
constructed using a rectangular box with shallow spherical caps adjoined [BR98].
Note that we make no assumptions regarding the hyperbolicity of the billiard flow
in the domain.
The Hamiltonian system we consider consists of the massive piston of mass M
located at positionQ, as well as n1+n2 gas particles, n1 inD1 and n2 inD2. Here n1
and n2 are fixed positive integers. For convenience, the gas particles all have unit
mass, though all that is important is that each gas particle has a fixed mass. We
denote the positions of the gas particles in Di by qi,j, 1 ≤ j ≤ ni. The gas particles
are ideal point particles that interact with ∂D and the piston by hard core, elastic
collisions. Although it has no effect on the dynamics we consider, for convenience
we complete our description of the Hamiltonian dynamics by specifying that the
piston makes elastic collisions with walls located at Q = 0, 1 that are only visible
to the piston. We denote velocities by dQ/dt = V = εW and dqi,j/dt = vi,j, and
we set
Ei,j = v
i,j/2, Ei =
Ei,j.
Our system has d(n1 + n2) + 1 degrees of freedom, and so its phase space is
(2d(n1 + n2) + 2)-dimensional.
We let
h(z) = h = (Q,W,E1,1, E1,2, · · · , E1,n1 , E2,1, E2,2, · · · , E2,n2),
so that h is a function from our phase space to Rn1+n2+2. We often abbreviate
h = (Q,W,E1,j , E2,j), and we refer to h as consisting of the slow variables because
these quantities are conserved when ε = 0. We let hε(t, z) = hε(t) denote the
actual motions of these variables in time for a fixed value of ε. Here z represents
the initial condition in phase space, which we usually suppress in our notation.
One should think of hε(·) as being a random variable that takes initial conditions
in phase space to paths (depending on the parameter t) in Rn1+n2+2.
4.1.2 The averaged equation
From the work of Neishtadt and Sinai [NS04], one can derive
= H̄(h) :=
d |D1(Q)|
− 2E2ℓ
d |D2(Q)|
− 2WE1,jℓ
d |D1(Q)|
2WE2,jℓ
d |D2(Q)|
(4.1)
as the averaged equation (with respect to the slow time τ = εt) for the slow
variables. Later, in Section 4.2.3, we will give another heuristic derivation of the
averaged equation that is more suggestive of our proof.
Neishtadt and Sinai [Sin99, NS04] pointed out that the solutions of Equa-
tion (1.3) have (Q,W ) behaving as if they were the coordinates of a Hamiltonian
system describing a particle undergoing motion inside a potential well. As in
Section 1.2, the effective Hamiltonian is given by
W 2 +
E1(0) |D1(Q(0))|2/d
|D1(Q)|2/d
E2(0) |D2(Q(0))|2/d
|D2(Q)|2/d
This can be seen as follows. Since
∂ |D1(Q)|
= ℓ = −∂ |D2(Q)|
d ln(Ei,j)/dτ = −(2/d)d ln(|Di(Q)|)/dτ , and so
Ei,j(τ) = Ei,j(0)
|Di(Q(0))|
|Di(Q(τ))|
By summing over j, we find that
Ei(τ) = Ei(0)
|Di(Q(0))|
|Di(Q(τ))|
and so
d2Q(τ)
E1(0) |D1(Q(0))|2/d
|D1(Q(τ))|1+2/d
E2(0) |D2(Q(0))|2/d
|D2(Q(τ))|1+2/d
Let h̄(τ, z) = h̄(τ) be the solution of
= H̄(h̄), h̄(0) = hε(0).
Again, think of h̄(·) as being a random variable.
4.1.3 The main result
The solutions of the averaged equation approximate the motions of the slow vari-
ables, hε(t), on a time scale O(1/ε) as ε → 0. Precisely, fix a compact set
V ⊂ Rn1+n2+2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1),W ⊂⊂ R, and Ei,j ⊂⊂ (0,∞)
for each i and j.1 We will be mostly concerned with the dynamics when h ∈ V.
Define
Qmin = inf
Q, Qmax = sup
Emin = inf
W 2 + E1 + E2, Emax = sup
W 2 + E1 + E2.
1 We have introduced this notation for convenience. For example, h ∈ V ⇒ Q ⊂⊂ (0, 1)
means that there exists a compact set A ⊂ (0, 1) such that h ∈ V ⇒ Q ∈ A, and similarly for
the other variables.
For a fixed value of ε > 0, we only consider the dynamics on the invariant subset
of phase space defined by
Mε = {(Q, V, qi,j, vi,j) ∈ R2d(n1+n2)+2 : Q ∈ [0, 1], qi,j ∈ Di(Q),
Emin ≤
V 2 + E1 + E2 ≤ Emax}.
Let Pε denote the probability measure obtained by restricting the invariant Liou-
ville measure to Mε. Define the stopping time
Tε(z) = Tε = inf{τ ≥ 0 : h̄(τ) /∈ V or hε(τ/ε) /∈ V}.
Theorem 4.1.1. If D is a gas container in d = 2 or 3 dimensions satisfying the
assumptions in Subsection 4.1.1 above, then for each T > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣→ 0 in probability as ε =M−1/2 → 0,
i.e. for each fixed δ > 0,
0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ ≥ δ
→ 0 as ε =M−1/2 → 0.
Remark 4.1.1. It should be noted that the stopping time in the above result is not
unduly restrictive. If the initial pressures of the two gasses are not too mismatched,
then the solution to the averaged equation is a periodic orbit, with the effective
potential well keeping the piston away from the walls. Thus, if the actual motions
follow the averaged solution closely for 0 ≤ τ ≤ T ∧ Tε, and the averaged solution
stays in V, it follows that Tε > T .
Remark 4.1.2. The techniques of this work should immediately generalize to prove
the analogue of Theorem 4.1.1 above in the nonphysical dimensions d > 3, although
we do not pursue this here.
Remark 4.1.3. As in Subsection 3.1.3, Theorem 4.1.1 can be easily generalized
to cover a system of N − 1 pistons that divide N gas containers, so long as, for
almost every fixed location of the pistons, the billiard flow of a single gas particle
on an energy surface in any of the N subcontainers is ergodic (with respect to the
invariant Liouville measure). The effective Hamiltonian for the pistons has them
moving like an (N − 1)-dimensional particle inside a potential well.
4.2 Preparatory material concerning a
two-dimensional gas container with only one
gas particle on each side
Our results and techniques of proof are essentially independent of the dimension
and the fixed number of gas particles on either side of the piston. Thus, we focus
D1 D2
ℓ ✲V = εW
M = ε−2 ≫ 1
Figure 4.2: A choice of coordinates on phase space.
on the case when d = 2 and there is only one gas particle on either side. Later,
in Section 4.4, we will indicate the simple modifications that generalize our proof
to the general situation. For clarity, in this section and next, we denote q1,1 by q1,
v2,1 by v2, etc. We decompose the gas particle coordinates according to whether
they are perpendicular to or parallel to the piston’s face, for example q1 = (q
1 , q
See Figure 4.2.
The Hamiltonian dynamics define a flow on our phase space. We denote this
flow by zε(t, z) = zε(t), where z = zε(0, z). One should think of zε(·) as being
a random variable that takes initial conditions in phase space to paths in phase
space. Then hε(t) = h(zε(t)). By the change of coordinates W = V/ε, we may
identify all of the Mε defined in Section 4.1 with the space
M = {(Q,W, q1, v1, q2, v2) ∈ R10 : Q ∈ [0, 1], q1 ∈ D1(Q), q2 ∈ D2(Q),
Emin ≤
W 2 + E1 + E2 ≤ Emax}.
and all of the Pε with the probability measure P on M, which has the density
dP = const dQdWdq⊥1 dq
(Throughout this work we will use const to represent generic constants that are
independent of ε.) We will assume that these identifications have been made, so
that we may consider zε(·) as a family of measure preserving flows on the same
space that all preserve the same probability measure. We denote the components
of zε(t) by Qε(t), q
1,ε(t), etc.
The set {z ∈ M : q1 = Q = q2} has co-dimension two, and so
zε(t){q1 =
Q = q2} has co-dimension one, which shows that only a measure zero set of initial
conditions will give rise to three particle collisions. We ignore this and other
measures zero events, such as gas particles hitting singularities of the billiard flow,
in what follows.
Now we present some background material, as well as some lemmas that will
assist us in our proof of Theorem 4.1.1. We begin by studying the billiard flow
of a gas particle when the piston is infinitely massive. Next we examine collisions
between the gas particles and the piston when the piston has a large, but finite,
mass. Then we present a heuristic derivation of the averaged equation that is
suggestive of our proof. Finally we prove a lemma that allows us to disregard
the possibility that a gas particle will move nearly parallel to the piston’s face
– a situation that is clearly bad for having the motions of the piston follow the
solutions of the averaged equation.
4.2.1 Billiard flows and maps in two dimensions
In this section, we study the billiard flows of the gas particles when M = ∞ and
the slow variables are held fixed at a specific value h ∈ V. We will only study
the motions of the left gas particle, as similar definitions and results hold for the
motions of the right gas particle. Thus we wish to study the billiard flow of a point
particle moving inside the domain D1 at a constant speed
2E1. The results of
this section that are stated without proof can be found in [CM06a].
Let T D1 denote the tangent bundle to D1. The billiard flow takes place in
the three-dimensional space M1h = M1 = {(q1, v1) ∈ T D1 : q1 ∈ D1, |v1| =√
2E1}/ ∼. Here the quotient means that when q1 ∈ ∂D1, we identify velocity
vectors pointing outside of D1 with those pointing inside D1 by reflecting through
the tangent line to ∂D1 at q1, so that the angle of incidence with the unit normal
vector to ∂D1 equals the angle of reflection. Note that most of the quantities
defined in this subsection depend on the fixed value of h. We will usually suppress
this dependence, although, when necessary, we will indicate it by a subscript h.
We denote the resulting flow by y(t, y) = y(t), where y(0, y) = y. As the billiard
flow comes from a Hamiltonian system, it preserves Liouville measure restricted
to the energy surface. We denote the resulting probability measure by µ. This
measure has the density dµ = dq1dv1/(2π
2E1 |D1|). Here dq1 represents area on
R2, and dv1 represents length on S
v1 ∈ R2 : |v1| =
There is a standard cross-section to the billiard flow, the collision cross-section
Ω = {(q1, v1) ∈ T D1 : q1 ∈ ∂D1, |v1| =
2E1}/ ∼. It is customary to parameter-
ize Ω by {x = (r, ϕ) : r ∈ ∂D1, ϕ ∈ [−π/2,+π/2]}, where r is arc length and ϕ
represents the angle between the outgoing velocity vector and the inward pointing
normal vector to ∂D1. It follows that Ω may be realized as the disjoint union
of a finite number of rectangles and cylinders. The cylinders correspond to fixed
scatterers with smooth boundary placed inside the gas container. If F : Ω 	 is the
collision map, i.e. the return map to the collision cross-section, then F preserves the
projected probability measure ν, which has the density dν = cosϕdϕ dr/(2 |∂D1|).
Here |∂D1| is the length of ∂D1.
We suppose that the flow is ergodic, and so F is an invertible, ergodic mea-
sure preserving transformation. Because ∂D1 is piecewise C3, F is piecewise C2,
although it does have discontinuities and unbounded derivatives near discontinu-
ities corresponding to grazing collisions. Because of our assumptions on D1, the
free flight times and the curvature of ∂D1 are uniformly bounded. It follows that
if x /∈ ∂Ω ∪ F−1(∂Ω), then F is differentiable at x, and
‖DF (x)‖ ≤ const
cosϕ(Fx)
, (4.2)
where ϕ(Fx) is the value of the ϕ coordinate at the image of x.
Following the ideas in Section 4.5, we induce F on the subspace Ω̂ of Ω cor-
responding to collisions with the (immobile) piston. We denote the induced map
by F̂ and the induced measure by ν̂. We parameterize Ω̂ by {(r, ϕ) : 0 ≤ r ≤
ℓ, ϕ ∈ [−π/2,+π/2]}. As νΩ̂ = ℓ/ |∂D1|, it follows that ν̂ has the density
dν̂ = cosϕdϕ dr/(2ℓ).
For x ∈ Ω, define ζx to be the free flight time, i.e. the time it takes the billiard
particle traveling at speed
2E1 to travel from x to Fx. If x /∈ ∂Ω ∪ F−1(∂Ω),
‖Dζ(x)‖ ≤ const
cosϕ(Fx)
. (4.3)
Santaló’s formula [San76, Che97] tells us that
Eνζ =
π |D1|
|v1| |∂D1|
. (4.4)
If ζ̂ : Ω̂ → R is the free flight time between collisions with the piston, then it
follows from Proposition 4.5.1 that
Eν̂ ζ̂ =
π |D1|
|v1| ℓ
. (4.5)
The expected value of
∣ when the left gas particle collides with the (immo-
bile) piston is given by
∣ = Eν̂
2E1 cosϕ =
∫ +π/2
cos2 ϕdϕ =
. (4.6)
We wish to compute limt→∞ t
∣2v⊥1 (s)
∣ δq⊥1 (s)=Qds, the time average of the
change in momentum of the left gas particle when it collides with the piston. If this
limit exists and is equal for almost every initial condition of the left gas particle,
then it makes sense to define the pressure inside D1 to be this quantity divided
by ℓ. Because the collisions are hard-core, we cannot directly apply Birkhoff’s
Ergodic Theorem to compute this limit. However, we can compute this limit by
using the map F̂ .
Lemma 4.2.1. If the billiard flow y(t) is ergodic, then for µ− a.e. y ∈ M1,
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds =
2 |D1(Q)|
Proof. Because the billiard flow may be viewed as a suspension flow over the
collision cross-section with ζ as the height function, it suffices to show that the
convergence takes place for ν̂ − a.e. x ∈ Ω̂. For an initial condition x ∈ Ω̂, define
N̂t(x) = N̂t = #
s ∈ (0, t] : y(s, x) ∈ Ω̂
. By the Poincaré Recurrence Theorem,
N̂t → ∞ as t→ ∞, ν̂ − a.e.
n=0 ζ̂(F̂
∣ (F̂ nx) ≤ 1
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds
≤ N̂t
∑N̂t−1
n=0 ζ̂(F̂
∣ (F̂ nx),
and so the result follows from Birkhoff’s Ergodic Theorem and Equations (4.5)
and (4.6).
Corollary 4.2.2. If the billiard flow y(t) is ergodic, then for each δ > 0,
y ∈ M1 :
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds−
2 |D1(Q)|
→ 0 as t→ ∞.
4.2.2 Analysis of collisions
In this section, we return to studying our piston system when ε > 0. We will
examine what happens when a particle collides with the piston. For convenience,
we will only examine in detail collisions between the piston and the left gas particle.
Collisions with the right gas particle can be handled similarly.
When the left gas particle collides with the piston, v⊥1 and V instantaneously
change according to the laws of elastic collisions:
1−M 2M
2 M − 1
In our coordinates, this becomes
1 + ε2
ε2 − 1 2ε
2ε 1− ε2
. (4.7)
Recalling that v1,W = O(1), we find that to first order in ε,
v⊥+1 = −v⊥−1 +O(ε), W+ =W− +O(ε). (4.8)
Observe that a collision can only take place if v⊥−1 > εW
−. In particular, v⊥−1 >
2Emax. Thus, either v
1 > 0 or v
1 = O(ε). By expanding Equation (4.7)
to second order in ε, it follows that
E+1 − E−1 = −2εW
∣+O(ε2),
W+ −W− = +2ε
∣+O(ε2).
(4.9)
Note that it is immaterial whether we use the pre-collision or post-collision values
of W and
∣ on the right hand side of Equation (4.9), because any ambiguity
can be absorbed into the O(ε2) term.
It is convenient for us to define a “clean collision” between the piston and the
left gas particle:
Definition 4.2.1. The left gas particle experiences a clean collision with the
piston if and only if v⊥−1 > 0 and v
1 < −ε
2Emax.
In particular, after a clean collision, the left gas particle will escape from the
piston, i.e. the left gas particle will have to move into the region q⊥1 ≤ 0 before
it can experience another collision with the piston. It follows that there exists
a constant C1 > 0, which depends on the set V, such that for all ε sufficiently
small, so long as Q ≥ Qmin and
∣ > εC1 when q
1 ∈ [Qmin, Q], then the
left gas particle will experience only clean collisions with the piston, and the time
between these collisions will be greater than 2Qmin/(
2Emax). (Note that when we
write expressions such as q⊥1 ∈ [Qmin, Q], we implicitly mean that q1 is positioned
inside the “tube” discussed at the beginning of Section 4.1.) One can verify that
C1 = 5
2Emax would work.
Similarly, we can define clean collisions between the right gas particle and
the piston. We assume that C1 was chosen sufficiently large such that for all ε
sufficiently small, so long as Q ≤ Qmax and
∣ > εC1 when q
2 ∈ [Q,Qmax], then
the right gas particle will experience only clean collisions with the piston.
Now we define three more stopping times, which are functions of the initial
conditions in phase space.
T ′ε = inf{τ ≥ 0 : Qmin ≤ q⊥1,ε(τ/ε) ≤ Qε(τ/ε) ≤ Qmax and
∣v⊥1,ε(τ/ε)
∣ ≤ C1ε},
T ′′ε = inf{τ ≥ 0 : Qmin ≤ Qε(τ/ε) ≤ q⊥2,ε(τ/ε) ≤ Qmax and
∣v⊥2,ε(τ/ε)
∣ ≤ C1ε},
T̃ε =T ∧ Tε ∧ T ′ε ∧ T ′′ε
Define H(z) by
H(z) =
∣ δq⊥1 =Q − 2
∣ δq⊥2 =Q
∣ δq⊥1 =Q
∣ δq⊥2 =Q
Here we make use of Dirac delta functions. All integrals involving these delta
functions may be replaced by sums.
The following lemma is an immediate consequence of Equation (4.9) and the
above discussion:
Lemma 4.2.3. If 0 ≤ t1 ≤ t2 ≤ T̃ε/ε, the piston experiences O((t2 − t1) ∨ 1)
collisions with gas particles in the time interval [t1, t2], all of which are clean
collisions. Furthermore,
hε(t2)− hε(t1) = O(ε) + ε
H(zε(s))ds.
Here any ambiguities arising from collisions occurring at the limits of integration
can be absorbed into the O(ε) term.
4.2.3 Another heuristic derivation of the averaged equa-
The following heuristic derivation of Equation (4.1) when d = 2 was suggested
in [Dol05]. Let ∆t be a length of time long enough such that the piston experiences
many collisions with the gas particles, but short enough such that the slow variables
change very little, in this time interval. From each collision with the left gas
particle, Equation (4.9) states that W changes by an amount +2ε
∣ + O(ε2),
and from Equation (4.6) the average change in W at these collisions should be
approximately επ
2E1/2 + O(ε2). From Equation (4.5) the frequency of these
collisions is approximately
2E1 ℓ/(π |D1|). Arguing similarly for collisions with
the other particle, we guess that
|D1(Q)|
− ε E2ℓ|D2(Q)|
+O(ε2).
With τ = εt as the slow time, a reasonable guess for the averaged equation for W
|D1(Q)|
− E2ℓ|D2(Q)|
Similar arguments for the other slow variables lead to the averaged equation (4.1),
and this explains why we used Pi = Ei/ |Di| for the pressure of a 2-dimensional
gas in Section 1.2.
There is a similar heuristic derivation of the averaged equation in d > 2 dimen-
sions. Compare the analogues of Equations (4.5) and (4.6) in Subsection 4.4.2.
4.2.4 A priori estimate on the size of a set of bad initial
conditions
In this section, we give an a priori estimate on the size of a set of initial conditions
that should not give rise to orbits for which sup0≤τ≤T∧Tε
∣hε(τ/ε)− h̄(τ)
∣ is small.
In particular, when proving Theorem 4.1.1, it is convenient to focus on orbits that
only contain clean collisions with the piston. Thus, we show that P{T̃ε < T ∧ Tε}
vanishes as ε → 0. At first, this result may seem surprising, since P{T ′ε ∧ T ′′ε =
0} = O(ε), and one would expect ∪T/εt=0zε(−t){T ′ε ∧ T ′′ε = 0} to have a size of order
1. However, the rate at which orbits escape from {T ′ε ∧ T ′′ε = 0} is very small, and
so we can prove the following:
Lemma 4.2.4.
P{T̃ε < T ∧ Tε} = O(ε).
In some sense, this lemma states that the probability of having a gas particle
move nearly parallel to the piston’s face within the time interval [0, T/ε], when one
would expect the other gas particle to force the piston to move on a macroscopic
scale, vanishes as ε → 0. Thus, one can hope to control the occurrence of the
“nondiffusive fluctuations” of the piston described in [CD06a] on a time scale
O(ε−1).
Proof. As the left and the right gas particles can be handled similarly, it suffices
to show that P{T ′ε < T} = O(ε). Define
Bε = {z ∈ M : Qmin ≤ q⊥1 ≤ Q ≤ Qmax and
∣ ≤ C1ε}.
Then {T ′ε < T} ⊂ ∪
t=0zε(−t)Bε, and if γ = Qmin/
8Emax,
zε(−t)Bε
 = P
zε(t)Bε
 = P
Bε ∪
((zε(t)Bε)\Bε)
≤ PBε + P
T/(εγ)
zε(kγ)
(zε(t)Bε)\Bε
≤ PBε +
(zε(t)Bε)\Bε
Now PBε = O(ε), so if we can show that P (
t=0(zε(t)Bε)\Bε) = O(ε2), then it
will follow that P{T ′ε < T} = O(ε).
If z ∈
t=0(zε(t)Bε)\Bε, it is still true that
∣ = O(ε). This is because
changes by at most O(ε) at the collisions, and if a collision forces
∣ > C1ε, then
the gas particle must escape to the region q⊥1 ≤ 0 before v⊥1 can change again, and
this will take time greater than γ. Furthermore, if z ∈
t=0(zε(t)Bε)\Bε, then at
least one of the following four possibilities must hold:
∣q⊥1 −Qmin
∣ ≤ O(ε),
• |Q−Qmin| ≤ O(ε),
• |Q−Qmax| ≤ O(ε),
∣Q− q⊥1
∣ ≤ O(ε).
It follows that P (
t=0(zε(t)Bε)\Bε) = O(ε2). For example,
1{|v⊥1 |≤O(ε), |q⊥1 −Qmin|≤O(ε)}dP
= const
{Emin≤W 2/2+v21/2+v22/2≤Emax}
1{|v⊥1 |≤O(ε)}dWdv
{Q∈[0,1], q1∈D1, q2∈D2}
1{|q⊥1 −Qmin|≤O(ε)}dQdq
= O(ε2).
4.3 Proof of the main result for two-dimensional
gas containers with only one gas particle on
each side
As in Section 4.2, we continue with the case when d = 2 and there is only one gas
particle on either side of the piston.
4.3.1 Main steps in the proof of convergence in probability
By Lemma 4.2.4, it suffices to show that sup0≤τ≤T̃ε
∣hε(τ/ε)− h̄(τ)
∣→ 0 in prob-
ability as ε =M−1/2 → 0. Several of the ideas in the steps below were inspired by
a recent proof of Anosov’s averaging theorem for smooth systems that is due to
Dolgopyat [Dol05].
Step 1: Reduction using Gronwall’s Inequality. Observe that h̄(τ) satisfies
the integral equation
h̄(τ)− h̄(0) =
H̄(h̄(σ))dσ,
while from Lemma 4.2.3,
hε(τ/ε)− hε(0) = O(ε) + ε
∫ τ/ε
H(zε(s))ds
= O(ε) + ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds+
H̄(hε(σ/ε))dσ
for 0 ≤ τ ≤ T̃ε. Define
eε(τ) = ε
∫ τ/ε
H(zε(s))− H̄(hε(s))ds.
It follows from Gronwall’s Inequality that
0≤τ≤T̃ε
∣hε(τ/ε)− h̄(τ)
O(ε) + sup
0≤τ≤T̃ε
|eε(τ)|
eLip(H̄|V)T . (4.10)
Gronwall’s Inequality is usually stated for continuous paths, but the standard
proof (found in [SV85]) still works for paths that are merely integrable, and
∣hε(τ/ε)− h̄(τ)
∣ is piecewise smooth.
Step 2: Introduction of a time scale for ergodization. Let L(ε) be a
real valued function such that L(ε) → ∞, but L(ε) ≪ log ε−1, as ε → 0. In
Section 4.3.2 we will place precise restrictions on the growth rate of L(ε). Think
of L(ε) as being a time scale that grows as ε → 0 so that ergodization, i.e. the
convergence along an orbit of a function’s time average to a space average, can
take place. However, L(ε) doesn’t grow too fast, so that on this time scale zε(t)
essentially stays on the submanifold {h = hε(0)}, where we have our ergodicity
assumption. Set tk,ε = kL(ε), so that
0≤τ≤T̃ε
|eε(τ)| ≤ O(εL(ε)) + ε
εL(ε)
∫ tk+1,ε
H(zε(s))− H̄(hε(s))ds
. (4.11)
Step 3: A splitting according to particles. Now H(z) − H̄(h(z)) divides
into two pieces, each of which depends on only one gas particle when the piston
is held fixed:
H(z)− H̄(h(z)) =
∣ δq⊥1 =Q −
|D1(Q)|
∣ δq⊥1 =Q +
|D1(Q)|
|D2(Q)| − 2
∣ δq⊥2 =Q
− WE2ℓ|D2(Q)| + 2W
∣ δq⊥2 =Q
We will only deal with the piece depending on the left gas particle, as the right
particle can be handled similarly. Define
G(z) =
∣ δq⊥1 =Q, Ḡ(h) =
2 |D1(Q)|
. (4.12)
Returning to Equation (4.11), we see that in order to prove Theorem 4.1.1, it
suffices to show that both
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
εL(ε)
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
converge to 0 in probability as ε→ 0.
Step 4: A splitting for using the triangle inequality. Now we let zk,ε(s)
be the orbit of the ε = 0 Hamiltonian vector field satisfying zk,ε(tk,ε) = zε(tk,ε).
Set hk,ε(t) = h(zk,ε(t)). Observe that hk,ε(t) is independent of t.
We emphasize that so long as 0 ≤ t ≤ T̃ε/ε, the times between collisions of a
specific gas particle and piston are uniformly bounded greater than 0, as explained
before Lemma 4.2.3. It follows that, so long as tk+1,ε ≤ T̃ε/ε,
tk,ε≤t≤tk+1,ε
|hk,ε(t)− hε(t)| = O(εL(ε)). (4.13)
This is because the slow variables change by at most O(ε) at collisions, and
dQε/dt = O(ε).
Also,
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
= O(εL(ε)2) +Wk,ε(tk,ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds,
and so
εL(ε)
∫ tk+1,ε
Wε(s)
G(zε(s))− Ḡ(hε(s))
≤ O(εL(ε)) + ε const
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
Thus, in order to prove Theorem 4.1.1, it suffices to show that
εL(ε)
∫ tk+1,ε
G(zε(s))− Ḡ(hε(s))ds
εL(ε)
|Ik,ε|+ |IIk,ε|+ |IIIk,ε|
converges to 0 in probability as ε → 0, where
Ik,ε =
∫ tk+1,ε
G(zε(s))−G(zk,ε(s))ds,
IIk,ε =
∫ tk+1,ε
G(zk,ε(s))− Ḡ(hk,ε(s))ds,
IIIk,ε =
∫ tk+1,ε
Ḡ(hk,ε(s))− Ḡ(hε(s))ds.
The term IIk,ε represents an “ergodicity term” that can be controlled by our
assumptions on the ergodicity of the flow z0(t), while the terms Ik,ε and IIIk,ε
represent “continuity terms” that can be controlled by controlling the drift of
zε(t) from zk,ε(t) for tk,ε ≤ t ≤ tk+1,ε.
Step 5: Control of drift from the ε = 0 orbits. Now Ḡ is uniformly
Lipschitz on the compact set V, and so it follows from Equation (4.13) that
IIIk,ε = O(εL(ε)2). Thus, ε
εL(ε)
k=0 |IIIk,ε| = O(εL(ε)) → 0 as ε→ 0.
Next, we show that for fixed δ > 0, P
εL(ε)
k=0 |Ik,ε| ≥ δ
→ 0 as ε → 0.
For initial conditions z ∈ M and for integers k ∈ [0, T/(εL(ε))− 1] define
Ak,ε =
|Ik,ε| >
and k ≤ T̃ε
εL(ε)
Az,ε = {k : z ∈ Ak,ε} .
Think of these sets as describing “poor continuity” between solutions of the ε = 0
and the ε > 0 Hamiltonian vector fields. For example, roughly speaking, z ∈ Ak,ε
if the orbit zε(t) starting at z does not closely follow zk,ε(t) for tk,ε ≤ t ≤ tk+1,ε.
One can easily check that |Ik,ε| ≤ O(L(ε)) for k ≤ T̃ε/(εL(ε))− 1, and so it
follows that
εL(ε)
|Ik,ε| ≤
+O(εL(ε)#(Az,ε)).
Therefore it suffices to show that P (#(Az,ε) ≥ δ(const εL(ε))−1) → 0 as ε → 0.
By Chebyshev’s Inequality, we need only show that
EP (εL(ε)#(Az,ε)) = εL(ε)
εL(ε)
P (Ak,ε)
tends to 0 with ε.
Observe that zε(tk,ε)Ak,ε ⊂ A0,ε. In words, the initial conditions giving rise to
orbits that are “bad” on the time interval [tk,ε, tk+1,ε], moved forward by time tk,ε,
are initial conditions giving rise to orbits which are “bad” on the time interval
[t0,ε, t1,ε]. Because the flow zε(·) preserves the measure, we find that
εL(ε)
εL(ε)
P (Ak,ε) ≤ constP (A0,ε).
To estimate P (A0,ε), it is convenient to use a different probability measure,
which is uniformly equivalent to P on the set {z ∈ M : h(z) ∈ V} ⊃ {T̃ε ≥ εL(ε)}.
We denote this new probability measure by P f , where the f stands for “factor.”
If we choose coordinates on M by using h and the billiard coordinates on the two
gas particles, then P f is defined on M by dP f = dh dµ1h dµ2h, where dh represents
the uniform measure on V ⊂ R4, and the factor measure dµih represents the in-
variant billiard measure of the ith gas particle coordinates for a fixed value of the
slow variables. One can verify that 1{h(z)∈V}dP ≤ const dP f , but that P f is not
invariant under the flow zε(·) when ε > 0.
We abuse notation, and consider µ1h to be a measure on the left particle’s initial
billiard coordinates once h and the initial coordinates of the right gas particle are
fixed. In this context, µ1h is simply the measure µ from Subsection 4.2.1. Then
P f(A0,ε)
dh dµ2h · µ1h
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
and we must show that the last term tends to 0 with ε. By the Bounded Con-
vergence Theorem, it suffices to show that for almost every h ∈ V and initial
condition for the right gas particle,
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
→ 0 as ε→ 0.
(4.14)
Note that if G were a smooth function and zε(·) were the flow of a smooth
family of vector fields Z(z, ε) that depended smoothly on ε, then from Gronwall’s
Inequality, it would follow that sup0≤t≤L(ε) |zε(t)− z0(t)| ≤ O(εL(ε)eLip(Z)L(ε)). If
this were the case, then
L(ε)−1
∫ L(ε)
G(zε(s))−G(z0(s))ds
= O(εL(ε)eLip(Z)L(ε)),
which would tend to 0 with ε. Thus, we need a Gronwall-type inequality for billiard
flows. We obtain the appropriate estimates in Section 4.3.2.
Step 6: Use of ergodicity along fibers to control IIk,ε. All that remains
to be shown is that for fixed δ > 0, P
εL(ε)
k=0 |IIk,ε| ≥ δ
→ 0 as ε→ 0.
For initial conditions z ∈ M and for integers k ∈ [0, T/(εL(ε))− 1] define
Bk,ε =
|IIk,ε| >
and k ≤ T̃ε
εL(ε)
Bz,ε = {k : z ∈ Bk,ε} .
Think of these sets as describing “bad ergodization.” For example, roughly speak-
ing, z ∈ Bk,ε if the orbit zε(t) starting at z spends the time between tk,ε and tk+1,ε
in a region of phase space where the function G(·) is “poorly ergodized” on the
time scale L(ε) by the flow z0(t) (as measured by the parameter δ/2T ). Note that
G(z) =
∣ δq⊥1 =Q is not really a function, but that we may still speak of the
convergence of t−1
G(z0(s))ds as t → ∞. As we showed in Lemma 4.2.1, the
limit is Ḡ(h0) for almost every initial condition.
Proceeding as in Step 5 above, we find that it suffices to show that for almost
every h ∈ V,
G(z0(s))ds− Ḡ(h0(0))
→ 0 as t→ ∞.
But this is simply a question of examining billiard flows, and it follows immediately
from Corollary 4.2.2 and our Main Assumption.
4.3.2 A Gronwall-type inequality for billiards
We begin by presenting a general version of Gronwall’s Inequality for billiard
maps. Then we will show how these results imply the convergence required in
Equation (4.14).
Some inequalities for the collision map
In this section, we consider the value of the slow variables to be fixed at h0 ∈ V.
We will use the notation and results presented in Section 4.2.1, but because the
value of the slow variables is fixed, we will omit it in our notation.
Let ρ, γ, and λ satisfy 0 < ρ≪ γ ≪ 1 ≪ λ <∞. Eventually, these quantities
will be chosen to depend explicitly on ε, but for now they are fixed.
Recall that the phase space Ω for the collision map F is a finite union of
disjoint rectangles and cylinders. Let d(·, ·) be the Euclidean metric on connected
components of Ω. If x and x′ belong to different components, then we set d(x, x′) =
∞. The invariant measure ν satisfies ν < const · (Lebesgue measure). For A ⊂ Ω
and a > 0, let Na(A) = {x ∈ Ω : d(x,A) < a} be the a-neighborhood of A.
For x ∈ Ω let xk(x) = xk = F kx, k ≥ 0, be its forward orbit. Suppose x /∈ Cγ,λ,
where
Cγ,λ =
∪λk=0F−kNγ(∂Ω)
∪λk=0F−kNγ(F−1Nγ(∂Ω))
Thus for 0 ≤ k ≤ λ, xk is well defined, and from Equation (4.2) it satisfies
d(x′, xk) ≤ γ ⇒ d(Fx′, xk+1) ≤
const
d(x′, xk). (4.15)
Next, we consider any ρ-pseudo-orbit x′k obtained from x by adding on an
error of size ≤ ρ at each application of the map, i.e. d(x′0, x0) ≤ ρ, and for k ≥ 1,
d(x′k, Fx
k−1) ≤ ρ. Provided d(xj , x′j) < γ for each j < k, it follows that
d(xk, x
k) ≤ ρ
const
≤ const ρ
const
. (4.16)
In particular, if ρ, γ, and λ were chosen such that
const ρ
const
< γ, (4.17)
then Equation (4.16) will hold for each k ≤ λ. We assume that Equation (4.17)
is true. Then we can also control the differences in elapsed flight times using
Equation (4.3):
|ζxk − ζx′k| ≤
const ρ
const
. (4.18)
It remains to estimate the size νCγ,λ of the set of x for which the above estimates
do not hold. Using Lemma 4.3.1 below,
νCγ,λ ≤ (λ+ 1)
νNγ(∂Ω) + νNγ(F−1Nγ(∂Ω))
≤ O(λ(γ + γ1/3)) = O(λγ1/3).
(4.19)
Lemma 4.3.1. As γ → 0,
νNγ(F−1Nγ(∂Ω)) = O(γ1/3).
This estimate is not necessarily the best possible. For example, for dispersing
billiard tables, where the curvature of the boundary is positive, one can show that
νNγ(F−1Nγ(∂Ω)) = O(γ). However, the estimate in Lemma 4.3.1 is general and
sufficient for our needs.
Proof. First, we note that it is equivalent to estimate νNγ(FNγ(∂Ω)), as F has
the measure-preserving involution I(r, ϕ) = (r,−ϕ), i.e. F−1 = I ◦F ◦I [CM06b].
Fix α ∈ (0, 1/2), and cover Nγ(∂Ω) with O(γ−1) starlike sets, each of diameter
no greater than O(γ). For example, these sets could be squares of side length γ.
Enumerate the sets as {Ai}. Set G = {i : FAi ∩ Nγα(∂Ω) = ∅}.
If i ∈ G, F |Ai is a diffeomorphism satisfying ‖DF |Ai‖ ≤ O(γ−α). See Equa-
tion (4.2). Thus diameter (FAi) ≤ O(γ1−α), and so
diameter (Nγ(FAi)) ≤ O(γ1−α).
Hence νNγ(FAi) ≤ O(γ2(1−α)), and νNγ(∪i∈GFAi) ≤ O(γ1−2α).
If i /∈ G, Ai ∩ F−1(Nγα(∂Ω)) 6= ∅. Thus Ai might be cut into many pieces
by F−1(∂Ω), but each of these pieces must be mapped near ∂Ω. In fact, FAi ⊂
NO(γα)(∂Ω). This is because outside F−1(Nγα(∂Ω)), ‖DF‖ ≤ O(γ−α), and so
points in FAi are no more than a distance O(γ/γα) away from Nγα(∂Ω), and γ <
γ1−α < γα. It follows that Nγ(FAi) ⊂ NO(γα)(∂Ω), and νNO(γα)(∂Ω) = O(γα).
Thus νNγ(F−1Nγ(∂Ω)) = O(γ1−2α + γα), and we obtain the lemma by taking
α = 1/3.
Application to a perturbed billiard flow
Returning to the end of Step 5 in Section 4.3.1, let the initial conditions of the slow
variables be fixed at h0 = (Q0,W0, E1,0, E2,0) ∈ V throughout the remainder of this
section. We can assume that the billiard dynamics of the left gas particle in D1(Q0)
are ergodic. Also, fix a particular value of the initial conditions for the right gas
particle for the remainder of this section. Then zε(t) and T̃ε may be thought of as
random variables depending on the left gas particle’s initial conditions y ∈ M1.
Now if hε(t) = (Qε(t),Wε(t), E1,ε(t), E2,ε(t)) denotes the actual motions of the slow
variables when ε > 0, it follows from Equation (4.13) that, provided εL(ε) ≤ T̃ε,
0≤t≤L(ε)
|h0 − hε(t)| = O(εL(ε)). (4.20)
Furthermore, we only need to show that
y ∈ M1 :
∫ L(ε)
G(zε(s))−G(z0(s))ds
and εL(ε) ≤ T̃ε
(4.21)
as ε→ 0, where G is defined in Equation (4.12).
For definiteness, we take the following quantities from Subsection 4.3.2 to de-
pend on ε as follows:
L(ε) = L = log log
γ(ε) = γ = e−L,
λ(ε) = λ =
ρ(ε) = ρ = const
(4.22)
The constant in the choice of ρ and ρ’s dependence on ε will be explained in the
proof of Lemma 4.3.3, which is at the end of this subsection. The other choices may
be explained as follows. We wish to use continuity estimates for the billiard map
to produce continuity estimates for the flow on the time scale L. As the divergence
of orbits should be exponentially fast, we choose L to grow sublogarithmically in
ε−1. Since from Equation (4.4) the expected flight time between collisions with
∂D1(Q0) when ε = 0 is Eνζ = π |D1(Q0)| /(
2E1,0 |∂D1(Q0)|), we expect to see
roughly λ/2 collisions on this time scale. Considering λ collisions gives us some
margin for error. Furthermore, we will want orbits to keep a certain distance,
γ, away from the billiard discontinuities. γ → 0 as ε → 0, but γ is very large
compared to the possible drift O(εL) of the slow variables on the time scale L. In
fact, for each C,m, n > 0,
= O(ε econstL2) → 0 as ε→ 0. (4.23)
Let X : M1 → Ω be the map taking y ∈ M1 to x = X(y) ∈ Ω, the location
of the billiard orbit of y in the collision cross-section that corresponds to the most
recent time in the past that the orbit was in the collision cross-section. We consider
the set of initial conditions
Eε = X−1(Ω\Cγ,λ)
x ∈ Ω :
ζ(F kx) > L
Now from Equations (4.19) and (4.22), νCγ,λ → 0 as ε → 0. Furthermore, by the
ergodicity of F ,
x ∈ Ω :
ζ(F kx) ≤ L
x ∈ Ω : λ−1
ζ(F kx) ≤ Eνζ/2
as ε → 0. But because the free flight time is bounded above, µX−1 ≤ const ·ν, and
so µEε → 1 as ε→ 0. Hence, the convergence in Equation (4.21) and the conclusion
of the proof in Section 4.3.1 follow from the lemma below and Equation (4.23).
Lemma 4.3.2 (Analysis of deviations along good orbits). As ε→ 0,
y∈Eε∩{εL≤T̃ε}
G(zε(s))−G(z0(s))ds
const
+O(L−1) → 0.
Proof. Fix a particular value of y ∈ Eε∩
εL ≤ T̃ε
. For convenience, suppose that
y = X(y) = x ∈ Ω. Let y0(t) denote the time evolution of the billiard coordinates
for the left gas particle when ε = 0. Then there is some N ≤ λ such that the
orbit xk = F
kx = (rk, ϕk) for 0 ≤ k ≤ N corresponds to all of the instances
(in order) when y0(t) enters the collision cross-section Ω = Ωh0 corresponding to
collisions with ∂D1(Q0) for 0 ≤ t ≤ L. We write Ωh0 to emphasize that in this
subsection we are only considering the collision cross-section corresponding to the
billiard dynamics in the domain D1(Q0) at the energy level E1,0. In particular, F
will always refer to the return map on Ωh0 .
Also, define an increasing sequence of times tk corresponding to the actual
times y0(t) enters the collision cross-section, i.e.
t0 = 0,
tk = tk−1 + ζxk−1 for k > 0.
Then xk = y0(tk). Furthermore, define inductively
N1 = inf {k > 0 : tk corresponds to a collision with the piston} ,
Nj = inf {k > Nj−1 : tk corresponds to a collision with the piston} .
Next, let yε(t) denote the time evolution of the billiard coordinates for the left
gas particle when ε > 0. We will construct a pseudo-orbit x′k,ε = (r
k,ε, ϕ
k,ε) of
points in Ωh0 that essentially track the collisions (in order) of the left gas particle
with the boundary under the dynamics of yε(t) for 0 ≤ t ≤ L.
First, define an increasing sequence of times t′k,ε corresponding to the actual
times yε(t) experiences a collision with the boundary of the gas container or the
moving piston. Define
N ′ε = sup
k ≥ 0 : t′k,ε ≤ L
N ′1,ε = inf
k > 0 : t′k,ε corresponds to a collision with the piston
N ′j,ε = inf
k > N ′j−1,ε : t
k,ε corresponds to a collision with the piston
Because L ≤ T̃ε(y)/ε, we know that as long as N ′j+1,ε ≤ N ′ε, then N ′j+1,ε−N ′j,ε ≥ 2.
See the discussion in Subsection 4.2.2. Then we define x′k,ε ∈ Ωh0 by
x′k,ε =
k,ε) if k /∈
N ′j,ε
F−1x′k+1,ε if k ∈
N ′j,ε
Lemma 4.3.3. Provided ε is sufficiently small, the following hold for each k ∈
[0, N∧N ′ε). Furthermore, the requisite smallness of ε and the sizes of the constants
in these estimates may be chosen independent of the initial condition y ∈ Eε ∩
εL ≤ T̃ε
and of k:
(a) x′k,ε is well defined. In particular, if k /∈
N ′j,ε
, yε(t
k,ε) corresponds to a
collision point on ∂D1(Q0), and not to a collision point on a piece of ∂D to
the right of Q0.
(b) If k > 0 and k /∈
N ′j,ε
, then x′k,ε = Fx
k−1,ε.
(c) If k > 0 and k ∈
N ′j,ε
, then d(x′k,ε, Fx
k−1,ε) ≤ ρ and the ϕ coordinate of
k,ε) satisfies ϕ(yε(t
k,ε)) = ϕ
k,ε +O(ε).
(d) d(xk, x
k,ε) ≤ const ρ(const/γ)k .
(e) k = N ′j,ε if and only if k = Nj.
(f) If k > 0, t′k,ε − t′k−1,ε = tk − tk−1 +O(ρ (const/γ)
We defer the proof of Lemma 4.3.3 until the end of this subsection. Assuming
that ε is sufficiently small for the conclusions of Lemma 4.3.3 to be valid, we
continue with the proof of Lemma 4.3.2.
Set M = N ∧ N ′ε − 1. Note that M ≤ λ ∼ L. From (f) in Lemma 4.3.3 and
Equations (4.22) and (4.23), we see that
∣tM − t′M,ε
∣t′k,ε − t′k−1,ε − (tk − tk−1)
∣ = O
constλ
→ 0 as ε → 0.
Because the flight times t′k,ε − t′k−1,ε and tk − tk−1 are uniformly bounded above,
it follows from the definitions of N and N ′ε that tM , t
M,ε ≥ L − const. But from
Subsection 4.2.2, the time between the collisions of the left gas particle with the
piston are uniformly bounded away from zero. Using (c) and Equation (4.20), it
follows that
G(zε(s))−G(z0(s))ds
= O(L−1) +
k∈{Nj :Nj≤M}
2E1,0 cosϕk −
2E1,ε(t
k,ε) cos(ϕ
k,ε +O(ε))
= O(L−1) +
k∈{Nj :Nj≤M}
2E1,0 cosϕk −
2E1,0 cosϕ
k,ε +O(εL)
= O(L−1) +O(εL2) +
2E1,0
k∈{Nj :Nj≤M}
∣cosϕk − cosϕ′k,ε
But using (d),
k∈{Nj :Nj≤M}
∣cosϕk − cosϕ′k,ε
O(ρ(const/γ)k) = O(ρ(const/γ)λ).
Since εL2 = O(ρ(const/γ)λ), this finishes the proof of Lemma 4.3.2.
Proof of Lemma 4.3.3. The proof is by induction. We take ε to be so small that
Equation (4.17) is satisfied. This is possible by Equation (4.23).
It is trivial to verify (a)-(f) for k = 0. So let 0 < l < N ∧N ′ε, and suppose that
(a)-(f) have been verified for all k < l. We have three cases to consider:
Case 1: l − 1 and l /∈
N ′j,ε
In this case, verifying (a)-(f) for k = l is a relatively straightforward application
of the machinery developed in Subsection 4.3.2, because for t′l−1,ε ≤ t ≤ t′l,ε, yε(t)
traces out the billiard orbit between x′l−1,ε and x
l,ε corresponding to free flight in
the domain D1(Q0). We make only two remarks.
First, as long as ε is sufficiently small, it really is true that x′l,ε = yε(t
corresponds to a true collision point on ∂D1(Q0). Indeed, if this were not the
case, then it must be that Qε(t
l,ε) > Q0, and yε(t
l,ε) would have to correspond
to a collision with the side of the “tube” to the right of Q0. But then x
l,ε =
Fx′l−1,ε ∈ Ωh0 would correspond to a collision with an immobile piston at Q0 and
would satisfy d(xk, x
k,ε) ≤ const ρ(const/γ)k ≤ const ρ(const/γ)λ = o(γ), using
Equations (4.16) and (4.23). But xk /∈ Nγ(∂Ωh0), and so it follows that when the
trajectory of yε(t) crosses the plane {Q = Q0}, it is at least a distance ∼ γ away
from the boundary of the face of the piston, and its velocity vector is pointed
no closer than ∼ γ to being parallel to the piston’s face. As Qε(t′l,ε) − Q0 =
O(εL) = o(γ), and it is geometrically impossible (for small ε) to construct a right
triangle whose sides s1, s2 satisfy |s1| ≥∼ γ, |s2| ≤ O(εL), with the measure of
the acute angle adjacent to s1 being greater than ∼ γ, we have a contradiction.
After crossing the plane {Q = Q0}, yε(t) must experience its next collision with
the face of the piston, which violates the fact that l /∈
N ′j,ε
Second, t′l,ε − t′l−1,ε = ζx′l−1,ε +O(εL), because v1,ε = v1,0 +O(εL). See Equa-
tion (4.20). From Equation 4.18,
∣ζxl−1 − ζx′l−1,ε
∣ ≤ O((ρ/γ) (const/γ)l−1). As
tl − tl−1 = ζxl−1 and εL = O((ρ/γ) (const/γ)l−1), we obtain (f).
Case 2: There exists i such that l = N ′i,ε:
For definiteness, we suppose that Qε(t
l,ε) ≥ Q0, so that the left gas particle collides
with the piston to the right of Q0. The case when Qε(t
l,ε) ≤ Q0 can be handled
similarly.
We know that xl−1, xl, xl+1 /∈ Nγ(∂Ωh0)∪Nγ(F−1Nγ(∂Ωh0)). Using the induc-
tive hypothesis and Equation (4.16), we can define
x′′l,ε = Fx
l−1,ε, x
l+1,ε = F
2x′l−1,ε,
and d(xl, x
l,ε) ≤ const ρ(const/γ)l, d(xl+1, x′′l+1,ε) ≤ const ρ(const/γ)l+1. In partic-
ular, x′′l,ε and x
l+1,ε are both a distance ∼ γ away from ∂Ωh0 . Furthermore, when
the left gas particle collides with the moving piston, it follows from Equation (4.8)
that the difference between its angle of incidence and its angle of reflection is
O(ε). Referring to Figure 4.3, this means that ϕ′l,ε = ϕ′′l,ε + O(ε). Geometric
arguments similar to the one given in Case 1 above show that the yε-trajectory
of the left gas particle has precisely one collision with the piston and no other
collisions with the sides of the gas container when the gas particle traverses the
region Q0 ≤ Q ≤ Qε(t′l,ε). Note that x′l,ε was defined to be the point in the
collision cross-section Ωh0 corresponding to the return of the yε-trajectory into
the region Q ≤ Q0. See Figure 4.3. From this figure, it is also evident that
d(r′l,ε, r
l,ε) ≤ O(εL/γ). Thus d(x′′l,ε, x′l,ε) = O(εL/γ), and this explains the choice
of ρ(ε) in Equation (4.22).
From the above discussion and the machinery of Subsection 4.3.2, (a)-(e) now
follow readily for both k = l and k = l + 1. Furthermore, property (f) follows in
much the same manner as it did in Case 1 above. However, one should note that
t′l,ε− t′l−1,ε = ζx′l−1,ε+O(εL)+O(εL/γ) and t′l+1,ε− t′l,ε = ζx′l,ε+O(εL)+O(εL/γ),
because of the extra distance O(εL/γ) that the gas particle travels to the right of
Q0. But εL/γ = O((ρ/γ) (const/γ)l−1), and so property (f) follows.
Case 3: There exists i such that l − 1 = N ′i,ε:
As mentioned above, the inductive step in this case follows immediately from our
analysis in Case 2.
r−coordinate Q0 Qε(t′l,ε)
D1(Q0)
γ/2 γ/2
O(εL)
O(εL/γ)
r′′l,ε
ϕ′′l,ε
r′l,ε
ϕ′l,ε
r′l−1,ε
ϕ′l−1,ε
r′l+1,ε
r′′l+1,ε
Figure 4.3: An analysis of the divergences of orbits when ε > 0 and the left
gas particle collides with the moving piston to the right of Q0. Note that the
dimensions are distorted for visual clarity, but that εL and εL/γ are both o(γ) as
ε→ 0.
Furthermore, ϕ′′l,ε ∈ (−π/2 + γ/2, π/2 − γ/2) and ϕ′l,ε = ϕ′′l,ε + O(ε), and so
r′l,ε = r
l,ε + O(εL/γ). In particular, the yε-trajectory of the left gas particle has
precisely one collision with the piston and no other collisions with the sides of the
gas container when the gas particle traverses the region Q0 ≤ Q ≤ Qε(t′l,ε)
4.4 Generalization to a full proof
of Theorem 4.1.1
It remains to generalize the proof in Sections 4.2 and 4.3 to the cases when n1, n2 ≥
1 and d = 3.
4.4.1 Multiple gas particles on each side of the piston
When d = 2, but n1, n2 ≥ 1, only minor modifications are necessary to generalize
the proof above. As in Subsection 4.2.2, one defines a stopping time T̃ε satisfying
T̃ε < T ∧ Tε
= O(ε) such that for 0 ≤ t ≤ T̃ε/ε, gas particles will only
experience clean collisions with the piston.
Next, define H(z) by
H(z) =
∣v⊥1,j
∣ δq⊥1,j=Q
∣v⊥2,j
∣ δq⊥2,j=Q
∣v⊥1,j
∣ δq⊥1,j=Q
∣v⊥2,j
∣ δq⊥2,j=Q
It follows that for 0 ≤ t ≤ T̃ε/ε, hε(t)−hε(0) = O(ε)+ε
H(zε(s))ds. From here,
the rest of the proof follows the same steps made in Subsection 4.3.1. We note
that at Step 3, we find that H(z) − H̄(h(z)) divides into n1 + n2 pieces, each of
which depends on only one gas particle when the piston is held fixed.
4.4.2 Three dimensions
The proof of Theorem 4.1.1 in d = 3 dimensions is essentially the same as the proof
in two dimensions given above. The principal differences are due to differences in
the geometry of billiards. We indicate the necessary modifications.
In analogy with Section 4.2.1, we briefly summarize the necessary facts for the
billiard flows of the gas particles when M = ∞ and the slow variables are held
fixed at a specific value h ∈ V. As before, we will only consider the motions of one
gas particle moving in D1. Thus we consider the billiard flow of a point particle
moving inside the domain D1 at a constant speed
2E1. Unless otherwise noted,
we use the notation from Section 4.2.1.
The billiard flow takes place in the five-dimensional space M1 = {(q1, v1) ∈
T D1 : q1 ∈ D1, |v1| =
2E1}/ ∼. Here the quotient means that when q1 ∈ ∂D1,
we identify velocity vectors pointing outside of D1 with those pointing inside D1
by reflecting orthogonally through the tangent plane to ∂D1 at q1. The billiard
flow preserves Liouville measure restricted to the energy surface. This measure
has the density dµ = dq1dv1/(8πE1 |D1|). Here dq1 represents volume on R3, and
dv1 represents area on S
v1 ∈ R3 : |v1| =
The collision cross-section Ω = {(q1, v1) ∈ T D1 : q1 ∈ ∂D1, |v1| =
2E1}/ ∼ is
properly thought of as a fiber bundle, whose base consists of the smooth pieces of
∂D1 and whose fibers are the set of outgoing velocity vectors at q1 ∈ ∂D1. This and
other facts about higher-dimensional billiards, with emphasis on the dispersing
case, can be found in [BCST03]. For our purposes, Ω can be parameterized as
follows. We decompose ∂D1 into a finite union ∪jΓj of pieces, each of which
is diffeomorphic via coordinates r to a compact, connected subset of R2 with
a piecewise C3 boundary. The Γj are nonoverlapping, except possibly on their
boundaries. Next, if (q1, v1) ∈ Ω and v1 is the outward going velocity vector, let
v̂ = v1/ |v1|. Then Ω can be parameterized by {x = (r, v̂)}. It follows that Ω it is
diffeomorphic to ∪jΓj × S2+, where S2+ is the upper unit hemisphere, and by ∂Ω
we mean the subset diffeomorphic to (∪j∂Γj × S2+)
(∪jΓj × ∂S2+). If x ∈ Ω,
we let ϕ ∈ [0, π/2] represent the angle between the outgoing velocity vector and
the inward pointing normal vector n to ∂D1, i.e. cosϕ = 〈v̂, n〉. Note that we no
longer allow ϕ to take on negative values. The return map F : Ω 	 preserves the
projected probability measure ν, which has the density dν = cosϕdv̂ dr/(π |∂D1|).
Here |∂D1| is the area of ∂D1.
F is an invertible, measure preserving transformation that is piecewise C2.
Because of our assumptions on D1, the free flight times and the curvature of ∂D1
are uniformly bounded. The bound on ‖DF (x)‖ given in Equation (4.2) is still
true. A proof of this fact for general three-dimensional billiard tables with finite
horizon does not seem to have made it into the literature, although see [BCST03]
for the case of dispersing billiards. For completeness, we provide a sketch of a
proof for general billiard tables in Section 4.6.
We suppose that the billiard flow is ergodic, so that F is ergodic. Again, we
induce F on the subspace Ω̂ of Ω corresponding to collisions with the (immobile)
piston to obtain the induced map F̂ : Ω̂ 	 that preserves the induced measure ν̂.
The free flight time ζ : Ω → R again satisfies the derivative bound given in
Equation (4.3). The generalized Santaló’s formula [Che97] yields
Eνζ =
4 |D1|
|v1| |∂D1|
If ζ̂ : Ω̂ → R is the free flight time between collisions with the piston, then it
follows from Proposition 4.5.1 that
Eν̂ ζ̂ =
4 |D1|
|v1| ℓ
The expected value of
∣ when the left gas particle collides with the (immo-
bile) piston is given by
∣ = Eν̂
2E1 cosϕ =
cos2 ϕdv̂1 =
As a consequence, we obtain
Lemma 4.4.1. For µ− a.e. y ∈ M1,
∣v⊥1 (s)
∣ δq⊥1 (s)=Qds =
3 |D1(Q)|
Compare the proof of Lemma 4.2.1.
With these differences in mind, the rest of the proof of Theorem 4.1.1 when
d = 3 proceeds in the same manner as indicated in Sections 4.2, 4.3 and 4.4.1
above. The only notable difference occurs in the proof of the Gronwall-type in-
equality for billiards. Due to dimensional considerations, if one follows the proof
of Lemma 4.3.1 for a three-dimensional billiard table, one finds that
νNγ(F−1Nγ(∂Ω)) = O(γ1−4α + γα).
The optimal value of α is 1/5, and so νNγ(F−1Nγ(∂Ω)) = O(γ1/5) as γ → 0.
Hence νCγ,λ = O(λγ1/5), which is a slightly worse estimate than the one in Equa-
tion (4.19). However, it is still sufficient for all of the arguments in Section 4.3.2,
and this finishes the proof.
4.5 Inducing maps on subspaces
Here we present some well-known facts on inducing measure preserving transfor-
mations on subspaces. Let F : (Ω,B, ν) 	 be an invertible, ergodic, measure
preserving transformation of the probability space Ω endowed with the σ-algebra
B and the probability measure ν. Let Ω̂ ∈ B satisfy 0 < νΩ̂ < 1. Define
R : Ω̂ → N to be the first return time to Ω̂, i.e. Rω = inf{n ∈ N : F nω ∈ Ω̂}.
Then if ν̂ := ν(· ∩ Ω̂)/νΩ̂ and B̂ := {B ∩ Ω̂ : B ∈ B}, F̂ : (Ω̂, B̂, ν̂) 	 defined
by F̂ω = FRωω is also an invertible, ergodic, measure preserving transforma-
tion [Pet83]. Furthermore Eν̂R =
Rdν̂ = (νΩ̂)−1.
This last fact is a consequence of the following proposition:
Proposition 4.5.1. If ζ : Ω → R≥0 is in L1(ν), then ζ̂ =
n=0 ζ ◦F n is in L1(ν̂),
Eν̂ ζ̂ =
Proof.
ζ ◦ F n dν̂ =
ζ ◦ F n dν =
Ω̂∩{R=k}
ζ ◦ F n dν
Fn(Ω̂∩{R=k})
ζ dν =
ζ dν,
because {F n(Ω̂ ∩ {R = k}) : 0 ≤ n < k <∞} is a partition of Ω.
4.6 Derivative bounds for the billiard map
in three dimensions
Returning to Section 4.4.2, we need to show that for a billiard table D1 ⊂ R3 with
a piecewise C3 boundary and the free flight time uniformly bounded above, the
billiard map F satisfies the following: If x0 /∈ ∂Ω ∪ F−1(∂Ω), then
‖DF (x0)‖ ≤
const
cosϕ(Fx0)
Fix x0 = (r0, v̂0) ∈ Ω, and let x1 = (r1, v̂1) = Fx0. Let Σ be the plane that
perpendicularly bisects the straight line between r0 and r1, and let r1/2 denote
the point of intersection. We consider Σ as a “transparent” wall, so that in a
neighborhood of x0, we can write F = F2 ◦ F1. Here, F1 is like a billiard map in
that it takes points (i.e. directed velocity vectors with a base) near x0 to points
with a base on Σ and a direction pointing near r1. (F1 would be a billiard map if
we reflected the image velocity vectors orthogonally through Σ.) F2 is a billiard
map that takes points in the image of F1 and maps them near x1. Let x1/2 =
F1x0 = F
2 x1. Then ‖DF (x0)‖ ≤ ‖DF1(x0)‖
∥DF2(x1/2)
It is easy to verify that ‖DF1(x0)‖ ≤ const, with the constant depending
only on the curvature of ∂D1 at r0. In other words, the constant may be chosen
independent of x0. Similarly,
∥DF−12 (x1)
∥ ≤ const. Because billiard maps pre-
serve a probability measure with a density proportional to cosϕ, detDF−12 (x1) =
cosϕ1/ cosϕ1/2 = cosϕ1. As Ω is 4-dimensional, it follows from Cramer’s Rule for
the inversion of linear transformations that
∥DF2(x1/2)
const
∥DF−12 (x1)
detDF−12 (x1)
≤ const
cosϕ1
and we are done.
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	Dedication
	Acknowledgements
	Abstract
	List of Figures
	Introduction
	The adiabatic piston
	Physical motivation for the results
	Background Averaging Material
	The averaging framework
	Some classical averaging results
	A proof of Anosov's theorem
	Moral
	Results for piston systems in one dimension
	Statement of results
	Heuristic derivation of the averaged equation for the hard core piston
	Proof of the main result for the hard core piston
	Proof of the main result for the soft core piston
	Appendix to Section 3.4
	The periodic oscillation of an adiabatic piston in two or three dimensions
	Statement of the main result
	Preparatory material concerning a two-dimensional gas container with only one gas particle on each side
	Proof of the main result for two-dimensional gas containers with only one gas particle on each side
	Generalization to a full proof of Theorem 4.1.1
	Inducing maps on subspaces
	Derivative bounds for the billiard map in three dimensions
	Bibliography
