Derivation of viscous Burgers equations
from weakly asymmetric simple exclusion
processes
Kenkichi TSUNODA
joint with M. Jara and C. Landim
Department of Mathematics, Osaka University

31, Jul, 2019

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Outline of the talk

Overview and Related Works
2 Model and Main Results
3 Comments on the Proof
1

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes1. Overview and Related Works

Kenkichi TSUNODA joint with M. Jara and C. LandimDepartment of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Overview
The macroscopic density of the (WASEP)n evolves
according to the nonlinear heat eq. as the system size n
grows to infinity (Hydrodynamic limit):∂t u = ∇ · [D(u)∇u] + ∇ · [σ(u)m] ,(1)where D, σ are d × d-matrices (Diﬀusivity and Mobility).
For small ε > 0, let us consider the first order correction
to (1) around a constant profile α0 ∈ (0, 1):!
∂t u ε = ∇ · [D(u ε )∇u ε ] + ε−1 ∇ · [σ(u ε )m] ,(2)
u ε (0, ·) = α0 + εv0 ,for some smooth function v0 .
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

The solution u ε should evolve as utε ∼ α0 + εvt .
Indeed, if σ ′ (α0 ) = 0, the sequence {ε−1 (u ε − α0 )}ε>0
converges to the solution to the Burgers eq. as ε ↓ 0(Incompressible limit):!
∂t v = ∇ · [D(α0 )∇v ] + (1/2)∇ · [v 2 σ ′′ (α0 )m] ,v (0, ·) = v0 (·) .
Main Result (rough version):
Taking ε = εn ↓ 0 (n → ∞), the correctly scaled densityof the WASEP evolves according to the Burgers eq.

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Related Works
Many results on hydrodynamic limits.
e.g. Guo-Papanicolaou-Varadhan, 88, Yau, 91.
Esposito-Marra-Yau, 94, 96 · · · Derivation of Burgers
equation and Navier-Stokes equation (d ≥ 3).
Quastel-Yau, 98 · · · Large deviations for theincompressible limits (d = 3).
Beltán-Landim, 08 · · · Derivation of Burgers equation
and Navier-Stokes equation in any dimensions but with(meso-scopically) big jumps.
Jara-Menezes, 19+ · · · Sharp entropy bound.
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes2. Model and Main Results

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

ModelEach particle moves on the d-dimensional discrete torus
Tdn = (Z/nZ)d = {1, 2, · · · , n}d , n ∈ N. Let Td be thed-dimensional torus Td = (R/Z)d = [0, 1)d .
Denote the number of particles at site x ∈ Tdn at time tdby ηtn (x) (ηtn = {ηtn : x ∈ Tdn } ∈ {0, 1}Tn ).
Some parameters:{εn }n∈N ⊂ R: a sequence converging to 0.
(cj )dj=1 : nonnegative local functions.
m = (m1 , . . . , md ): a vector in Rd .

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

dLet ηtn be a Markov process on {0, 1}Tn with theAgenerator Ln = n2 LSn + ε−1
n nLn with(LSn f ) (η)=d""x∈Tdn j=1(LAn f ) (η)=d""x∈Tdn

j=1cj (τx η) {f (σ x,x+ej η) − f (η)} ,mj cj (τx η) ηx+ej (1 − ηx )× {f (σ x,x+ej η) − f (η)} .

dfor a function f : {0, 1}Tn → R.
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

The dynamics of our particle system is as follows:
Each particle can jump from x to x + 1 or x − 1 at given
rates only if the site x is occupied and the site x + 1 orx − 1 is vacant.

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

For a continuous function u0 : Td → [0, 1], let νun0 be thedproduct Bernoulli measure on {0, 1}Tn :νun0 (η : η(x) = 1) = u0 (x/n) ,x ∈ Tdn .

Gradient condition: For each j, there exist finitely
supported signed measures mj,p , p = 1, . . . , nj and local
functions gj,p such thatcj (η)[η0 − ηej ] ="nj""mj,p (y )gj,p (τy η) ,p=1 y ∈Zdmj,p (y ) = 0 .

y ∈ZdKenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Classical case (εn = 1)
dAssume that η0n = νun0 for some continuous function
u0 : Td → [0, 1].
Hydrodynamic limit: For any t ≥ 0 and any smoothfunction H : Td → R,%⎤⎡%%%&"%%n ⎣% 1n⎦lim E % dH(x/n)ηt (x) −H(x)u(t, x)dx %% = 0,n→∞ndT% x∈Td%n

where u : [0, ∞) × Td → [0, 1] is the unique weaksolution of the Cauchy problem!
∂t u = ∇ · [D(u)∇u] + ∇ · [σ(u)m] ,u(0, ·) = u0 (·) .
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Incompressible case (εn ↓ 0)
Fix α0 ∈ (0, 1) with σ ′ (α0 ) = 0 and assume thatdη0n = ναn0 +εn v0 for some function v0 ∈ C 3+ (Td ).
Let v : [0, ∞) × Td → R be the unique weak (classical)
solution of the Burgers eq.:!
∂t v = ∇ · [D(α0 )∇v ] + (1/2)∇ · [v 2 σ ′′ (α0 )m] ,v (0, ·) = v0 (·) .
For each t ≥ 0, let utn = α0 + εn vt , νtn = νuntn and let µnt
be the distribution of ηtn .
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Main Results

Theorem 1 (Jara-Landim-T., 19+)
Assume that n2 ε4n ≤ C0 gd (n) for some constant C0 , where
gd (n) = n, log n, 1 if d = 1, d = 2, d ≥ 3, respectively. Then,for any T > 0, there exists a constant C1 = C1 (T , v0 , C0 ) suchthat for any 0 ≤ t ≤ T ,H(µnt |νtn ) ≤ C1 nd−2 gd (n) ,where H(µ|ν) is the relative entropy of µ w.r.t. ν.

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Corollary 2 (Jara-Landim-T., 19+)
Assume that n2 ε4n ≤ C0 gd (n) and ε2n n2 gd (n)−1 ↑ ∞. For anyt ≥ 0 and any smooth function H : Td → R,%⎤⎡%%%&"%%n ⎣% 1n⎦
lim E % dH(x/n)[ηt (x) − α0 ] −H(x)v (t, x)dx %% = 0,n→∞
Td%εn n x∈Td%nRemarks

dInitial distribution: The assumption η0n = ναn0 +εn v0 can bereplaced with the entropy bound at time 0.
σ ′ (α0 ) = 0: In the case of general α ∈ (0, 1), introducing′
the Galilean transformation α + εn v (t, x − ε−1n σ (α)mt),
we can obtain a similar result.
Kenkichi TSUNODA joint with M. Jara and C. LandimDepartment of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes3. Comments on the ProofKenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Following [Jara-Menezes, 19+], we shall compute theentropy production. Let Ht = H(µnt |νtn ). Then, we have&) ∗,ν n*d2nSnHt ≤ −n D(gt , Ln , νt ) +Ln t 1 − ∂t log νtn dµnt .
dt∗,ν nWe need to compute the integrand Ln t 1 − ∂t log νtn
explicitly. Indeed, it can be expressed in terms of the“Fourier coeﬃcients” of gj,p (but quite messy...).
We also need to expand several terms in εn properly:e.g. Eνtn [τ−ej gj,p − gj,p ].
Together with these calculations, we shall apply
techniques developed in [Jara-Menezes, 19+].
Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes

Thank you for your attention.

Kenkichi TSUNODA joint with M. Jara and C. Landim

Department of Mathematics, Osaka University

Derivation of viscous Burgers equations from weakly asymmetric simple exclusion processes