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 1 Overview and Related Works 2 Model and Main Results 3 Comments on the Proof 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. Landim Department 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 (Di ff usivity and Mobility). For small ε > 0, let us consider the first order correction to (1) around a constant profile α 0 ∈ (0 , 1) : � ∂ t u ε = ∇ · [ D ( u ε ) ∇ u ε ] + ε − 1 ∇ · [ σ ( u ε ) m ] , (2) u ε (0 , · ) = α 0 + ε v 0 , for some smooth function v 0 . 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 u ε t ∼ α 0 + ε v t . Indeed, if σ ′ ( α 0 ) = 0, the sequence { ε − 1 ( u ε − α 0 ) } ε > 0 converges to the solution to the Burgers eq. as ε ↓ 0 (Incompressible limit) : � ∂ t v = ∇ · [ D ( α 0 ) ∇ v ] + (1 / 2) ∇ · [ v 2 σ ′′ ( α 0 ) m ] , v (0 , · ) = v 0 ( · ) . Main Result (rough version) : Taking ε = ε n ↓ 0 ( n → ∞ ), the correctly scaled density of 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 the incompressible limits ( d = 3). Belt´ an-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
Model Each particle moves on the d -dimensional discrete torus n = ( Z / n Z ) d = { 1 , 2 , · · · , n } d , n ∈ N . Let T d be the T d d -dimensional torus T d = ( R / Z ) d = [0 , 1) d . Denote the number of particles at site x ∈ T d n at time t n } ∈ { 0 , 1 } T d by η n t ( x ) ( η n t = { η n t : x ∈ T d n ). Some parameters : { ε n } n ∈ N ⊂ R : a sequence converging to 0. ( c j ) d j =1 : nonnegative local functions. m = ( m 1 , . . . , m d ) : a vector in R d . 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
t be a Markov process on { 0 , 1 } T d n with the Let η n generator L n = n 2 L S n + ε − 1 n nL A n with d � � ( L S c j ( τ x η ) { f ( σ x , x + e j η ) − f ( η ) } , n f ) ( η ) = x ∈ T d j =1 n d � � ( L A n f ) ( η ) = m j c j ( τ x η ) η x + e j (1 − η x ) x ∈ T d j =1 n × { f ( σ x , x + e j η ) − f ( η ) } . for a function f : { 0 , 1 } T d n → 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 or x − 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 u 0 : T d → [0 , 1], let ν n u 0 be the product Bernoulli measure on { 0 , 1 } T d n : ν n x ∈ T d u 0 ( η : η ( x ) = 1) = u 0 ( x / n ) , n . Gradient condition: For each j , there exist finitely supported signed measures m j , p , p = 1 , . . . , n j and local functions g j , p such that n j � � c j ( η )[ η 0 − η e j ] = m j , p ( y ) g j , p ( τ y η ) , p =1 y ∈ Z d � m j , p ( y ) = 0 . y ∈ Z d 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
Classical case ( ε n = 1) d Assume that η n = ν n u 0 for some continuous function 0 u 0 : T d → [0 , 1]. Hydrodynamic limit: For any t ≥ 0 and any smooth function H : T d → R , ⎡ � � ⎤ � � 1 � n →∞ E n � H ( x / n ) η n ⎦ = 0 , � � lim t ( x ) − T d H ( x ) u ( t , x ) dx � � ⎣ n d � � x ∈ T d � � n where u : [0 , ∞ ) × T d → [0 , 1] is the unique weak solution of the Cauchy problem � ∂ t u = ∇ · [ D ( u ) ∇ u ] + ∇ · [ σ ( u ) m ] , u (0 , · ) = u 0 ( · ) . 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 ( ε n ↓ 0) Fix α 0 ∈ (0 , 1) with σ ′ ( α 0 ) = 0 and assume that d η n = ν n α 0 + ε n v 0 for some function v 0 ∈ C 3+ ( T d ). 0 Let v : [0 , ∞ ) × T d → R be the unique weak (classical) solution of the Burgers eq.: � ∂ t v = ∇ · [ D ( α 0 ) ∇ v ] + (1 / 2) ∇ · [ v 2 σ ′′ ( α 0 ) m ] , v (0 , · ) = v 0 ( · ) . For each t ≥ 0, let u n t = α 0 + ε n v t , ν n t = ν n t and let µ n u n t be the distribution of η n t . 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 n 2 ε 4 n ≤ C 0 g d ( n ) for some constant C 0 , where g d ( n ) = n , log n , 1 if d = 1 , d = 2 , d ≥ 3 , respectively. Then, for any T > 0 , there exists a constant C 1 = C 1 ( T , v 0 , C 0 ) such that for any 0 ≤ t ≤ T, H ( µ n t | ν n t ) ≤ C 1 n d − 2 g d ( 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+) n n 2 g d ( n ) − 1 ↑ ∞ . For any Assume that n 2 ε 4 n ≤ C 0 g d ( n ) and ε 2 t ≥ 0 and any smooth function H : T d → R , ⎡ � � ⎤ � 1 � � n →∞ E n � H ( x / n )[ η n ⎦ = 0 , � � lim t ( x ) − α 0 ] − T d H ( x ) v ( t , x ) dx ⎣ � � ε n n d � � x ∈ T d � � n Remarks d Initial distribution: The assumption η n = ν n α 0 + ε n v 0 can be 0 replaced with the entropy bound at time 0. σ ′ ( α 0 ) = 0: In the case of general α ∈ (0 , 1), introducing the Galilean transformation α + ε n v ( t , x − ε − 1 n σ ′ ( α ) m t ), we can obtain a similar result. 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
3. Comments on the Proof 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
Following [Jara-Menezes, 19+], we shall compute the entropy production. Let H t = H ( µ n t | ν n t ). Then, we have d � � L ∗ , ν n dt H t ≤ − n 2 D ( g n t , L S n , ν n 1 − ∂ t log ν n d µ n � t ) + t . t n t We need to compute the integrand L ∗ , ν n 1 − ∂ t log ν n t n t explicitly. Indeed, it can be expressed in terms of the “Fourier coe ffi cients” of g j , p (but quite messy...). We also need to expand several terms in ε n properly: e.g. E ν n t [ τ − e j g j , p − g j , 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
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