数理解析研究所講究録 1275 巻 2002 年 141-154

141
Application of Renormalization Group Method to
Kinetic Equations: roles of initial condition
Y. Hatta and T. Kunihiro 2
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan

2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract
The s0-called renormalization group (RG) method is applied to derive Boltzmann
equation in classical mechanics and Fokker-Planck equation from the respective
microscopic equations. Utilizing the formulation of the RG method which elucidates
the important role played by the choice of the initial conditions, the general structure
and the underlying assumptions in the derivation of kinetic equations in the RG
method is clarified.

1Introduction
In [1], the reduction-theoretical aspect of the so called renormalization group (RG) method[2] and the improved formulation given in [19] were reformulated mathematically with
the notion of invariant manifolds familiar in the theory of dynamical systems [6]. The
perturbative RG method can be used to construct invariant manifolds successively as theinitial values of evolution equations using the Wilsonian RG [16, 20, 21]; the would-be
integral constants, which have one-t0-0ne correspondence with the initial values, in theunperturbed solution, constitute natural coordinates of the invariant manifold. It was also
shown that the RG equation determines the slow motion of the would-be integral constantson the invariant manifold of the dynamical system, hence areduction of evolution equationis achieved.
We apply the RG [16] method to derive and reduce kinetic equations to aslower
dynamics [2, 18, 19, 1]: the Boltzmann equation is derived from the BBGKY (BogoliubovBorn-Green-Kirkwood-Yvon) hierarchy[3] and the Fokker-Planck equation is derived fromthe Langevin equation. It seems that the basic notions to implement the reduction are
given by (1) the coarse-grained time-derivative and (2) the choice of the initial conditions
in solving the microscopic equations, respectively:(1) In an attempt to characterize hydr0-dynamical processes microscopically, Mori pointed
out that time derivatives appearing in equations which define transport coefficients are“the average” of time derivatives describing microscopic dynamics [10]. His definition of
the macroscopic derivative of an observable $F$ is,$\frac{\delta}{\delta t}\langle F\rangle(t)\equiv\frac{1}{\tau}\{\langle F\rangle(t+\tau)-\langle F\rangle(\mathrm{t})\}=\frac{1}{\tau}\int_{0}^{\tau}ds\frac{d}{ds}\langle F\rangle(t+s)$(1.1)

where is some time scale between microscopic (mean free) time and macroscopic (relaxation) time. An important point is that is finite. The idea of the coarse-graining oftime in kinetic and transport equations were first given by Kirkwood[ll]; see also [12] forarigorous formulation$\tau$$\tau$

142(2) The importance of the choice of the initial condition in the derivation of kinetic equation is noticed and emphasized in the literature[4, 5, 13, 14, 15]. For instance, Kawasaki
clarifies in an excellent monograph[15] that the initial value of the microscopic distribution function before averaging must be given solely in terms of the averaged distribution
to obtain aclosed equation for the distribution function of macroscopic slow variables,
which is equivalent to the construction of an invariant manifold mentioned above. He also
clarifies that by this initial condition, the dominating class of states (”typical states”) areselected which leads to an increase of the entropy, while exceptional states from which
the entropy would increase could be unstabilized to the dominating typical states by amechanism producing chaotic behavior.
We shall show that the straightforward application of the RG method as formulated
in $[19, 1]$ naturally leads to the choice for the initial value of the microscopic distributionfunction at an arbitrary time to be on the averaged distribution, which is an implementation of (2) in the RG method, thereby leads to time-irreversible equations even from atime-reversible equation. The averaged distribution function may be thought as an integral constant of the solution of microscopic evolution equation. The RG equation gives
the slow dynamics of the would-be initial constant, which is actually the kinetic equation
governing the averaged distribution function. It will be further shown that the averaging
as given above automatically gives rise to acoarse-graining of the time-derivative, whichis expressed with the initial time to. This shows that the initial time
has amacr0ic nature in contrast to the time appearing in the microscopic equations, which isan implementation of (1) in our method.
It should be noticed here that the RG equation has been already applied to kinetic
equations in [22]: In [22], it was ascertained that Boltzmann equation is arenormalizationgroup equation on the basis of the work by. Green[23] on the uniform system whichshows that the perturbative solution for the BBGKY hierarchy exhibit asecular term, andasketch was given to derive the Fokker-Planck equation from asimple Langevin equation
noticing again an appearance of asecular term. On the basis of these two examples, theyclaimed that all other kinetic equations are also RG equations. These models dealt in [22]will be retreated in our formulation, and implicit assumptions in their treatment will bemade explicit so that the roles of the initial conditions and the scale transformation of
the time erivative will become clear for the RG method to lead to kinetic or transport$t_{0}$$t_{0}$$\mathrm{s}\mathrm{c}\mathrm{o}\dot{\mathrm{p}}$$t$$\mathrm{M}.\mathrm{S}$equations.
In section 2, we shall deal with the Langevin equation and derive the Fokker-Planck
equation as atypical problem of dynamical reduction leading to akinetic equation in
the RG method. We shall summarize the basic structure of the reduction given by the
RG method. One will see that asimilar definition to Eq.(l.l) of the macroscopic timederivative naturally emerges in the RG method. In section 3, the Boltzmann equation
is derived from the Liouville equation of the classical mechanics; we shall clarify thedifference between the present method and the one by Bogoliubov

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2Reduction of Langevin equation to Fokker-Planck
equationIn this section, the RG equation is applied to obtain the Fokker-Planck(FP) equation[9]from the stochastic Liouville equation [25] corresponding to Langevin equation[9]. The
present derivation is thought to be atypical one for the reduction of evolution equations
appearing in non-equilibrium physics[15]. We shall clarify that the initial values of theare naturally chosen to be on thestochastic distribution function at arbitrary time
averaged distribution function for the RG equation derives the FP equation governing
the averaged distribution function. We shall also notice that the time derivative in theRG equation which will be converted to the derivative in the FP equation is with respect
to amacroscopic time, hence the coarse-graining of time is automatically built in in thepresent RG method.
$t_{0}$

2.1

From generic Langevin equation to Fokker-Planck equation

Let us consider the following generic Langevin equation withstochastic variables;$\frac{du}{dt}=h(u)+\hat{g}(u)R$$R_{i}$$(i=1,2, \ldots, n)$

being
(2.1)

,

a times matrix and $(\hat{g}(u)R)_{i}=$where $u={}^{t}(u_{1}, u_{2}, \ldots, u_{n})$ ,. We remark that the noise enters multiplicatively. Here we assume without lossof generality that the noise has the vanishing average,$h={}^{t}(h_{1}, h_{2}, \ldots, h_{n}),\hat{g}$

$n$

$n$

$\sum_{j}g_{ij}R_{j}$$\langle R(t)\rangle=0$

(2.2)

,

with respect to the noise $R$ . Let
where (C)(t) denotes the average ofdistribution function with $R(t)$ given; the continuity equation reads$\mathcal{O}(t)$$\rangle$$\frac{\partial f(u,t)}{\partial t}+\nabla_{u}\cdot(vf(u, t))=0$$f(u, \mathrm{t})$be the(2.3),

and $v=du/dt$ is the velocity of , which is given in (2.1).
where
Inserting (2.1) into (2.3), one has the Kubo’s stochastic Liouville equation[25],$u$$\nabla_{u}=\Sigma_{i}\partial/\partial u_{i}$$\frac{\partial f}{\partial t}=-\nabla_{u}\cdot[(h+\hat{g}R)f]$

(2.4)

.

Although it is arather easy task to derive the FP equation in an exact way if the noise is
Gaussian, it is formidably difficult if the noise is non-Gaussian[9]. The present approachis admittedly based on the perturbation theory and of approximate nature. Nevertheless,it will be found that the first order calculation suffices to derive the exact FP equation
when the noise in Gaussian, and furthermore that the method is applicable even to nonGaussian noises without difficulties.
The solution to (2.4) with the initial condition given at $t=t_{0}$ is formally given by [25]$\tilde{f}(u, t;t_{0})=T\exp[\int_{t_{0}}^{t}dsL(s)]\tilde{f}(u, t_{0;}t_{0})$

,

(2.5)

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$L(s)=-\nabla_{u}\cdot(h(u)+\hat{g}R(s))$

,

(2.6)

and
denotes the time ordering operator. The initial distribution
will bespecified later and found to play asignificant role in the present method.
Now we are interested in the averaged distribution function
which is defined
as an average of $f(u, t;t_{0})$ with respect to the noise $R$ , i.e.,$T$$\tilde{f}(u, t_{0};t_{0})$$\tilde{P}(u, t;t_{0})$$\tilde{P}(u, t;t_{0})=$( $T \exp[\int_{t_{0}}^{t}dsL(s)]f(u$ , to)).

We take an interaction picture dividing the “Hamiltonian”

We first define,
tonian”(2.7)

as follows;

$L$

,

$L$

$=$

$L_{0}+L_{1}$$L_{0}$$=$$-\nabla_{u}\cdot h$,$L_{1}=-\nabla_{u}\hat{g}R$

(2.8)
(2.9)

.

as the time volution operator governed by the unperturbed “Hamil-$U_{0}(t)$$L_{0}$$U_{0}(t, t_{0})=T \exp[\int_{t_{\mathrm{O}}}^{t}dsL_{0}(s)]$,(2.10)

which satisfies the evolution equation$\frac{\partial}{\partial t}U_{0}(t, t_{0})=L_{0}U_{0}(t;t_{0})$,(2.11)

with the initial condition$U_{0}(t_{0}, t_{0})=1$

Then to incorporateby$L_{1}$.

(2.12), we define another microscopic distribution function$f(u, t;t_{0})=U_{0}(t, t_{0})\rho_{1}(u, t;t_{0})$.

$\rho_{1}(u, t;t_{0})$(2.13)We remark that the initial values of andcoincides with each other, which wetake to be equal to the averaged distribution function $P(u, t_{0})$ at $t=t_{0}$ ;$f$$\rho_{1}t=t_{0}$$\tilde{f}(u, t=t_{0};t_{0})=\rho_{1}(u, t=t_{0};t_{0})=P(u, t_{0})$.

(2.14)

One will recognize that this choice of the initial condition is inevitable for the RG equation
to be identified with the Fokker-Planck equation.
One can easily verify that $(u, t;t_{0})$ is formally solved to be$\rho_{1}$[$\rho_{1}(u, t;\mathrm{t}_{0})=T\exp$ $\int_{t_{0}}^{t}ds\mathcal{L}_{1}(s;$to)]pi$(\mathrm{u}, t_{0;}t_{0})$,(2. 15)where$\mathcal{L}_{1}(t;t_{0})=U_{0}^{-1}(t, t_{0})L_{1}(t)U_{0}(t, t_{0})$,(2.10)

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is an “interaction Hamiltonian” in the “interaction picture”.
as follows,
Thus we obtains the compact form of$\tilde{P}(u, t;t_{0})$$\tilde{P}(u, t; \mathrm{t}_{0})$(2.17),$=$$\langle U_{0}(\mathrm{t}, t_{0})\rho_{1}(u, t;t_{0})\rangle$$=$$U_{0}(t, t_{0}) \langle T\exp[\int_{t_{0}}^{t}ds\mathcal{L}_{1}(s;t_{0})]\rangle P(u, t_{0})$$\equiv$$U_{0}(t, t_{0})S(t; \mathrm{t}_{0})P(u, t_{0})$where we have used the fact that

(2.18),(2.19),and the abbreviation$\rho_{1}(u, \mathrm{t}=\mathrm{t}_{0}, \mathrm{t}_{0})=P(u, t_{0})$$S(t;t_{0}) \equiv\langle T\exp[\int_{t_{0}}^{t}ds\mathcal{L}_{1}(s;t_{0})]\rangle$.

The computation may be performed in aperturbative way:$S(t; t_{0})$$=$$1+T \int_{t_{0}}^{t}ds\langle \mathcal{L}(s)\rangle+\frac{1}{2}T\int_{t_{\mathrm{O}}}^{t}ds_{1}\int_{t_{0}}^{t}ds_{2}\langle \mathcal{L}(s_{1})\mathcal{L}(s_{2})\rangle+\ldots$$=$$1+ \frac{1}{2}T\int_{t_{0}}^{t}ds_{1}\int_{t_{0}}^{t}ds_{2}\Gamma(s_{1}, s_{2})+\ldots$(2.20)
where we have put$\Gamma(s_{1}, s_{2})\equiv\langle \mathcal{L}_{1}(s_{1})\mathcal{L}_{1}(s_{2})\rangle$(2.21).

will be afunction of the
If the noise is stationary, which we shall assume from now,
;furthermore, owing to the time-reversible invariance of the microscopic
difference.
,,
becomes a function of the absolute value
law,,
Then one has for$\Gamma(s_{1}, s_{2})$$s_{1}-s_{2}$$|s_{1}-s_{2}|$$\Gamma(s_{1},52)$$t$$\mathrm{i}.\mathrm{e}.$$\Gamma(s_{1}, s_{2})=\Gamma(|s_{1}-s_{2}|)$$>t_{0}$(2.22)$S(t;t_{0})=1+(t -t_{0})G(t-t_{0})+\cdots$ ,where we have put for $t>0$$G(t)= \int_{0}^{t}ds\Gamma(s)$(2.23).

If we stop at the second order approximation, we have$\tilde{P}(u, t;t_{0})=U(t;t_{0})[1+(t-t_{0})G(t -t_{0})]P(u, t_{0})$.

(2.24)

Notice the appearance of the secular term which indicates that the above formula is only
valid for around .
Now we apply the RG equation to (2.24) which reads$\mathrm{t}$$t_{0}$$\partial\tilde{P}(u, t;t_{0})/\partial t_{0}|_{t_{0}=t}=0$,

which leads to$\partial_{t_{0}}U_{0}(t, t_{0})|{}_{t0=t}P(u, t)+\partial_{t}P(u, t)-G(0)P(u, t)$ $=0$,

146
where use has been made that $U_{0}(t_{0}, t_{0})=1$ ,. Noticing thath, we arrive at the Fokker-Planck equation,$\forall t_{0}$$\partial_{t_{0}}U_{0}(t, t_{0})|_{t_{0}=t}=-L_{0}=$$-\nabla_{u}\cdot$$\partial {}_{t}P(u, t)=-\nabla_{u}\cdot$$hP(u, t)+G(0)P(u, t)$ .

(2.25)

The concrete form of $G(0)$ depends on the character of the noise $R(t)$ . This is one of themain results of this section.
To see that (2.25) is the desired equation, let us evaluate $G(0)$ for asimple Gaussian
noise given by$\langle R_{\dot{*}}(t)R_{j}(t’)\rangle=2\delta_{\dot{l}j}D_{i}\delta(t-t’)$.

(2.26)For this case, one has$\Gamma(s)=U_{0}^{-1}\partial_{i}g_{ij}\partial_{k}g_{kl}2D_{j}\delta_{jl}\delta(s)$where$\partial_{i}=\partial/\partial u_{i}$,. Then $G(t)$ is evaluated as follows;$G(t)$$\equiv$$\int_{0}^{t}ds\Gamma(s)=\frac{1}{2}U_{0}^{-1}\partial_{i}g_{\dot{l}j}\partial_{k}g_{kl}2D_{j}\delta_{jl}$$=$$G(0)$,.

(2.27)

Here we have used the identity $\theta(0)=1/2$ , in accordance with the Stratonovich scheme[9].
Notice that $G(t)$ in this case is independent of .
Inserting $G(0)$ thus obtained into (2.25), one has the familiar form of the Fokker-Planck
equation for the multiplicative Gaussian noise,$\mathrm{t}$$\partial_{t}P(u, t)=-\nabla_{u}\cdot hP(u, t)+D_{j}\partial_{i}g_{ij}\partial_{k}g_{kj}P(u, t)$.

(2.28)

This shows that the initial distribution $P(u,t_{0})$ satisfies the Fokker-Planck equation andjustifies the identification of the initial distribution with the averaged one made in Eq.
(2.14).

2.2

Discussion

Firstly, it is noteworthy that we have been naturally led to identify the initial values ofthe microscopic distribution function $f(u, t_{0}, t_{0})$ before averaging with the averaged value$P(u, t_{0})$ at an arbitrary initial time $t=t_{0}$ .
As mentioned in \S 1, the necessity to takesuch an initial condition to achieve reduction of evolution equation was advocated by
Bogoliubov[5] and others $[14, 15]$ including Boltzmann[4]. Secondly, this means that thenature of the initial time in the RG method is completely different from that of thetime in the stochastic equation (microscopic equation); represents the coarse-grained
time describing the variation of the averaged quantity, and the derivativein the RG
equation is a macroscopic time-derivative. Again as mentioned in \S 1, this coa se-grainingof time was also noted by others [11, 10, 12] in different approaches.
This automatic averaging and the appearance of the macroscopic time-derivative givenin the RG method may be generically understood as ageneralization of the scheme given
in \S 2 of [1]: First discretize the variable $uarrow u_{i}$ and write as $P(u, t)(u_{i}, t)=X_{/}.(t)$ anduse avector notation $X=$
. Thus the discretized stochastic Liouville equation$t_{0}$$t$$t_{0}$$\partial_{t_{0}}$$(X_{1}, X_{2}, \ldots)$

147
the
with the initial value $X(t_{0})$ at an arbitrary time may be solved perturbatively, and, which satisfies the initial condition
solution is denoted as$t_{0}$$\tilde{X}(t;t_{0}, X(t_{0}))$$\tilde{X}(t;t_{0}, X(t_{0}))=X(t_{0})$(2.29).

We could solve the same equation with the initial condition given at ashifted initial$\mathrm{t}=t_{0}+\triangle t$time;$\tilde{X}(t; t_{0}+\triangle t, X(t_{0}+\triangle t))=X(t_{0}+\triangle t)$(2.30).

is macroscopically small but microscopically so
We suppose that the time difference,,
andlarge that it may be taken as infinity. For the time between
in theandIf$t_{0}<t<t_{0}+\triangle t$ , the perturbation should be valid.
microscopic scale, we may anticipate that the system is relaxed to the averaged trajectoryand have$\triangle t$$t$$t_{0}+\triangle t$$t_{0}$$\mathrm{t}$$\mathrm{i}.\mathrm{e}.$$\triangle tarrow\infty$$-\mathrm{t}_{0}$$X(\mathrm{t})$$\tilde{X}(t;t_{0}+\triangle t, X(t_{0}+\triangle t))\simeq\tilde{X}(t;t_{0}, X(t_{0}))$which implies that the macroscopic time derivative

0

$=$

$=$

$\delta/\delta t_{0}$,vanishes,,$\frac{\delta\tilde{X}}{\delta t_{0}}\equiv\cdot\frac{\tilde{X}(t,t_{0}+\triangle t,X(t_{0}+\triangle t))-\tilde{X}(\mathrm{t},t_{0},X(\mathrm{t}_{0}))}{\triangle t_{0}}.$$\frac{\partial\tilde{X}}{\partial t_{0}}|_{t_{\mathrm{O}}=t}+\frac{\partial\tilde{X}}{\partial X}\cdot\frac{dX}{dt_{0}}$(2.31)(2.32).

should be taken for
Notice that in the macroscopic scale, the equality
granted. This is the RG equation underlying the derivation of the Fokker-Planck equation
and also other transport equations including kinetic equations as will be shown in the next
section.
$\mathrm{t}_{0}\simeq t\simeq t_{0}+\triangle t$

3Reduction of BBGKY hierarchy to Boltzmann equation
As is well known, Bogoliubov first derived the Boltzmann equation from the BBGKY
hierarchy in his classic paper[5]. His derivation starts from an ansatz that the many
particle distribution function depends on time only through the one-particle distributionfunction and uses aspecial perturbative expansion method. His approach is actually anapplication and generalization of the asymptotic theory by Krylov and Bogoliubov (KB)
successful to non-linear oscillators[8]. In this section, we apply the RG method to derive
the Boltzmann equation. We do not use that ansatz and start from the naive perturbation
theory. We will see how the ansatz given by KB can be incorporated in the RG method.
The importance of the initial condition again emerges. This implies that the appearance ofasecular term[22] does not constitutes the final story for the derivation of the Boltzmann
equation.

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3.1

Derivation of the Boltzmann equation

Let us consider asystem of $N$ identical classical particles enclosed in avolume . We
shall adopt the notation of [26]; the -th particle’s phase space coordinate is represented
by $x_{i}=(r_{i},p_{i})$ . The Hamiltonian of the system reads
$V$

$i$

.

(3.1)

$H= \sum_{i=1}^{N}p^{2}\mathrm{i}+\frac{1}{2}\sum_{\dot{l}\neq j}2mU(|r_{i}-r_{j}|)$We suppose that the potential depends only on the relative distance of two particles andthat its range is much shorter than the mean free path 1. The $N$-particle distributionfunction
is normalized as
$U$

$d$

$f_{N}(x_{1}, \cdots, x_{N}, t)$$\int f_{N}(x_{1}, \cdots, x_{N}, t)\frac{\prod_{\dot{l}=1}^{N}dr_{i}dp_{i}}{N!}=1$

.

(3.2)

We define the -particle distribution function by
$s$

.

(3.3)

$f_{s}(x_{1}, \cdots x_{s}, t)=\int f_{N}(x_{1}, \cdots, x_{N}, t)\frac{dx_{s+1}\cdots dx_{N}}{(N-s)!}$Then the normalization condition for$f_{s}$

becomes$\int f_{s}(x_{1}, \cdots, x_{s}, t)dx_{1}\cdots dx_{s}=\frac{N!}{(N-s!)}\simeq N^{s}$ffom which we see that$n\ll 1$.

$f_{s}$

is of -th order in the particle density
$s$

,

(3.4)

$n= \frac{N}{V}$. We assume that

The kinetic equation for is obtained by integrating the Liouville equationover,,. Equations for and read$f_{s}$$x_{s+1}$$\cdots$$x_{N}$$f_{1}$$\frac{d}{dt}f_{N}=0$$f_{2}$,(3.5)$\frac{d}{dt}f_{1}(x_{1}, t)$$=$$( \frac{\partial}{\partial t}+\dot{\iota}L_{1}^{0})f_{1}(x_{1}, t)=-\int dx_{2}L_{12}’f_{2}(x_{1}, x_{2}, t)$$\frac{d}{dt}f_{2}(x_{1}, x_{2}, t)$$=$$( \frac{\partial}{\partial t}+\dot{\iota}L_{12})f_{2}(x_{1}, x_{2}, t)=-\int dx_{3}(iL_{13}’+iL_{23}’)f_{3}(x_{1}, x_{2}, x_{3}, t)(3.6)$

where$L_{i}^{0}=-i \frac{p_{i}}{m}\cdot\frac{L_{1}\partial}{\partial r_{i}’}2=L_{1}^{0}+L_{2}^{0},+LL_{ij}=i’\frac{\partial U(r_{i}-r_{j})\prime 12}{\partial r_{j}}\cdot(\frac{\partial}{\partial p_{j}}-\frac{\partial}{\partial p}\dot{.})$

.

(3.7)

These are the first two equations of the BBGKY hierarchy which is aseries of equations
relating the evolution of. Our goal is to derive an equation (or equations)towhich captures the essence of the system’s dynamics described by the BBGKY hierarchy.
In the language of the the theory of dynamical systems[6], we wish to construct alowdimensional invariant manifold in the (practically) infinite-dimensional functional spacespanned byand derive the reduced equations of motion on it$f_{s}$$\{f_{s}\}$$f_{s\dagger 1}$

149
Whereas the Liouville equation or, equivalently, the BBGKY hierarchy describes microscopic collisions between particles in detail, what interests us is the macroscopic variation of the system caused by the accumulation of many collisions. More concretely, wewish to know the variation of the system over the space-time scale much longer than thecollision radius and the collision time and much shorter than the mean free path and themean free time. Such scale is called the mesoscale. The derivatives appearing in (3.5) and(3.6) are, so to speak, microscopic derivatives, while those appearing in kinetic equations
are macroscopic derivatives. We must take into account their difference when deriving
kinetic equations.
Following [1], suppose that we have found the solution to the BBGKY hierarchy
we try to solve
up to an arbitrary time . With the initial condition
expansion)
to obtain(3.5) and (3.6) by the perturbative expansion in the density (virial
is of -th order in the density, we. Recalling thataroundasolution
expand as follows.
$\{f_{s}(, t_{0})\}$$t_{0}$$\{f_{s}(, \mathrm{t})\}$$t\sim t_{0}$$\tilde{f}_{s}(t;t_{0})$$f_{s}$$\mathrm{s}$$\tilde{f}_{1}(x_{1}, t)$$=$$\tilde{f}_{1}^{0}(x_{1}, t)+\tilde{f}_{1}^{1}(x_{1}, t)+\tilde{f}_{1}^{2}(x_{1}, t)+\cdots$$\tilde{f}_{2}(x_{1}, x_{2}, t)$$=$$\tilde{f}_{2}^{0}(x_{1}, x_{2}, t)+\tilde{f}_{2}^{1}(x_{1}, x_{2}, t)+\cdots$$\tilde{f}_{3}(x_{1}, x_{2}, x_{3}, t)$$=$$\tilde{f}_{3}^{0}(x_{1}, x_{2}, x_{3}, t)+\cdots$

(3.8)
(3.9)
(3.10),,,

is of $(i+j)$ -th order in the density. Substituting the above expansion
wherein (3.5) and (3.6), we get$\tilde{f_{i}}^{j}(x_{1}, \cdots, x_{i}, t)$$\frac{d}{dt}\tilde{f}_{1}^{0}(x_{1}, t)=0$(3.11),$\frac{d}{d\mathrm{t}}\tilde{f}_{2}^{0}(x_{1}, x_{2}, t)=0$(3.12),,$( \frac{\partial}{\partial t}+\frac{p_{1}}{m}\cdot\frac{\partial}{\partial r_{1}})\tilde{f}_{1}^{1}(x_{1}, t)=\int dx_{2}\frac{\partial}{\partial r_{1}}U(|r_{1}-r_{2}|)\cdot\frac{\partial}{\partial p_{1}}\tilde{f}_{2}^{0}(x_{1}, x_{2}, t)$(3.13)

where we have dropped terms which result in the surface integral. We also expand theinitial condition$f_{1}(x_{1}, t_{0})$$=$$f_{1}^{0}(x_{1}, t_{0})+f_{1}^{1}(x_{1}, t_{0})+\cdots$$f_{2}(x_{1}, x_{2}, t_{0})$$=$$f_{2}^{0}(x_{1}, x_{2}, t_{0})+\cdots$,.

(3.14)

Equation (3.11) and (3.12) are easily integrated:$\tilde{f}_{1}^{0}(x_{1}, t)$$\tilde{f}_{2}^{0}(x_{1}, x_{2}, t)$,$=$$e^{-iL_{1}^{0}(t-t_{0})}f_{1}^{0}(x_{1}, t_{0})$$=$$e^{-iL_{12}(t-t_{0})}f_{2}^{0}(x_{1}, x_{2}, t_{0})=f_{2}^{0}(x_{10}, x_{20}, t_{0})$,(3.15)

where$x_{i0}(x_{1}, x_{2}, t, \mathrm{t}_{0})$,(3.16)$i=1,2$are positions and momenta of the particles 1and 2at time$t_{0}$

under the influence of the2-body Hamiltonia$H^{(2)} \equiv\frac{p_{1}^{2}}{2m}+\frac{p_{2}^{2}}{2m}+U(|r_{1}-r_{2}|)$.

(3.17)

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The initial values
andmay be considered as the integration constants of the lowest-0rder equation. In the RG method as formulated in $[19, 1]$ , theintegration constants will constitute the coordinates of the zeroth invariant manifold[6].
The decisive step of the present approach is to choose the initial condition as follows$f_{1}^{0}(x_{1}, t_{0})$$f_{2}^{0}(x_{1}, x_{2}, t_{0})$$f_{2}^{0}(x_{1}, x_{2}, t_{0})=f_{1}^{0}(x_{1}, t_{0})f_{1}^{0}(x_{2}, t_{0})$,(3.18)

irrespective of the distance between
and . The underlying picture of this choice isthat the system is so dilute that the two particles at an arbitrary time are most probably
located at distance much longer than the collision radius , so that the correlation of thetwo particles is negligible andcan be set to the product of one-particle distributionfunctions. We remark that aprobabilistic nature enters at this point[27].
The integration of (3.13) from to with( is the average velocity), which
implies thatis small in the macroscopic scale, gives$r_{1}$$r_{2}$$t_{0}$$d$$f_{2}$$t_{0}$$t$$\frac{l}{v}\gg t-t_{0}$$v$$\mathrm{t}-t_{0}$$\tilde{f}_{1}^{1}(x_{1}, t)$$=$$e^{-iL_{1}^{0}(t-t_{0})}f_{1}^{1}(x_{1}, t_{0})$$\dagger\int_{t_{0}}^{t}dt’e^{-iL_{1}^{0}(t-t’)}\int dx_{2}\frac{\partial}{\partial r_{1}}U(|r_{1}-r_{2}|)\cdot\frac{\partial}{\partial p_{1}}f_{1}^{0}(x_{10}’, t_{0})f_{1}^{0}(x_{20}’, t_{0}\mathrm{X}\partial\cdot 19)$

where we have used (3.15) and (3.18), andand$tarrowt’$
replacement. We remark that the condition
expansion in the density to be valid [5]. In (3.19), only
to the integral. In this region, we can write$x_{10}’$$x_{20}’$are given by (3.16) with theis also required for the$( \frac{l}{v}\gg t-t_{0})$$r_{i0}’ \sim r_{i}-\frac{p_{i0}}{m}(t’-t_{0})$$r_{2}$

for $|r_{1}-r_{2}|\leq d$ contributes.

(3.20)

for amicroscopically large period. Here we have neglected vectors whose
magnitudes are of order . Then the perturbative solution in the mesoscopic regimeis$t’-t_{0} \gg\frac{d}{v}$$d$$\frac{l}{v}\gg t$$-t_{0}>> \frac{d}{v}$$\tilde{f}_{1}(x_{1}, t)$$=$$\tilde{f}_{1}^{0}(x_{1}, t)+\tilde{f}_{1}^{1}(x_{1}, t)$$=$$e^{-:L_{1}^{0}(t-t_{0})}f_{1}^{0}(x_{1}, t_{0})+ \int_{t_{0}}^{t}dt’e^{-iL_{1}^{0}(t-t’)}\int dx_{2}\frac{\partial}{\partial r_{1}}U(|r_{1}-r_{2}|)$.

$\frac{\partial}{\partial p_{1}}f_{1}^{0}(r_{1}-\frac{p_{10}}{m}(t’-t_{0}),p_{10}, t_{0})f_{1}^{0}(r_{2}-\frac{p_{20}}{m}(t’-t_{0}),p_{20}, t_{0})$(3.21).

Note that $p_{i0}=p_{i0}’$ :The magnitudes of aare of course different for different
systems. For adilute gas system, typical values areand,respectively. The second term of the r.h.s. of (3.22) is the secular term. Indeed, it can beshown that in the spatially homogeneous case it is proportional to. Accordingly,
we have chosen
to be zero following the prescription given in [1]. The RG equation
reads$\frac{l}{v}$$\mathrm{d}$$\frac{d}{v}$$10^{-8}\sim 10^{-9}\mathrm{s}$$10^{-12}\sim 10^{-13}\mathrm{s}$$t-\mathrm{t}_{0}[23]$$f_{1}^{1}(, t_{0})$$\frac{\partial}{\partial t_{0}}\tilde{f}_{1}(x_{1}, t)|_{t=t_{0}}=0$$\Rightarrow$,$\frac{\partial}{\partial t}f_{1}^{0}(x_{1}, t)$(3.22)$+$$\frac{p_{1}}{m}\cdot\frac{\partial}{\partial \mathrm{r}_{1}}f_{1}^{0}(x_{1}, t)$$=$$\int dx_{2}\frac{\partial}{\partial r_{1}}U(|r_{1}-\mathrm{r}_{2}|)\cdot\frac{\partial}{\partial p_{1}}f_{1}^{0}(r_{1},p_{10}, t)f_{1}^{0}(r_{1},p_{20}, t)(3.23)$

151
In (3.22) we have imposed that $t=t_{0}$ although the expression (3.22) is valid for. This manipulation can be justified by the same logic given in the last part in \S 2 and willappear also in the case of field theory discussed in the following section: The t-derivative isthe microscopic derivative and the -derivative is the macroscopic one. Through the RG
equation, we can automatically go over to the mesoscopic physics from the microscopic
physics. Thus the mesoscopic nature of the Boltzmann equation is transparent in our
approach.
(3.23) is the kinetic equation we have been seeking for. In the language of the RG
method, it is the renormalization group equation describing the slow motion on the invari. To obtain the usual Boltzmann equation
ant manifold with the coordinate
which contains the gain minus loss term, we have to manipulate the r.h.s. ignoring the
spatial dependence. The result is$\mathrm{t}-t_{0}\gg\frac{d}{v}$$t_{0}$$f_{1}^{0}(x_{1}, t)[22]$$\frac{\partial}{\partial t}f_{1}^{0}(x_{1}, t)$$+$$\frac{p_{1}}{m}\cdot\frac{\partial}{\partial r_{1}}f_{1}^{0}(x_{1}, t)$$=$$\int_{0}^{\infty}\rho d\rho\int_{0}^{2\pi}d\phi\int d\mathrm{p}_{2}v_{12}\{f_{1}^{0}(r_{1},p_{1}’, t)f_{1}^{0}(r_{1}, \mathrm{p}_{2}’, t)-f_{1}^{0}(r_{1},p_{1}, t)f_{1}^{0}(r_{1}, \mathrm{p}_{2}, t)\}$,(3.24)

where we have introduced the cylindrical coordinate pointing the direction of the relative
velocity vi2 $=(p_{2}-\mathrm{p}_{1})/m$ . Comparing (3.11) and (3.23), we see the change of theequation by including the lowest contribution of the collision.

3.2

Role of the initial condition

It was Bogoliubov [5] who first pointed out that the Boltzmann equation represents themesoscopic physics and derived it from the Liouville equation. The decisive assumption
in his derivation is that the system can be described only in terms of the one-particle
distribution function. That is, starting from an arbitrary initial condition, the system
will rapidly reach the state in which $f_{s}(s\geq 2)$ are functional of .
$f_{1}$$f_{s}(x_{1}, \cdots, x_{s}, t)=f_{s}(x_{1}, \cdots, x_{s};f_{1}(, t))s\geq 2$.

(3.25)) In fact, this assumption is a basis of any kinetic( depends on time only throughtheory. The special expansion method based on this assumption leads to aspecial solution
to the BBGKY hierarchy. In the RG method, we started with anaive perturbative
expansion without any knowledge about kinetic theory. Physics enters when we choose a
special initial condition (3.18) and with this choice we can construct an invariant manifold
spanned by the one-particle distribution function in the infinite-dimensional functional
space, which was originally envisaged by Bogoliubov[5].
$f_{s}$$f_{1}.$

4Summary and Concluding Remarks
In this report, we have described an attempt to apply the s0-called renormalizationgroup(RG) method to derive and reduce kinetic equations; the Boltzmann equation, andthe Fokker-Planck equation. In contrast to the previous work[22], our main purpose was152
to elucidate the general structure of the reduction of the dynamical equations in the hierarchy of the evolution equations. We have noticed that the significance of the choice of
the initial value on the attractive manifold which is also an invariant manifold[6] in deriving kinetic equations is fully recognized and emphasized by Bogoliubov[5], Lebowitz[14],KubO[13] and Kawasaki[15], for instance. The notion of coarse-grained time derivative
was also noticed by Mori[10] and others[ll, 12]. Our point was that these basic ingredients naturally appear in the-theoretical derivation of kinetic equations when properly
formulated so as to respect the role played by the initial condition as formulated in $[19, 1]$ .
This report is based on a part of [28], in which adetailed account of this report and ap
plications to other kinetic equations including aquantum field theoretical model may
be$\mathrm{R}\mathrm{G}$seen.

References[1] S.-I. Ei, K. Fujii and T. Kunihiro, Ann. Phys. 280 (2000), 236.
[2] N. Goldenfeld, O. Martin and Y. Oono, J. Sci. Comp. 4(1989),4; N. Goldenfeld,
O. Martin, Y. Oono and F. Lin,Phys. Rev. Lett. 64 (1990), 1361; N. D. Goldenfeld, “Lectures on Phase Ransitions and the Renormalization Group,”AddisonWesley, Reading, Mass., 1992. L. Y. Chen, N. Goldenfeld, Y. Oono and G. Paquette, Physica A204(1994)111. G. Paquette, L. Y. Chen, N. Goldenfeld and Y.
Oono,Piys. Rev. Lett. 72(1994)76; L. Y. Chen, N. Goldenfeld and Y. oono,Phys.
Rev. Lett.73(1994)1311; L. Y. Chen, N. Goldenfeld and Y. oono,Phys. Rev. E54(1996), 376. Attempts to straighten the logic and to give a proper interpretation ofwhat is done by Illinois group may be seen in [17]; see also [19]. Most of the referenceson the RG method published unti11999 may be found in [1].
[3] L.E. Reichl, Modern Course in Statistical Physics 2nd ed. (John Wiley and Sons,
1998).
[4] L. Boltzmann, “Lectures on Gas Theory” University of California Press, Berkeley,

1964.

[5] N. N. Bogoliubov, in “Studies in Statistical Mechanics, vol. 1,” edited by J. de Boerand G. E. Uhlenbeck, North-Holland, Amsteredam, 1962.
[6] See for example, J. Guckenheimer and P. Holmes, Nonlinear Oscillators, Dynamical
Systems, and Bifurcations of Vector Fields,(Springer-Verlag, 1983);
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,(Springer-Verlag, New York, 1990);
J. D. Crawford, Rev. Mod. Phys. 63 (1991), 991.

[7] S. Chapman and T.G. Cowling, “The Mathematical Theory of Non-Uniform Gases”(3rd ed.) Cambridge U.P., 1970.
[8] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of
Nonlinear Oscillations, Gordon and Breach, 1961

153
[9] R. L. Stratonovich,Topics in the Theory of Random Noise, vol. 1 and 2, (Gordon
and Breach, 1963). C. W. Gardiner, Handbook of Stochastic Methods for Physics,
Chemistry and the Natural Sciences, 2nd ed. (Springer, 1985). H. Risken, The FokkerPlanck Equation– Methods of Solution and Applications, 2nd ed. (Springer, 1989).
N.G. van Kampen, Stochastic Processes in Physics and Chemistry, rev. and enlargeded. (North-Holland, 1992).

[10] H. Mori, J. Phys. Soc. Jpn. 11 (1956), 1029; Phys. Rev. 112 (1958), 1829; 115(1959), 298.
[11] J. G. Kirkwood, J. Chem. Phys. 14 (1946), 180[12] I. Ojima, J. Stat. Phys. 56(1989), 203.

[13] R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II Non-equilibrium Statistical Mechanics (Springer-Verlag, 1985).
[14] J. L. Lebowitz, PhysicaA (1993),[15] K. Kawasaki, Non-equilibrium and Phase Transition –Statistical Physics in Meso
Scales, chap. 7(Asakura Shoten, 2000), in Japanese.
[16] E.C.G. Stueckelberg and A. Petermann, Helv. Phys. Acta 26(1953), 499;
M. Gell-Mann and F. E. Low, Phys. Rev. 95 (1953), 1300.
K. Wilson, Phys. Rev. D3(1971), 1818, S.Weinberg, in” Asymptotic Realms of
Physics” (A. H. Guth et al. Ed.), MIT Press, 1983.
As review articles, S.K. Ma, ”Modern Theory of Critical Phenomen\"a, W. A. Benjamin, New York, 1976. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989.
.
D. V. Shirkov,$\mathrm{h}\mathrm{e}\mathrm{p}-\mathrm{t}\mathrm{h}/9602024;\mathrm{h}\mathrm{e}\mathrm{p}-\mathrm{t}\mathrm{h}/9903073$[17] G.C. Paquette, Physica A276 (2000), 122;
B. Mudavanhu and R.E. O’Malley, Studies in Applied Mathematicsl07(2001), 63.
[18] J. Bricmont and A. Kupiainen,Commun. Math. Phys. 150 (1992), 193; J. Bricmont,
A. Kupiainen and G. Lin, Cooun. Pure. Appl. Math. 47(1994), 893; J. Bricmont and.
A. Kupiainen,$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{o}-\mathrm{d}\mathrm{y}\mathrm{n}/9411015$[19] T. Kunihiro,Prog. Theor. Phys. 94(1995), 503; (E) ibid., 95(1996)835; Jpn. J.
Phys.
Ind. Appl. Math. 14 (1997), 51, Prog. Theor. Phys. 97(1997),179;
also
459;
see
(1998),
131
Suppl.
Phys.
Rev. D57 (1998), R2035; Prog. Theor.
.
$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{t}-\mathrm{s}\mathrm{o}\mathrm{l}/979003$[20] K.G.Wilson and M.E.Fisher,Phys. Rev. Lett. 28 (1972),240; K.G. Wilson, Phys.
Rev. Lett. 28(1972), 548; K.G. Wilson and J. Kogut,Piys. Rep. 12C(1974), 75.
[21] F. Wegner and A. Houghton,Phys. Rev. A8 $(1973),401$ .

[22] O. Pashko and Y. Oono, Int. J. Mod. Phys. B14 (2000), 555

154[23] M. S. Green, J. Chem. Phys. 25 (1956), 836.
[24] G. I. Barlenblatt, Similarity, Self-Similarity and Intermediate Asymptotics (Consultants Bureau, N.Y., 1979).

[25] R. Kubo, J. Math. Phys.4(1963), 174;
R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II Nonequilibrium Statistical Mechanics (Springer-Verlag, 1985).

[26] D. N. Zubarev, V. G. Morozov and G. Ropke, Statistical Mechanics ofNonequilibrium
Processes Vol.I,(Akademie Verlag, Berlin, 1996).
[27] E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics, (Butterworth-Heinemann, 1981)

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, (Wiley, New
York, 1975).
L.E. Reichl, Modern Course in Statistical Physics, chap. 11, 2nd ed. (John Wileyand Sons, 1998).
[28] Y. Hatta and T. Kunihiro,$\mathrm{h}\mathrm{e}\mathrm{p}-\mathrm{t}\mathrm{h}/0108159$