Out of equilibrium Stat. Mech.: TheKardar-Parisi-Zhang equation
Bertrand Delamotte, University Paris VI

Kyoto, september 2011

The Kardar-Parisi-Zhang equation

Collaborators
L. Canet (Grenoble, France)
H. Chaté (Saclay, France)
N. Wschebor (Univ. Montevideo, Uruguay)
and also
F. Benitez (Univ. Montevideo, Uruguay)
J.-P. Blaizot (Saclay, France)
I. Dornic (Saclay, France)
M. Munoz (Granada, Spain)

The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?

The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?
What does non equil. mean?

The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?
What does non equil. mean?
• Relaxation towards equil. (dyn. expo. z)

The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?
What does non equil. mean?
• Relaxation towards equil. (dyn. expo. z)• Continuous phase transitions in syst. in a NESS

The Kardar-Parisi-Zhang equation

“Branching and annihilating random walks”: (BARW)Partciles (A) diffusing (rate D) on a lattice and that undergo
reactions.
Directed Percolation:
2A → ∅rate λA → 2Arate σGeneralized Voter model:
2A → ∅rate λA → 3Arate σPhase transition between active and absorbing phase?
Universality classes? Exponents? etc.
The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?
What does non equil. mean?
• Relaxation towards equil. (dyn. expo. z)• Continuous phase transitions in syst. in a NESS
• Systems showing generic scaling (in a NESS)

The Kardar-Parisi-Zhang equation

Scaling out of thermal equilibrium= Paradise for RG?
What does non equilibrium mean?
• Relaxation towards equil. (dyn. expo. z)• Continuous phase transitions in syst. in a NESS
• Systems showing generic scaling (in a NESS)• Scaling away from stationarity: short time critical dynamics,
coarsening,...

The Kardar-Parisi-Zhang equation

Criticality out of thermal equilibrium
Systems studied:particles diffusing and reacting
orsystems coupled to a “stochastic bath”(deposition of particles on a surface)⇓Not necessarily an Hamitonian evolution⇓
No Boltzmann weight for the stationary distribution⇓
Difficulty: need to describe the whole dynamics
Models studied: Langevin equations for a N-body system
Questions: long-time and large-scale properties of the system?
The Kardar-Parisi-Zhang equation

Langevin equations∂t ϕ(~x , t) = −F [ϕ] + N[ϕ]ζ(~x , t),where ζ is a gaussian (white) noise:hζ(~x , t)i = 0,hζ(~x , t)ζ(~x 0 , t 0 )i = 2δ (d) (~x − ~x 0 )δ(t − t 0 ).
For BARW: ϕ(~x , t) = density (of particles) field.
In general (when detailed balance is violated) the probability
distribution of the stationary states is not known.

The Kardar-Parisi-Zhang equation

Averages of O(φ(t, ~x )) over the noise distribution:
ZhO(φ)i = Dζ P(ζ) O(φζ )hO(φ)i =R

Dζ P(ζ)R

Dφ δ(φ − φζ ) O(φ)=

RDζ P(ζ)

RDφ δ(∂t φ + F (φ) − N(φ)ζ) J (φ) O(φ)
RDφD[i φ̃] e t,~x −φ̃(∂t φ+F (φ)−N(φ)ζ) J (φ) O(φ)
δF(φ) δN(φ)−ζ.
J (φ) = det ∂t +δφδφ
If J = 1 (Ito’s prescription)
ZRZ[j, j̃] = Dφ D[i φ̃] e −S[φ,φ̃]+ t,~x jφ+j̃ φ̃=RDζ P(ζ)

withRZ
S[φ, φ̃] =φ̃ (∂t φ + F (φ)) − N 2 (φ)φ̃2 .

t,~xThe Kardar-Parisi-Zhang equation

Langevin equation for Directed Percolation:

2A → ∅rate λA → 2Arate σ∂t ϕ(~x , t) = D∇2 ϕ + σϕ − λϕ2 +pσϕ − λϕ2 ζ(~x , t)

Mean-field (mass action law)∂t ϕ(t) = σϕ − λϕ2Only one stable stationary state: ϕ = σ/λ ⇒ active phase.
⇒ no phase transition... WRONGThe Kardar-Parisi-Zhang equation

6σ/D543d=6d=5d=4d=3d=2d=1210024681012λ/D

The Kardar-Parisi-Zhang equation

14

Kardar-Parisi-Zhang equation
M. Kardar, G. Parisi, Y.-C. Zhang, PRL (1986)∂t h(~x , t) = ν∇2 h +λ(∇h)2 + σζ(~x , t)
2describes:- surface growth through ballistic deposition of particles- disordered systems at equilibrium (directed polymers inrandom media),~ = velocity field),- Burgers equation (~v = ∇h- magnetic flux lines in superconductors, etc...
shows generic scaling : h[h(~x , t) − h(~0, 0)]2 i ∼ x 2χ f (t/x z ).
with z + χ = 2.

The Kardar-Parisi-Zhang equation

2
λ∂h(~x , t)
= ν ∇2 h(~x , t) +∇h(~x , t) + ζ(~x , t)∂t
2
becomes a multiplicative noise equationvia the Cole-Hopf
λtransformation w (~x , t) = exp 2ν
h(~x , t) :∂t w (~x , t) = ∇2 w (~x , t) +λ D 1/2w (~x , t) ζ(~x , t)2ν 3/2shows two phases for d > 2 (d + 1 space-time dimenions):- smooth phase for small non-linearities (gaussian fluctuations)- rough phase for large non-linearities.
is underlied by symmetries : (gauged) Galilean symmetry,(gauged) shift of h, time reversal symmetry in d=1, and otherdiscrete symmetries (non-linearly realized).
is perturbatively trivial in the Cole-Hopf representation, but...
BUT... the rough phase is unreachable perturbatively.
The Kardar-Parisi-Zhang equation

Field theory associated with KPZ:
ZZ
Z[j, j̃] =D[h, i h̃] exp −S[h, h̃] + (j h + j̃ h̃)xZ  λ2
2S[h, h̃] =h̃ ∂t h − ν ∆h − (∇h) − D h̃
2xSymmetries:(i) “invariance” of S under the gauged Galilean transformation
h(x) → h(~x + λ~v (t), t) + ~x .∂t ~v (t)h̃(x) → h̃(~x + λ~v (t), t)(ii) “invariance” of S under the gauged shift symmetryh(x) → h(x) + f (t) where f (t) is arbitrary;(iii) in d = 1, additional time-reversal invarianceνh(t) → h(−t), h̃(t) → h̃(−t) + 2D∆h(−t).
and nonlinearly realized discrete symmetries...
The Kardar-Parisi-Zhang equation

Aim: compute the long-time and large-distance physics ⇒derivative expansion?
BUT
Problem: interaction=derivative term ⇒ necessary to take into
account the momentum dependence of the two-point correlation
functions ⇒ BMW?
BUT
direct implementation is hindered by the symmetries (Wardidentities).

The Kardar-Parisi-Zhang equation

Blaizot-Mendez-Wschebor (BMW) approximation in a nutshell:(2)
Exact equation on Γk (p, φ) (for uniform field φ):Z(2)∂k Γk (p, φ) = ∂k Rk (q)G (q)2 .

q

(4)
(3)
(3)
Γk (p, q, −p − q)G (p + q)Γk (−p, −q, p + q) − 12 Γk (p, −p, q, −q)
(3)

(4)

(2)

Truncate the q-dependence of Γk , Γk → closed equation on ΓkZ(2)∂k Γk (p, φ) = ∂k Rk (q)G (q)2 .
q


(3)
(3)
(4)
Γk (p, 0, −p)G (p + q)Γk (−p, 0, p) − 21 Γk (p, −p, 0, 0)
Final result:1  2 (2) (2)(2) 2J2 (0, φ)∂k Γk (p, φ) = ∂φ ΓkJ3 (p, φ) −∂ Γ2 φ kwhereZJn (p, φ) =∂k Rk (q) G (p + q, φ)G (q, φ)n−1q
The Kardar-Parisi-Zhang equation

Back to KPZ
Two difficulties to implement BMW here:- no frequency-dependent regulator (forbidden by Galilean sym) ⇒is BMW justified?
- Ward identities / BMW approximation ⇒ delicate interplay
Defineϕ = hhi ; ϕ̃ = hh̃i(m,n)Γk({qi }, ϕu , ϕ̃u ) =δ m+n Γkδϕq1 ...δϕqm δ ϕ̃qm+1 ...δ ϕ̃qm+n ϕu ,ϕ̃uAn example of Ward identity (gauged-Galilean):iω∂ (2,1)
Γ(ω, ~p = ~0; ω1 , ~p1 ) = λ~p1
∂~p κ


(1,1)
~~× Γ(1,1)(ω+ω,p)−Γ(ω,p),1111κ
κ
The Kardar-Parisi-Zhang equation

Galilean symmetry = root of the problem
but
Galilean symmetry = part of its solution!
iω∂ (2,1)Γ(ω, ~p = ~0; ω1 , ~p1 ) = λ~p1∂~p κ
(1,1)~~× Γ(1,1)(ω+ω,p)−Γ(ω,p),1111κ
κ(2,1)⇒ Γκ(1,1)

in terms of Γκ

doesn’t need ω = 0 !

Nevertheless, difficult to implement directly BMW.
Solution: devise an ansatz close in spirit to BMW but thatcircumvents the above difficulties.
A priori difficult. But...

The Kardar-Parisi-Zhang equation

Geometric formulation of KPZ
Call scalar (under Galilean transfos) a quantity such that:δf (x) = tλ~v · ∇f (x).
⇒Rd d ~x f is Galilean-invariant.

- h̃, ∇i ∇j h are scalars, ∇(scalar) = scalar.
- h and ∂t (scalar) are not scalars.
⇒ as in fluid mechanics build a covariant time derivativeD̃t ≡ ∂t − λ∇h(x) · ∇⇒ D̃t (scalar) = scalar.
Covariant derivative of h: Dt h(x) ≡ ∂t h(x) − λ2 (∇h(x))2Scalars at our disposal: h̃, ∇i ∇j h, Dt h, ∇(scalar), D̃t (scalar).

The Kardar-Parisi-Zhang equation

Truncation:- full momentum dependence (derivative interaction),- full frequency dependence (comparison with exact results ind = 1),- minimal field content.
full frequency dependence⇒ non polynomial dependence in D̃t (D̃t ≡ ∂t − λ∇ϕ · ∇)⇒ non polynomial dependence in ϕ
⇒ minimal = minimal in ϕ̃, ∇2 ϕ and Dt ϕ.
Our Ansatz:ZΓκ [ϕ, ϕ̃] =ϕ̃ fκλ (−D̃t2 , −∇2 ) Dt ϕ − ϕ̃ fκD (−D̃t2 , −∇2 ) ϕ̃xi
ν h 2 ν∇ ϕ fκ (−D̃t2 , −∇2 ) ϕ̃ + ϕ̃ fκν (−D̃t2 , −∇2 ) ∇2 ϕ .
−2DThe Kardar-Parisi-Zhang equation

• For d = 1, time reversal sym. ⇒ λ 2 2 fκ (ω , p ) = 1
fκν (ω 2 , p 2 ) = fκD (ω 2 , p 2 )⇒ only (!) one function (of ω 2 and p 2 ).
• For d > 1, 3 functions and 3-dimensional integrals ⇒ simplify !
Aim: zero frequency sector ⇒ neglect all ω-dependence (but thebare one).
Moreover, since fκλ (p 2 = 0) = 1 ⇒ impose fκλ (p 2 ) = 1, ∀~p .
⇒ two functions: fκν (p 2 ) and fκD (p 2 ).
Two additional inputs:• causality has to be preserved (and Ito’s prescription),
 2

q0
νκ q 2• Rκ (~q ) = r, r (x) = α/(exp(x) − 1)νκ q 2 −2Dκκ2And now, turn the crank...
The Kardar-Parisi-Zhang equation

Results with the simplified ansatz (∀d):
L.Canet, H. Chaté, B.D. and N. Wschebor, PRL (2009), Arxiv (2011)- three fixed points: gaussian FEW , transition FTR and strong
coupling FSC ,- generic scaling at FSC (no phase transition associated with thisFP),- asymptotic safety (FTR ),- exact results for exponents recovered in d = 0 and d = 1 at FSC ,- exponents not so bad at FSC for d ≤ 3.
BUT
- exact results for exponents not recovered for FTR (in d > 2)
- strange behavior of the critical exponents for d & 3.5 ⇒ noprediction as for the existence of an upper critical dimension.

The Kardar-Parisi-Zhang equation

(a)

(b)

02~
γ~ (p)~

3 g vd0χ0,6v

0,40,2vvv

v

0s-20(d)

v

0-40v
v

1v

v

v

v

v

v

v

3202v
v

v

g~ vdp~10v

10(c)

v

FEWv

40v

v

10-2v

FRTv

10-3v

1v

-210v

2v

-110vv

FSCv

v

10v

0v v v
246d80d0246

The Kardar-Parisi-Zhang equation

8The d = 1 case
Exact results in d = 1:M. Praehofer and H. Spohn, PRL (2000), J. Stat. Phys. (2004)
J. Baik and E.M. Rain, J. Stat. Phys. (2000)
Experimental results: K.A. Takeuchi and M. Sano, PRL (2010)
Results with the full ansatz for d = 1
L.Canet, H. Chaté, B.D. and N. Wschebor, Arxiv (2011)
2fk (ω 2 , p 2 ) ⇒ excellent data collapse at k → 0 ⇒ fk=0 ( ωp3 )⇓existence of scaling

The Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang equation

Conclusion and outlook:

A lot remains to be done for KPZ...
Everything remains to be done for Navier-Stokes!

The Kardar-Parisi-Zhang equation