aa r X i v : . [ m a t h . P R ] O c t Hydrodynamic limit for two-species exclusion processes
Makiko Sasada
Graduate School of Mathematical Sciences,The University of Tokyo, Komaba, Tokyo 153-8914, Japane-mail: [email protected]
Abstract
We consider two-species exclusion processes on the d -dimensional discrete torus takingthe effects of exchange, creation and annihilation into account. The model is, ingeneral, of nongradient type. We prove that the (charged) particle density convergesto the solution of a certain nonlinear diffusion equation under the diffusive rescalingin space and time. We also prove a lower bound on the spectral gap for the generatorof the process confined in a finite volume. Keywords: hydrodynamic limit; interacting particle systems; two-species exclusion pro-cesses
The aim of this paper is to obtain the hydrodynamic behavior of two-species exclusionprocesses. Our results can be applied to establish the hydrodynamic limit for the evolutionof height differences in interfaces governed by the 1-dimensional SOS dynamics.The two-species exclusion process describes the evolution of a system of mechanicallydistinguishable particles, say +particles and − particles moving on a discrete lattice spaceunder the constraint that at most one particle can occupy each site. The state space ofthe process is given by {− , , } T dN where T dN stands for the d -dimensional discrete toruswith side-length N and its elements (called configurations) are denoted by η = ( η ( x ) , x ∈ T dN ), with η ( x ) = 0 or 1 or − x ∈ T dN is empty or occupiedby a +particle or a − particle, respectively. Each ± particle moves to a neighboringempty site with the constant jump rate C ± >
0, respectively. Two different types ofneighboring particles exchange their locations with the constant rate C E ≥
0. Also theyannihilate simultaneously when they are neighboring with the constant rate C A ≥
0, andtwo different types of particles are created with the constant rate C C ≥ C A > C C >
0, so that the process has aunique conserved quantity P x ∈ T Nd η ( x ). We prove the hydrodynamic limit for the profileassociated with this quantity and obtain the explicit expression of the diffusion coefficient. Tel.: +81-3-5465-7001; Fax: +81-3-5465-7011 (M.Sasada).
MSC: primary 60K35, secondary 82C22.
1e classify the dynamics into three types as the case where C A > C C > C A > C C = 0 (Case 2) and C A = 0 and C C > C + = C − > C A = C C = C E = 0,so it is not included in our cases.The SOS dynamics describe the evolution of the integer-valued heights of interfaceson the discrete lattice. In the 1-dimensional case, the height difference of SOS dynamicsand the configuration of the two-species exclusion process have one-to-one correspondence,see, e.g. [1].This paper is organized as follows: In Section 2 we introduce our model and state themain results for three types of models respectively. In Section 3, we give the proof of themain theorem for Case 1. In the proof, we give a spectral gap estimate and characterizethe class of closed forms. In Sections 4 and 5, we give the proofs of the main theoremsfor Cases 2 and 3, respectively. In Section 6, we state the uniqueness results for nonlinearparabolic equations whose diffusion coefficient matrices are diagonal. The two-species exclusion process is a Markov process η t on the configuration space χ dN = {− , , } T dN , where T dN = ( Z /N Z ) d is the d -dimensional discrete torus. The dynamics aredefined by means of an infinitesimal generator L N acting on functions f : χ dN → R as( L N f )( η ) = X b ∈ ( T dN ) ∗ L b f ( η ) , where ( T dN ) ∗ stands for the set of all directed bonds b = ( x, y ), i.e., the ordered pairs of x, y ∈ T dN such that | x − y | = 1 where | x − y | = P ≤ i ≤ d | x i − y i | is the sum norm in R d .Here, for each bond b ∈ ( T dN ) ∗ ,(2.1) L b f ( η ) = c b ( η )( π b f )( η ) , where( π b f )( η ) = [1 { η ( x )=1 ,η ( y )=0 } + 1 { η ( x )=0 ,η ( y )= − } + 1 { η ( x )= − ,η ( y )=1 } ]( f ( η x,y ) − f ( η ))+ 1 { η ( x )=1 ,η ( y )= − } ( f ( η x =0 ,y =0 ) − f ( η )) + 1 { η ( x )=0 ,η ( y )=0 } ( f ( η x = − ,y =1 ) − f ( η )) , and c b ( η ) = C + { η ( x )=1 ,η ( y )=0 } + C − { η ( x )=0 ,η ( y )= − } + C E { η ( x )= − ,η ( y )=1 } + C A { η ( x )=1 ,η ( y )= − } + C C { η ( x )=0 ,η ( y )=0 } .
2n the above formula, η x,y , η x = − ,y =1 and η x =0 ,y =0 ∈ χ dN stand for η x,y ( z ) = η ( z ) if z = x, y , η ( y ) if z = x , η ( x ) if z = y , η x = − ,y =1 ( z ) = η ( z ) if z = x, y , − z = x ,1 if z = y ,and η x =0 ,y =0 ( z ) = η ( z ) if z = x, y ,0 if z = x ,0 if z = y ,respectively, and we assume that C + and C − are positive constants and C A , C C and C E are nonnegative constants.We will use the following simplified notations η ( x,y ) = η x,y if ( η ( x ) , η ( y )) = (1 ,
0) or (0 , −
1) or ( − , η x =0 ,y =0 if ( η ( x ) , η ( y )) = (1 , − η x = − ,y =1 if ( η ( x ) , η ( y )) = (0 , x,ya,b ( η ) = 1 { η ( x )= a,η ( y )= b } , and define r b for b ∈ ( T dN ) ∗ by r b ( η ) = Ψ x,y , ( η ) + Ψ x,y , − ( η ) + Ψ x,y − , ( η ) + Ψ x,y , − ( η ) + Ψ x,y , ( η ) . With these notations, L b and π b can be rewritten as L b f ( η ) = c b ( η )( f ( η ( x,y ) ) − f ( η )) ,π b f ( η ) = r b ( η )( f ( η ( x,y ) ) − f ( η )) , respectively.The process is reversible with respect to the following one parameter family of trans-lation invariant product measures ν ρ . Definition 2.1.
For each fixed ρ ∈ [ − , , let ν ρ be a product measure on χ dN withmarginals given by ν ρ { η ( x ) = 1 } = 1 − Φ( ρ ) + ρ ,ν ρ { η ( x ) = 0 } = Φ( ρ ) ,ν ρ { η ( x ) = − } = 1 − Φ( ρ ) − ρ , or all x ∈ T dN , where Φ( ρ ) = − √ β + ρ − βρ − β if β = − ρ if β = with β = C C /C A . Especially, if C A > , C C = 0 , then ν ρ { η ( x ) = 1 } = ρ ∨ ,ν ρ { η ( x ) = 0 } = 1 − | ρ | ,ν ρ { η ( x ) = − } = | ρ ∧ | , and if C A = 0 , C C > , then ν ρ { η x = 1 } = 1 + ρ ,ν ρ { η x = 0 } = 0 ,ν ρ { η x = − } = 1 − ρ . The index ρ stands for the density of particles with charge, namely E ν ρ [ η (0)] = ρ . Wewill abuse the same notation ν ρ for the product measures on the configuration spaces χ dN or χ d = {− , , } Z d on the torus or on the infinite lattice. The expectation with respectto ν ρ will be sometimes denoted by Z f ( η ) ν ρ ( dη ) = h f i ρ . From the definition, our model satisfies the detailed balance condition, namely, forany directed bond b = ( x, y ),(2.2) c b ( η ) ν ρ ( η ) = c b ′ ( η ( x,y ) ) ν ρ ( η ( x,y ) )holds, where b ′ = ( y, x ) is the reversed bond of b .Here and after, we call f a cylinder function on χ d if f depends on the configurationsonly through a finite set of coordinates. For any directed bond b = ( x, y ) and cylinderfunctions f , g , let us define D b ( ν ρ ; f, g ) and D b ( ν ρ ; f ) by D b ( ν ρ ; f, g ) := h− ( L b + L b ′ ) f, g i ρ and D b ( ν ρ ; f ) := D b ( ν ρ ; f, f ) , where b ′ = ( y, x ) and h· , ·i ρ stands for the inner product in L ( ν ρ ). The reversibility (2.2)implies(2.3) D b ( ν ρ ; f, g ) = h c b ( π b f )( π b g ) i ρ . Let τ x be the shift operator acting on the set A ⊂ Z d and cylinder functions f as wellas configurations η as follows: 4 x A := x + A, τ x f ( η ) = f ( τ x η ) , ( τ x η )( z ) := η ( z − x ) , z ∈ Z d . For every cylinder function g : χ d → R , consider the formal sumΓ g := X x ∈ Z d τ x g which does not make sense but for which the gradient π Γ g = ( π ,e Γ g , ..., π ,e d Γ g )is well defined.We are now in a position to define the diffusion coefficient. For each ρ ∈ [ − , d ( ρ ) = 1 χ ( ρ ) inf g D ,e ( ν ρ ; η (0) + Γ g )where inf g is taken over all cylinder functions g and e is a unit vector of arbitrary direction.In this formula χ ( ρ ) stands for the so-called static compressibility which in our case is equalto χ ( ρ ) = h η (0) i ρ − h η (0) i ρ = 1 − Φ( ρ ) − ρ Notice that d ( ρ ) does not depend on the choice of a unit vector e i , ≤ i ≤ d .For a probability measure µ N on χ dN , we denote by P µ N the distribution on the pathspace D ( R + , χ dN ) of the Markov process η t = { η t ( x ) , x ∈ T dN } with generator N L N , whichis accelerated by a factor N , and the initial measure µ N . Hereafter E µ N stands for theexpectation with respect to P µ N .With these notations our main theorems are stated as follows: Theorem 2.1.
Assume C A > and C C > . Let ( µ N ) N ≥ be a sequence of probabilitymeasures on χ dN such that the corresponding initial density fields satisfy lim N →∞ µ N [ | N d X x ∈ T dN G ( xN ) η ( x ) − Z T d =[0 , d G ( u ) ρ ( u ) du | > δ ] = 0 , for every δ > , every continuous function G : T d → R and some measurable function ρ : T d → [ − , . Then, for every t > , lim sup N →∞ P µ N [ | N d X x ∈ T dN G ( xN ) η t ( x ) − Z T d G ( u ) ρ ( t, u ) du | > δ ] = 0 , for every δ > and every continuous function G : T d → R , where ρ ( t, u ) is the uniqueweak solution of the following nonlinear parabolic equation: (2.4) ∂ t ρ ( t, u ) = ∆( ˜ d ( ρ ( t, u ))) (cid:16) = d X i =1 ∂∂u i (cid:8) d ( ρ ( t, u )) ∂ρ∂u i ( t, u ) (cid:9)(cid:17) ρ (0 , · ) = ρ ( · ) , where ˜ d ( ρ ) = Z ρ − d ( γ ) dγ. Remark 2.1.
If we assume C + + C − − C A − C E = 0 , then our model turns out to be agradient system. In this case, d ( ρ ) = − Φ ′ ( ρ )2 ( C + − C − ) + ( C + + C − ) holds. In particular,we can compute the diffusion coefficient d ( ρ ) explicitly from the concrete values of C + , C − and β . Remark 2.2.
Generalized exclusion process with κ = 2 is corresponding to our model with C + = C − = C A = C C = 1 and C E = 0 . Theorem 2.2.
Assume C A > , C C = 0 and the gradient condition C + + C − − C A − C E =0 . Let ( µ N ) N ≥ satisfy the same assumption as in Theorem 2.1. Then, for every t > , lim sup N →∞ P µ N [ | N d X x ∈ T dN G ( xN ) η t ( x ) − Z T d G ( u ) ρ ( t, u ) du | > δ ] = 0 , for every δ > and every continuous function G : T d → R , where ρ ( t, u ) is the uniqueweak solution of the following nonlinear parabolic equation: (2.5) ∂ t ρ ( t, u ) = ∆( P ( ρ ( t, u ))) (cid:16) = d X i =1 ∂ ∂u i P ( ρ ( t, u )) (cid:17) ρ (0 , · ) = ρ ( · ) , where (2.6) P ( ρ ) = C + ρ { ρ> } − C − ρ { ρ< } . Remark 2.3.
Equation (2.5) is the weak or enthalpy formulation of the following two-phases Stefan problem: ∂ t ρ ( t, u ) = C + ∆ ρ ( t, u ) on L ( t ) = { ρ ( t, u ) > } ∂ t ρ ( t, u ) = C − ∆ ρ ( t, u ) on S ( t ) = { ρ ( t, u ) < } n · (cid:16) C + ∇ ( ρ ( t, u ) ∨ − C − ∇ ( ρ ( t, u ) ∧ (cid:17) on Σ( t ) = { ρ ( t, u ) = 0 } ρ (0 , · ) = ρ ( · ) where n denotes the unit normal vector on Σ( t ) directed to L ( t ) and ∇ ( ρ ( t, u ) ∨ (re-spectively ∇ ( ρ ( t, u ) ∧ ) is the limit of the gradient of ρ ∨ (respectively ρ ∧ ) at u ∈ Σ( t ) when approached from L ( t ) (respectively S ( t ) ), see [2]. Theorem 2.3.
Assume C A = 0 , C C > and the gradient condition C + + C − − C E = 0 .Let ( µ N ) N ≥ satisfy the same assumption as in Theorem 2.1. Then, for every t > , lim sup N →∞ P µ N [ | N d X x ∈ T dN G ( xN ) η t ( x ) − Z T d G ( u ) ρ ( t, u ) du | > δ ] = 0 , or every δ > and every continuous function G : T d → R , where ρ ( t, u ) is the uniqueweak solution of the heat equation: (2.7) ∂ t ρ ( t, u ) = C E ∆ ρ ( t, u ) (cid:16) = C E d X i =1 ∂ ∂u i ρ ( t, u ) (cid:17) ρ (0 , · ) = ρ ( · ) . In this section, we consider Case 1, namely the dynamics with C A > C C > We start with considering a class of martingales associated with the empirical measure.We take
T > H : T d → R , let M H,N ( t ) = M H ( t ) be the martingale defined by M H ( t ) = h π Nt , H i − h π N , H i − Z t N L N h π Ns , H i ds, where π Nt stands for the empirical measure associated with η t , namely(3.1) π Nt ( du ) = 1 N d X x ∈ T dN η t ( x ) δ xN ( du ) , ≤ t ≤ T, u ∈ T d , and h π Nt , f i stands for the integration of f with respect to π Nt .A simple computation shows that the expected value of the quadratic variation of M H ( t ) vanishes as N ↑ ∞ , and therefore by Doob’s inequality, for every δ >
0, we havelim N →∞ P µ N [ sup ≤ t ≤ T | M H ( t ) | ≥ δ ] = 0 . A spatial summation by parts permits to rewrite the martingale M H ( t ) as M H ( t ) = h π Nt , H i − h π N , H i− d X i =1 Z t N − d X x ∈ T dN ( ∂ Nu i H )( xN ) τ x W ,e i ( η s ) ds, (3.2) 7here W ,e i represents the instantaneous current from 0 to e i : W ,e i ( η ) = C + (Ψ ,e i , ( η ) − Ψ ,e i , ( η )) + ( C A + 2 C E )(Ψ ,e i , − ( η ) − Ψ ,e i − , ( η ))+ C − (Ψ ,e i , − ( η ) − Ψ ,e i − , ( η ))and ∂ Nu i H represents the discrete derivative of H in the i -th direction:( ∂ Nu i H )( xN ) = N [ H ( x + e i N ) − H ( xN )] . Next we show that the current W ,e i can be decomposed into a linear combination ofthe gradients { η ( e j ) − η (0) , ≤ j ≤ d } and a function in the range of the generator L N : W ,e i + P dj =1 D i,j ( ρ )[ η ( e j ) − η (0)] = L N f for a certain cylinder function f and some matrix D i,j ( ρ ) that depends on the density, see Theorem 3.1 and Corollary 3.2 for more precisestatement.Denote by { D i,j ( ρ ) , ≤ i, j ≤ d } the unique symmetric matrix such that a ∗ D ( ρ ) a = 1 χ ( ρ ) inf g n d X i =1 D ,e i ( ν ρ , a i η (0) + Γ g ) o , for every vector a in R d where inf g is taken over all cylinder functions g .For positive integers l, N, a function H in C ( T d ) and a cylinder function f on χ d , let X f ,iN,l ( H, η ) = N − d X x ∈ T dN H ( xN ) τ x V f ,li ( η ) , where V f ,li ( η ) = W ,e i ( η ) + d X j =1 D i,j ( η l (0))[ η l ( e j ) − η l (0)] − L N f ( η ) , and η l ( x ) = 1(2 l + 1) d X | y − x |≤ l η ( y ) , x ∈ T dN . Theorem 3.1.
Fix ρ ∈ ( − , arbitrarily. Then, for every function H in C ( T d ) and ≤ i ≤ d , we have inf f ∈C lim sup ε → lim sup N →∞ N d log E ν Nρ [exp { N d | Z T X f ,iN,εN ( H, η s ) ds |} ] = 0 , where C stands for the set of cylinder functions on χ d . The proof of Theorem 3.1 is postponed to the next subsection. This theorem impliesthe following corollary. For a positive integer l and a function H in C ( T d ), let Y iN,l ( H, η ) = N − d X x ∈ T dN H ( xN ) { W x,x + e i ( η ) + d X j =1 D i,j ( η l ( x ))[ η l ( x + e j ) − η l ( x )] } . orollary 3.2. For every function H in C ( T d ) and ≤ i ≤ d , lim sup ε → lim sup N →∞ E µ N [ | Z T Y iN,εN ( H, η s ) ds | ] = 0 . To prove this corollary, we can use the method in [3] straightforwardly. In particular,the L N f term is negligible. We have now all elements to prove the hydrodynamic behaviorof our nongradient system. Proof of Theorem 2.1.
Recall that the empirical measure π Nt is defined by (3.1). Denoteby Q µ N the distribution on the path space D ([0 , T ] , M ( T d )) of the process π Nt where M ( T d ) stands for the space of signed measures on T d endowed with the weak topology.Following the same argument as for the generalized exclusion process in [3] it is easyto prove that the sequence { Q µ N , N ≥ } is weakly relatively compact and that everylimit points Q ∗ is concentrated on absolutely continuous paths π t ( du ) = π ( t, u ) du withdensity bounded by 1 and -1 from above and below respectively: − ≤ π ( t, u ) ≤ { Q µ N , N ≥ } are concentrated on absolutely continuous trajectories π ( t, du ) = π ( t, u ) du whose densities are weak solutions of the equation (2.4).Fix a smooth function H : T d → R and recall the definition of the martingale M H ( t ).Applying Corollary 3.2 to the last integral term in the formula (3.2) of M H ( t ), we obtainthat for every δ > ε → lim sup N →∞ P µ N [ |h π NT , H i − h π N , H i + d X i,j =1 Z T N − d X x ∈ T dN ( ∂ Nu i H )( xN ) τ x V i,j,εN ( η s ) ds | > δ ] = 0 , where V i,j,εN ( η ) = D i,j ( η εN (0))[ η εN ( e j ) − η εN (0)] . Denote by ˜ D i,j the integral of D i,j : ˜ D i,j ( ρ ) = R ρ − D i,j ( γ ) dγ . Since H is smooth and D i,j is continuous by Theorem 3.8 stated below, with the help of Taylor’s expansion and aspatial summation by parts, we havelim sup ε → lim sup N →∞ P µ N [ |h π NT , H i − h π N , H i− d X i,j =1 Z T N − d X x ∈ T dN ( ∂ u i ,u j H )( xN ) ˜ D i,j ( η εNs ( x )) ds | > δ ] = 0 . Therefore, for every limit point Q ∗ of the sequence Q µ N ,lim sup ε → Q ∗ [ |h π T , H i − h π , H i− d X i,j =1 Z T ds Z T d du ( ∂ u i ,u j H )( u ) ˜ D i,j (( π s ∗ ι ε )( u )) | > δ ] = 09here ι ε ( · ) := (2 ε ) − d [ − ε,ε ] d ( · )and ∗ represents the convolution. Since each limit point Q ∗ is concentrated on absolutelycontinuous paths π t = π ( t, u ) du with − ≤ π ( t, u ) ≤
1, for each fixed 0 ≤ s ≤ T , ( π s ∗ ι ε )( u )converges to π ( s, u ) for almost u in T d as ε ↓
0. From this remark and the continuity of { ˜ D i,j , ≤ i, j ≤ d } , we obtain that Q ∗ [ (cid:12)(cid:12)(cid:12) h π T , H i − h π , H i − d X i,j =1 Z T ds Z T d du ( ∂ u i ,u j H )( u ) ˜ D i,j ( π ( s, u )) (cid:12)(cid:12)(cid:12) > δ ] = 0for all H in C ( T d ). The fact D ( ρ ) = d ( ρ ) I , namely ˜ D ( ρ ) = ˜ d ( ρ ) I proved in Theorem3.10 permits to rewrite the last expression as Q ∗ [ h π T , H i = h π , H i + Z T ds Z T d du ∆ H ( u ) ˜ d ( π ( s, u ))] = 1 . Denote by { t n } n ∈ N a dense subset of [0 , T ] and repeat the same argument as we have doneup to this point for any fixed t n , then Q ∗ [ h π t n , H i = h π , H i + Z t n ds Z T d du ∆ H ( u ) ˜ d ( π ( s, u )) for every n ∈ N ] = 1 . Since Q ∗ is the probability measure on D space and ˜ d is bounded function on [ − ,
1] and Q ∗ is concentrated on paths π t = π ( t, u ) du with − ≤ π ( t, u ) ≤ Q ∗ is concentrated onthe weak solution of (2.4) which concludes the proof of the theorem. To state the main theorem of this subsection, first we introduce some notation. For a fixedpositive integer l we denote by Λ l a cube in Z d of side-length 2 l + 1 centered at the origin:Λ l := {− l, − l + 1 , ..., l − , l } d . We denote the set of cylinder functions on χ d by C . ForΨ in C , denote by Λ Ψ the smallest d -dimensional rectangle that contains the support ofΨ and by s Ψ the smallest positive integer s such that Λ Ψ ⊂ Λ s . Let C be the space ofcylinder functions with mean zero with respect to all canonical invariant measures: C = { g ∈ C ; h g i Λ g ,K = 0 for all − | Λ g | ≤ K ≤ | Λ g | } . Here, for a finite subset Λ of Z d , we denote by | Λ | the cardinality of Λ and by h·i Λ ,K theexpectation with respect to the canonical measure ν Λ ,K := ν α ( · | P x ∈ Λ η ( x ) = K ) for −| Λ | ≤ K ≤ | Λ | which is indeed independent of the choice of α . For a rectangle Λ anda canonical measure ν Λ ,K , denote by h· , ·i Λ ,K (resp. h· , ·i α ) the inner product in L ( ν Λ ,K )(resp. L ( ν α )).It is known that to conclude the proof of Theorem 3.1 it is enough to show that(3.3) inf f ∈C lim l →∞ sup K (2 l ) d h ( − L Λ l ) − ˜ V f ,li , ˜ V f ,li i l,K = 010here ˜ V f ,li ( η ) = (2 l ′ + 1) − d X | y |≤ l ′ τ y W ,e i ( η )+ d X j =1 D i,j ( η l (0))[ η l ′ ( e j ) − η l ′ (0)] − (2 l f + 1) − d X y ∈ Λ l f ( τ y L N f )( η ) ,l ′ = l − l f = l − s f − τ y L N f is F Λ l -measurable for every y in Λ l f . Thisfollows from Theorem 3.4 and Corollary 3.9 below.For the beginning of the proof we obtain a variational formula for this variance. Westart with introducing a semi-norm on C , which is closely related to the central limittheorem variance. For 1 ≤ k ≤ d denote U k = ( U k , ..., U kd ) the d -dimensional cylinderfunction with coordinates defined by( U k ) i ( η ) = δ i,k ∇ ,e k η (0) for all 1 ≤ i ≤ d. Here δ i,j stands for the delta of Kronecker. For cylinder functions g , h in C and 1 ≤ i ≤ d ,let ≪ g, h ≫ ρ, = X x ∈ Z d h g, τ x h i ρ and ≪ g ≫ ρ,j = X x ∈ Z d x j h g, η ( x ) i ρ , where x j stands for the j -th coordinate of x ∈ Z d . Both ≪ g, h ≫ ρ, and ≪ g ≫ ρ,j arewell defined because g and h belong to C and therefore all but a finite number of termsvanish. For h in C , define the semi-norm ≪ h ≫ ρ by ≪ h ≫ ρ (3.4) = sup g ∈C ,a ∈ R d { ≪ g, h ≫ ρ, +2 d X i =1 a i ≪ h ≫ ρ,i − d X i =1 h (cid:0) d X j =1 a j ( U j ) i + ∇ ,e i Γ g (cid:1) i ρ } = sup g ∈C ,a ∈ R d { ≪ g, h ≫ ρ, +2 d X i =1 a i ≪ h ≫ ρ,i − d X i =1 D ,e i ( ν ρ ; a i η (0) + Γ g ) } , where a = ( a i ) di =1 .We investigate in the next section several properties of the semi-norm ≪ · ≫ ρ , whilein this section we prove that the variance(2 l ) − d h ( − L Λ l ) − X | x |≤ l ψ τ x ψ, X | x |≤ l ψ τ x ψ i l,K l of any cylinder function ψ in C converges to ≪ ψ ≫ ρ , as l ↑ ∞ and K l (2 l ) d → ρ . Here l ψ stands for l − s ψ so that the support of τ x ψ is included in Λ l for every x ≤ l ψ . Byelementary computations relying on an adequate change of variables, the norm ≪ · ≫ ρ may be rewritten as ≪ h ≫ ρ = sup g ∈C ,a ∈ R d { ≪ g, h ≫ ρ, +2 d X i =1 a i ≪ h ≫ ρ,i
11 2 d X i =1 a i ≪ W ,e i , g ≫ ρ, −k a k h ( ∇ ,e η (0)) i ρ − hk∇ Γ g k i ρ } . We are now in a position to state the main result of this section.
Proposition 3.3.
Consider a cylinder function ψ in C and a sequence of integers K l such that − (2 l + 1) d ≤ K l ≤ (2 l + 1) d and lim l →∞ K l (2 l ) d = ρ . Then, lim l →∞ (2 l ) − d h ( − L Λ l ) − X | x |≤ l ψ τ x ψ, X | x |≤ l ψ τ x ψ i l,K l = ≪ ψ ≫ ρ . Once Theorem 3.18, which is stated below, is established the proof of Proposition 3.3is the same as that of Theorem 7.4.1 of [3] since the proof does not depend on the specificform of D b .We conclude this section proving that for each ψ in C the function ≪ ψ ≫ : [ − , → R + that associates to each density ρ the value ≪ ψ ≫ ρ is continuous and that theconvergence of the finite volume variances to ≪ · ≫ ρ is uniform on [ − , l in N and − (2 l +1) d ≤ K ≤ (2 l +1) d , denote by V ψl (cid:0) K (2 l +1) d (cid:1) the variance of (2 l +1) − d P | x |≤ l ψ τ x ψ with respect to ν l,K : V ψl (cid:16) K (2 l + 1) d (cid:17) = (2 l ) − d h ( − L Λ l ) − X | x |≤ l ψ τ x ψ, X | x |≤ l ψ τ x ψ i l,K We may interpolate linearly to extend the definition of V ψl to the all interval [ − , V ψl is continuous. Proposition 3.3 asserts that V ψl converges, as l ↑ ∞ ,to ≪ ψ ≫ ρ , for any sequence K l such that K l (2 l +1) d → ρ . In particular, lim l →∞ V ψl ( ρ l ) = ≪ ψ ≫ ρ for any sequence ρ l → ρ . This implies that ≪ ψ ≫ ρ is continuous and that V ψl ( · )converges uniformly to ≪ ψ ≫ · as l ↑ ∞ . We have thus proved the following theorem. Theorem 3.4.
For each fixed h in C , ≪ h ≫ ρ is continuous as a function of the density ρ on [ − , . Moreover, the variance (2 l ) − d h ( − L Λ l ) − X | x |≤ l h τ x h, X | x |≤ l h τ x h i l,K l converges uniformly to ≪ h ≫ ρ as l ↑ ∞ and K l (2 l +1) d → ρ . In particular, lim l →∞ sup − (2 l +1) d ≤ K ≤ (2 l +1) d (2 l ) − d h ( − L Λ l ) − X | x |≤ l h τ x h, X | x |≤ l h τ x h i l,K = sup − ≤ ρ ≤ ≪ h ≫ ρ . We investigate here the main properties of the semi norm ≪ · ≫ ρ introduced in the previ-ous section. We first define from ≪ · ≫ ρ a semi-inner product on C through polarization:(3.5) ≪ g, h ≫ ρ = 14 {≪ g + h ≫ ρ − ≪ g − h ≫ ρ } .
12t is easy to check that (3.5) defines a semi-inner product on C . Denote by N ρ thekernel of the semi-norm ≪ · ≫ ρ on C . Since ≪ · ≫ ρ is a semi-inner product on C , thecompletion of C | N ρ , denoted by H ρ , is a Hilbert space.Simple computations show that the linear space generated by the currents { W ,e i , ≤ i ≤ d } and L C = { Lg ; g ∈ C } are subsets of C . The first main result of this sectionconsists in showing that H ρ is the completion of L C | N ρ + { W ,e i , ≤ i ≤ d } , in otherwords, that all elements of H ρ can be approximated by P ≤ i ≤ d a i W ,e i + Lg for some a in R d and g in C . To prove this result we derive two elementary identities:(3.6) ≪ h, Lg ≫ ρ = − ≪ h, g ≫ ρ, and ≪ h, W ,e i ≫ ρ = − ≪ h ≫ ρ,i for all h, g in C and 1 ≤ i ≤ d .By Proposition 3.3 and (3.5), the semi-inner product ≪ h, g ≫ ρ is the limit of thecovariance (2 l ) − d h ( − L Λ l ) − P | x |≤ l g τ x g, P | x |≤ l h τ x h i l,K l as l ↑ ∞ and K l (2 l ) d → ρ . In partic-ular, if g = Lg , for some cylinder function g , the inverse of the generator cancels withthe generator. Therefore, ≪ h, Lg ≫ ρ is equal to − lim l →∞ (2 l ) − d h X | x |≤ l g τ x g , X | x |≤ l h τ x h i l,K l = ≪ g , h ≫ ρ, . The second identity is proved in a similar way.It follows from the first identity of (3.6) that the gradients { η ( e i ) − η (0) , ≤ i ≤ d } areorthogonal to the space L C , while the second identity permits to compute inner productof cylinder functions with the current: ≪ η ( e i ) − η (0) , Lh ≫ ρ = 0 , (3.7) ≪ η ( e i ) − η (0) , W ,e j ≫ ρ = − χ ( ρ ) δ i,j , (3.8)and(3.9) ≪ W ,e i , W ,e j ≫ ρ = h ( ∇ ,e η (0)) i ρ δ i,j . for all 1 ≤ i, j ≤ d and h ∈ C . In this formula χ ( ρ ) stands for the static compressibilityand is equal to h η (0) i ρ − h η (0) i ρ . Furthermore,(3.10) ≪ d X j =1 a j W ,e j + Lg ≫ ρ = d X i =1 h{∇ ,e i ( a i η (0) + Γ g ) } i ρ for a in R d and g in C . In particular, the variational formula for ≪ h ≫ ρ writes(3.11) ≪ h ≫ ρ = sup g ∈C ,a ∈ R d {− ≪ h, d X i =1 a i W ,e i + Lg ≫ ρ − ≪ d X i =1 a i W ,e i + Lg ≫ ρ } . Proposition 3.5.
Recall that we denote by L C the space { Lg ; g ∈ C } . Then, for each − ≤ ρ ≤ , we have H ρ = L C | N ρ ⊕ { W ,e i , ≤ i ≤ d } . roof. We can apply the proof of Proposition 7.5.2 in [3] straightforwardly.
Corollary 3.6.
For each g ∈ C , there exists a unique vector a ∈ R d such that g − d X j =1 a j W ,e j ∈ L C in H ρ . We now start to describe the diffusion coefficient D of the hydrodynamic equation.From Corollary 3.6, there exists a matrix { Q i,j , ≤ i, j ≤ d } such that(3.12) η ( e i ) − η (0) + d X j =1 Q i,j W ,e j ∈ L C in H ρ . Notice that the matrix Q = Q ( ρ ) depends on the density ρ because the inner productdepends on ρ . It is easily shown that Q is symmetric and strictly positive.Denote by D = D ( ρ ) the inverse of Q , which is also symmetric and strictly positive.We will see below that D ( ρ ) is the diffusion coefficient of the hydrodynamic equation (2.4).Since D is the inverse of Q , we have that W ,e i + d X j =1 D i,j [ η ( e j ) − η (0)] ∈ L C in H ρ . for 1 ≤ i ≤ d . This relation provides a variational characterization of the diffusioncoefficient D . Indeed, for all vectors a ∈ R d ,(3.13) inf g ∈C {≪ d X i =1 a i W ,e i + d X i,j =1 a i D i,j [ η ( e j ) − η (0)] − Lg ≫ ρ } = 0 . Since gradients are orthogonal to the space L C , ≪ η ( e j ) − η (0) , W ,e i ≫ ρ = − χ ( ρ ) δ i,j , and ≪ η ( e j ) − η (0) , η ( e k ) − η (0) ≫ ρ = χ ( ρ ) Q j,k = χ ( ρ )[ D − ] j,k , the last identity reduces toinf g ∈C {− χ ( ρ ) a ∗ Da + ≪ d X i =1 a i W ,e i − Lg ≫ ρ } = 0 , where a ∗ stands for the transposition of a . We have thus obtained a variational formulafor D ( ρ ). Theorem 3.7.
The diffusion coefficient D ( ρ ) is such that (3.14) a ∗ Da = 1 χ ( ρ ) inf g ∈C ≪ d X i =1 a i W ,e i − Lg ≫ ρ = 1 χ ( ρ ) inf g ∈C d X i =1 h{ ( ∇ ,e i ( a i η (0) − Γ g ) } i ρ for all a ∈ R d . D since D is symmetric.It is now easy to prove the diffusion coefficient is continuous including at the boundaryof [ − , χ ( ρ ), h ( ∇ ,e η (0)) i ρ and h Ψ ,e i ρ , we have that D ( ρ ) converges to C + I as ρ ↑ C − I as ρ ↓ − Theorem 3.8.
The diffusion coefficient D ( ρ ) is continuous on [ − , . Moreover it con-verges to C + I as ρ ↑ and C − I as ρ ↓ − . From the continuity of the diffusion coefficient we have
Corollary 3.9.
Let D be the matrix defined in Theorem 3.7. Then, for each ≤ i ≤ d , inf f ∈C sup − ≤ ρ ≤ ≪ W ,e i + d X j =1 D i,j ( ρ )[ η ( e j ) − η (0)] − L f ( η ) ≫ ρ = 0 . This result together with (3.3), the definition of ˜ V f ,li and Theorem 3.4 concludes theproof of Theorem 3.1.We conclude this section proving that the diffusion coefficient D is a diagonal matrixand it has the same diagonal component, therefore D ( ρ ) = d ( ρ ) I . Theorem 3.10.
There exists a continuous function d ( ρ ) on [ − , such that D ( ρ ) = d ( ρ ) I and χ ( ρ )4 h Ψ ,e i ρ ≤ d ( ρ ) ≤ h ( ∇ ,e η (0)) i ρ χ ( ρ ) . Proof.
Because of the symmetry of the dynamics, it is obvious that D has the samediagonal component. It remains to show that D is a diagonal matrix.According to [7], the diffusion coefficient matrix defined by the variational formula(3.14) coincides with the diffusion coefficient matrix defined by the Green-Kubo formulabased on the current-current correlation function: a ∗ D ( ρ ) a := 1 χ ( ρ ) n d X i =1 a i h ( ∇ ,e i η (0)) i ρ − Z ∞ X x ∈ Z d E ν ρ [ W a e Lt τ x W a ] dt o where W a := P i a i W ,e i . Therefore, we have only to prove that Z ∞ X x ∈ Z d E ν ρ [ W ,e i e Lt τ x W ,e j ] dt = 0for all i = j . In [5], Kipnis and Varadhan proved some equivalent relation about thecentral limit theorem variance. We can use one of them. It holds that Z ∞ X x ∈ Z d E ν ρ [ W ,e i e Lt τ x W ,e j ] dt = lim λ → X x E ν ρ [ W ,e i τ x g jλ ]where g jλ is a solution of the resolvent equation λg jλ − Lg jλ = W ,e j .15enote by θ i the reflection operator with respect to e i along the e i direction, namelyfor x ∈ Z d , θ i x = ( x , x , ..., x i − , − x i +1 , x i +1 , ..., x d ). We may extend θ i to configurationsin χ d and to functions on χ d naturally:( θ i η )( x ) := η ( θ i x ) ( θ i f )( η ) := f ( θ i η ) . Then, for i = j , λ θ i τ x g jλ − L θ i τ x g jλ = θ i τ x W ,e j = τ θ i x − e i W ,e j . Therefore, since ν ρ is translation invariant and a product measure, E ν ρ [ W ,e i τ x g jλ ] = E ν ρ [ θ i W ,e i θ i τ x g jλ ] = E ν ρ [ − W ,e i τ θ i x − e i g jλ ] . Since the map x → ( θ i x − e i ) is a bijection, X x E ν ρ [ W ,e i τ x g jλ ] = X x E ν ρ [ − W ,e i τ x g jλ ] . Thus, P x E ν ρ [ W ,e i τ x g jλ ] = 0 for all λ . Remark 3.1.
If we assume the gradient condition C + + C − − C A − C E = 0 , then W ,e i = h ( η (0)) − h ( η ( e i )) with h ( −
1) = C + , h (0) = 0 and h ( −
1) = − C − . In this case, ≪ W ,e i , Lg ≫ ρ = 0 holds. Therefore d ( ρ ) = h ( ∇ ,e i η (0)) i ρ = − Φ ′ ( ρ )2 ( C + − C − )+ ( C + + C − ) . In this section, we prove the spectral gap for the two-species exclusion process on finite d -dimensional cubes. For a positive integer N , we denote by Ω N the box { , ..., N } d and by Y N the space of configurations {− , , } Ω N . Let L Ω N be the generator of the two-speciesexclusion process on Ω N with free boundary conditions: L Ω N f ( η ) = X x,y ∈ Ω N , | x − y | =1 L xy f ( η )where L xy was defined in (2.1).For −| Ω N | ≤ K ≤ | Ω N | , we denote by Y N,K the hyperplane { η ; P x ∈ Ω N η ( x ) = K } and by µ N,K the product measure ν ρ on Y N conditioned on the hyperplane Y N,K : µ N,K ( · ) = ν ρ ( · | X x ∈ Ω N η ( x ) = K ) . As in the previous sections, expected values with respect to the measure µ N,K are denotedby h·i
N,K : h f i N,K := Z Y N,K f ( η ) µ N,K ( dη ) . In the main theorem of this section we prove that the generator L Ω N in L ( µ N,K ) hasa spectral gap of order at least N − . 16 heorem 3.11. There exists a positive constant C , which only depends on the constants C + , C − , C A , C C and C E , such that for every positive integer N , every integer −| Ω N | ≤ K ≤ | Ω N | and every function f in L ( µ N,K ) satisfying h f i N,K = 0 , h f i N,K ≤ CN h− L Ω N f, f i N,K . We start with showing that h− L Ω N f, f i N,K is bounded below by C h− ˜ L Ω N f, f i N,K with some constant C where ˜ L Ω N acting on functions as˜ L Ω N f ( η ) = X x,y ∈ Ω N , | x − y | =1 ˜ L xy f ( η )and ˜ L xy f ( η ) = [Ψ x,y , ( η ) + Ψ x,y , − ( η ) + Ψ x,y − , ( η )]( f ( η x,y ) − f ( η ))+ Ψ x,y , − ( η )( f ( η x =0 ,y =0 ) − f ( η )) + β Ψ x,y , ( η )( f ( η x = − ,y =1 ) − f ( η )) . Notice that ˜ L Ω N is the generator of two-species exclusion process with C + = C − = C A = C E = 1 and C C = β . The probability measures µ N,K are also reversible for the Markovprocess with generator ˜ L Ω N . Lemma 3.12.
If we assume that C + , C − , C A , C C are all positive constants and C E is anonnegative constant, there exists a positive constant C such that for every positive integer N , every integer −| Ω N | ≤ K ≤ | Ω N | , every function f in L ( µ N,K ) and every directedbond b = ( x, y ) we have h ( − ˜ L xy − ˜ L yx ) f, f i N,K ≤ C h ( − L xy − L yx ) f, f i N,K
Proof.
It is enough to prove the lemma assuming C E = 0. Especially, we only have tobound the term h Ψ x,y − , ( η )( f ( η x,y ) − f ( η )) i N,K by the term C h ( − L xy − L yx ) f, f i N,K withsome constant C . By the Cauchy-Shwartz inequality, we have h Ψ x,y − , ( η )( f ( η x,y ) − f ( η )) i N,K = h Ψ x,y , − ( η )( f ( η x,y ) − f ( η )) i N,K ≤ h Ψ x,y , − ( η )[( f ( η x,y ) − f ( η x =0 ,y =0 )) + ( f ( η x =0 ,y =0 ) − f ( η )) ] i N,K and the last expression is written as2 β h Ψ x,y , ( η )( f ( η x = − ,y =1 ) − f ( η )) i N,K + 2 h Ψ x,y , − ( η )( f ( η x =0 ,y =0 ) − f ( η )) i N,K by change of variables. Therefore, we can obtain the desirable estimate with the constant C := min { C + , C − , C A } .Now, to conclude the proof of Theorem 3.11, we have only to prove the theorem asfollows: Theorem 3.13.
There exists a positive constant C such that for every positive integer N ,every −| Ω N | ≤ K ≤ | Ω N | and every function f in L ( µ N,K ) satisfying h f i N,K = 0 , h f i N,K ≤ CN h− ˜ L Ω N f, f i N,K . Y N , whose generator˜ L m Ω N acting on functions f as˜ L m Ω N f ( η ) = 1 | Ω N | X x,y ∈ Ω N ˜ L xy f ( η ) . Notice that the probability measures µ N,K are also reversible for the Markov process withgenerator ˜ L m Ω N . This generator has a spectral gap in L ( µ N,K ) of order at least 1 as statedin the next theorem.
Theorem 3.14.
There exists a finite constant C such that for every positive integer N ,every integer −| Ω N | ≤ K ≤ | Ω N | and every function f in L ( µ N,K ) satisfying h f i N,K = 0 , h f i N,K ≤ C h− ˜ L m Ω N f, f i N,K . Before proving Theorem 3.14, we show that Theorem 3.13 is an easy corollary of thisresult.
Proof of Theorem 3.13.
For each pair { x, y } ∈ Ω N × Ω N , we determine a path inside Ω N which connects x = ( x , ..., x d ) and y = ( y , ..., y d ) as follows: First we connect x and( y , x , ..., x d ) only by changing the first coordinate one by one. Then, ( y , x , x ..., x d )and ( y , y , x ..., x d ) are connected by changing the second coordinate and this proce-dure is continued. We denote the sequence of bonds appearing in this path by b =( z , w ) , b , ..., b M = ( z M , w M ) and the set of these bonds by B ( x, y ). For a configuration η satisfying r x,y ( η ) = 0, let define a sequence of configurations ( ξ i ) ≤ i ≤ M − such that ξ = η , ξ M − = η ( x,y ) as follows : ξ := η , ξ j := ( ξ j − ) z j ,w j for j ≤ M − ξ M = ( ξ M − ) ( z M ,w M ) and ξ j := ( ξ j − ) z M − j ,w M − j for M + 1 ≤ j ≤ M −
1. Then, we have h Ψ x,y , ( η )( f ( η ( x,y ) ) − f ( η )) i N,K = h Ψ x,y , ( η )[ X ≤ i ≤ M − ( f ( ξ i +1 ) − f ( ξ i ))] i N,K ≤ (2 M − X ≤ i ≤ M − h r z i +1 ,w i +1 ( η )( f ( η z i +1 ,w i +1 ) − f ( η )) i N,K ≤ dN X b ∈ B ( x,y ) h ( − ˜ L b − ˜ L b ′ ) f, f i N,K . Similarly, we have h ( − ˜ L xy − ˜ L yx ) f, f i N,K ≤ CN X b ∈ B ( x,y ) h ( − ˜ L b − ˜ L b ′ ) f, f i N,K for some positive constant C .Applying Theorem 3.14, for all functions f in L ( µ N,K ) satisfying h f i N,K = 0 weobtain that h f i N,K ≤ C | Ω N | X x,y ∈ Ω N h− ˜ L xy f, f i N,K ≤ C N | Ω N | X x,y ∈ Ω N X b ∈ B ( x,y ) h ( − ˜ L b − ˜ L b ′ ) f, f i N,K C N | Ω N | X b ∈ (Ω N ) ∗ h− ˜ L b f, f i N,K × { ( x, y ) ∈ Ω N × Ω N ; b or b ′ ∈ B ( x, y ) }≤ CN X b ∈ (Ω N ) ∗ h− ˜ L b f, f i Λ N ,K where a constant C changes each line. Proof of Theorem 3.14.
There is a duality between +particles and − particles, i.e., +particlesevolve with the same dynamics as − particles do under the generator ˜ L u Ω N . Therefore, weassume that 0 ≤ K ≤ | Ω N | .Let X ( η ) denote the number of sites occupied by − particles in the configuration η : X ( η ) := X x ∈ Ω N { η ( x )= − } . We first project f on the σ -field generated by X and on its orthogonal:(3.15) h f i N,K = h ( f − E [ f | X ]) i N,K + h ( E [ f | X ]) i N,K . We consider the two terms separately. Let us define L xy for each ordered pair ( x, y ) by( L xy f )( η ) = [Ψ x,y , ( η ) + Ψ x,y , − ( η ) + Ψ x,y − , ( η )]( f ( η x,y ) − f ( η ))and define L N by L N f ( η ) = 1 | Ω N | X x,y ∈ Ω N L xy f ( η ) . To bound the first term in (3.15) by the Dirichlet form h− L N f, f i , we use a general resultconcerning the spectral gap for multispecies exclusion processes.We introduce some notation. For positive integers r , N and nonnegative integers K , ...K r such that P ri =1 K i ≤ N , define Σ rN,K ,...K r as the hyperplane of all configurationsof Σ rN := { , , ...r } N with K i sites occupied by the i -particles:Σ rN,K ,...K r = { η ∈ Σ rN ; N X j =1 { η ( j )= i } = K i ≤ i ≤ r } and m N,K ,...K r as the uniform probability measure on Σ rN,K ,...K r . As before we denoteby h·i N,K ,...K r , the expectation with respect to the measure m N,K ,...K r . Consider theprocess that exchanges the value of configurations between any two sites at a fixed rate.Its generator L rN is given by L rN f ( η ) = 1 N X ≤ j,k ≤ N ( f ( η j,k ) − f ( η ))where η j,k ( z ) = η ( z ) if z = x, yη ( k ) if z = jη ( j ) if z = k .A simple computation shows that the uniform measures m N,K ,...K r are reversible for thisprocess. We prove that the spectral gap of the generator L rN is of order O (1).19 roposition 3.15. There exists a positive constant C = C ( r ) such that for every positiveinteger N , every set of nonnegative integers K , ...K r such that P ri =1 K i ≤ N and everyfunction f in L ( m N,K ,...K r ) satisfying h f i N,K ,...K r = 0 , (3.16) h f i N,K ,...K r ≤ C h− L rN f, f i N,K ,...K r . The proof of this proposition is postponed to the last part of this section. To applythe estimate in (3.16), we rewrite the first term of in the right hand side of (3.15) as h ( f − E [ f | X ]) i N,K = | Ω N |− K X l =0 µ N,K ( { X = l } ) h f l i Λ N ,K,l where h·i Λ N ,K,l stands for the expectation with respect to the uniform measure on the set ofconfigurations η ∈ {− , , } Ω N satisfying P x ∈ Ω N { η ( x )=1 } = K + l and P x ∈ Ω N { η ( x )= − } = l , and f l stands for the function on this set defined by f l ( η ) = f ( η ) − E [ f | X ]( η ). By Propo-sition 3.15, we have that h ( f − E [ f | X ]) i N,K ≤ C | Ω N |− K X l =0 µ N,K ( { X = l } ) h− L N f l , f l i Λ N ,K,l = C h− L N f, f i N,K ≤ C h− ˜ L Ω N f, f i N,K , notice that f l can be replaced by f to have the second line.Next, let us consider the second term of (3.15). Let η t be the Markov process withthe generator ˜ L m Ω N . Since the original geometry of the process evolving according to thegenerator L Ω N is lost, X ( η t ) is a Markov process and the state space of this Markov processis χ := { , , ..., | Ω N |− K } . A simple computation shows that its generator is given by L N,K f ( l ) = r ( l, l − f ( l − − f ( l )) + r ( l, l + 1)( f ( l + 1) − f ( l )) ∀ l ∈ χ where r ( l, l −
1) = l ( K + l ) | Ω N | and r ( l, l + 1) = ( | Ω N |− K − l )( | Ω N |− K − l − β | Ω N | . For fixed N and K ,denote by ˜ m N,K the probability measure µ N,K X − on χ :˜ m N,K ( l ) := µ N,K ( { X = l } ) . For X -measurable function f , define a function ˜ f : χ → R by ˜ f ( l ) := f ( η ) for some η suchthat X ( η ) = l . A simple computation shows that ˜ L mN f = L N,K ˜ f and h f i N,K = h ˜ f i ˜ m N,K .Therefore, h− ˜ L uN f, f i N,K = h−L N,K ˜ f , ˜ f i ˜ m N,K and h f i N,K = h ˜ f i ˜ m N,K hold. To concludethe proof of Theorem 3.14, we have only to prove Lemma 3.16 below.
Lemma 3.16.
There exists a constant C β such that for any integer N and K satisfying ≤ K ≤ | Ω N | h f i ˜ m N,K ≤ C β h−L N,K f, f i ˜ m N,K for all functions f : χ → R satisfying h f i ˜ m N,K = 0 where ˜ m N,K , L N,K and χ were definedabove. Proposition 3.17.
Let ( X t ) be a Markov process with generator denoted by L on a count-able state space E reversible with respect to a probability measure m , where L acting onfunctions as Lf ( x ) = P y ∈ E r ( x, y )( f ( y ) − f ( x )) for x ∈ E . Suppose that there exists apoint e ∈ E , a positive constant C and a ramification { γ ( e , x ); x ∈ E } satisfying thefollowing assumption (H): | γ ( e , x ) | ≤ C [ r ( x, p ( x )) − P y ∈ s ( x ) r ( x, y )] . Then, for every f in L ( m ) , we have h ( f − h f i m ) i m ≤ C h− Lf, f i m Proof of Lemma 3.16.
For K = 2 | Ω N | − | Ω N | the process has only one possiblestate. Therefore, we may assume that 0 ≤ K ≤ | Ω N | − ξ t the Markov process on χ with generator L N,K . The proof consists infinding a state e and a ramification { γ ( e , x ) , x ∈ χ n } satisfying the assumption ( H )required in Proposition 3.17 for some strictly positive constant C .The natural candidate as root of the ramification is the point where the drift of theparticle is 0. Define D ( x ) as the mean drift of the particle at x : D ( x ) = r ( x, x + 1) − r ( x, x − ≤ K ≤ | Ω N | −
2, a simple computation shows that D (0) ≥ D ([ | Ω N |− K ]) ≤
0. Let ¯ e be the unique root of D ( x ) in [0 , [ | Ω N |− K ]] and define e as thenearest integer of ¯ e .Once e is defined, there is only one possible ramification of the state space. For x ≥ e we have to define the path from e to x as γ ( e , x ) = ( e , e + 1 , ..., x ). In the sameway for x ≤ e the path from e to x has to be γ ( e , x ) = ( e , e − , ..., x ). With thisramification, for a fixed x ≥ e the parent of x is x − x + 1.In the same way for a fixed x ≤ e the parent of x is x + 1 and there is only one child x −
1. Therefore, in order to prove that this ramification satisfies assumption (H) of theProposition 3.17, we have to show that r ( x, x + 1) − r ( x, x − ≥ C ( e − x ) for x < e and r ( x, x − − r ( x, x + 1) ≥ C ( x − e ) for x > e .A simple computation shows that D ′ ( x ) ≤ − C Nβ where C Nβ = | Ω N |− β | Ω N | for β ≥ and C Nβ = 2 β | Ω N |− | Ω N | for β ≤ . Therefore, for x > e and large N, r ( x, x − − r ( x, x + 1) = − D ( x ) ≥ C Nβ ( x − ¯ e ) ≥ C Nβ ( x − e ) − C Nβ | e − ¯ e |≥ C Nβ x − e ) . The last inequality follows from the definition of e since | e − ¯ e | ≤ and from the factthat x ≥ e + 1. In the same way we can prove that for x < e , r ( x, x + 1) − r ( x, x − ≥ C Nβ e − x ) . Since there exists some finite constant C β such that C Nβ ≤ C β for all N , we conclude theproof of the Lemma 3.16. 21 roof of Proposition 3.15. The idea of the proof consists in using the means of the math-ematical induction with respect to r . Assume r = 1. Then, the process is the uniformsymmetric simple exclusion process for which Quastel proved a spectral gap in [8].Next, consider the general positive integer r . First of all, we suppose without lossof generality that K ≤ K i for 0 ≤ i ≤ r where K := N − P ri =1 K i . This implies that K ≤ Nr +1 . For 0 ≤ i ≤ r , define π : Σ rN → { , } N as a function on the configurationspace which do not distinguish sites occupied by some particle: π ( η ) = ξ ∈ { , } N where ξ ( j ) = ( η ( j ) = 00 otherwise.For a function f in L ( m N,K ,...K r ) satisfying h f i N,K ,...K r = 0, define f as the conditionalexpectation of f with respect to the σ -field generated by π . Then, similar to the proofof Theorem 3.14, we project f on this σ -field and on its orthogonal:(3.17) h f i m N,K ,...Kr = h ( f − f ) i m N,K ,...Kr + h f i m N,K ,...Kr . We first consider the second term. The arguments are similar to the ones used in thesecond step of the proof of Theorem 3.14.We may think f as a function defined on { , } N . The generator L rN acting on { , } N is the generator of the usual uniform symmetric exclusion process for which Quastel [8]proved a spectral gap. Therefore, we have h f i m N,K ,...Kr ≤ h L rN f , f i m N,K ,...Kr ≤ h L rN f, f i m N,K ,...Kr . We now turn to the first term of (3.17). For a subset B ⊂ Λ N := { , ..., N } such that B = K , define f ,B as a function on Σ r − N − B,K ,K ,...,K r − as follows: f ,B ( ξ ) = f ( ξ B ) − f ( ξ B ) ξ ∈ Σ r − N − B,K ,K ,...,K r − . In this formula, ξ B stands for the configuration η ∈ Σ rN,K ,K ,...,K r such that η ( i ) = ( i ∈ Bj + 1 if ξ ( i ) = j. With this notation, we rewrite the first term of (3.17) as follows: h ( f − f ) i m N,K ,...Kr = X B ⊂ Λ N m N,K ,...K r ( { η ; η ( i ) = 0 for all i ∈ B } ) h f ,B i m N − K ,K ,...Kr − . The generator L rN acting on Σ r − N − B,K ,K ,...,K r − is the operator N − K N L r − N − K . By theinduction assumption, there exists some constant C r − such that h f ,B i m N − K ,K ,...Kr − ≤ C r − h L r − N − K f ,B , f ,B i m N − K ,K ,...Kr − ≤ C r − NN − K h L rN f, f i m N,K ,...Kr . By the assumption K ≤ Nr +1 , therefore we have NN − K ≤ r +1 r and h f ,B i m N − K ,K ,...Kr − ≤ C r h L rN f, f i m N,K ,...Kr for some constant C r . 22 .5 Closed Forms In this section, we introduce the notion of closed forms associated to our model and thoseof the generalized exclusion process. We have one-to-one correspondence between them,so algebraic characterization of the closed forms can be reduced to that for the generalizedexclusion process.Let H x,x + e i be a subspace of χ = {− , , } Z d such that H x,x + e i := { η ; Ψ x,x + e i , ( η ) + Ψ x,x + e i , − ( η ) + Ψ x,x + e i − , ( η ) + Ψ x,x + e i , − ( η ) + Ψ x,x + e i , ( η ) = 1 } . For two configurations η and ξ ∈ χ , define D ( η, ξ ) as follows: D ( η, ξ ) = 1 if thereexists a unique point x ∈ Z d and a unique direction i such that η ( z ) = ξ ( z ) for all z = x, x + e i and η ( x ) + η ( x + e i ) = ξ ( x ) + ξ ( x + e i ) and η = ξ , and D ( η, ξ ) = 0 otherwise.A path Γ( η, ξ ) = ( η = η , η , ..., η m − , η m = ξ ) from η to ξ is a sequence of configurations η j such that every two consecutive configurations satisfies D ( η j , η j +1 ) = 1.Consider a family of continuous functions u = ( u ix ) ≤ i ≤ d,x ∈ Z d where u ix : H x,x + e i → R .For an ordered pair ( η, ξ ) satisfying D ( η, ξ ) = 1, define a path integral I η,ξ ) of u by I η,ξ ) ( u ) := 1 √ C + (cid:0) u ix ( η )Ψ x,x + e i , ( η ) − u ix ( ξ )Ψ x,x + e i , ( ξ ) (cid:1) + 1 √ C − (cid:0) u ix ( η )Ψ x,x + e i , − ( η ) − u ix ( ξ )Ψ x,x + e i − , ( ξ ) (cid:1) + 1 √ C E (cid:0) u ix ( η )Ψ x,x + e i − , ( η )Ψ x,x + e i , − ( ξ ) − u ix ( ξ )Ψ x,x + e i , − ( η )Ψ x,x + e i − , ( ξ ) (cid:1) + 1 √ C A (cid:0) u ix ( η )Ψ x,x + e i , − ( η )Ψ x,x + e i , ( ξ ) − u ix ( ξ )Ψ x,x + e i , ( η )Ψ x,x + e i , − ( ξ ) (cid:1) + 1 √ C C (cid:0) u ix ( η )Ψ x,x + e i , ( η )Ψ x,x + e i − , ( ξ ) − u ix ( ξ )Ψ x,x + e i − , ( η )Ψ x,x + e i , ( ξ ) (cid:1) A path integral can be naturally extended to paths of any length as I η,ξ ) ( u ) := m − X j =0 I η j ,η j +1 ) ( u ) . A family of continuous functions ( u ix ) ≤ i ≤ d,x ∈ Z d is called an I -closed form if for allclosed path Γ( η, ξ ), I η,ξ ) ( u ) = 0 where a path Γ( η, ξ ) is called closed if η = ξ .Next, let us recall the path integral and the closed form of the generalized exclusionprocesses. Let ^ H x,x + e i be a subspace of χ = {− , , } Z d such that ^ H x,x + e i := { η ; Ψ x,x + e i , ( η ) + Ψ x,x + e i , − ( η ) + Ψ x,x + e i , − ( η ) + Ψ x,x + e i , ( η ) = 1 } . For two configurations η and ξ ∈ χ , define ˜ D ( η, ξ ) as follows: ˜ D ( η, ξ ) = 1 if thereexists a unique point x ∈ Z d and a unique direction i such that η ( z ) = ξ ( z ) for all z = x, x + e i and η ( x ) − ξ ( x ) and η ( x + e i ) + 1 = ξ ( x + e i ) or η ( x ) + 1 = ξ ( x )23nd η ( x + e i ) − ξ ( x + e i ), and ˜ D ( η, ξ ) = 0 otherwise. A path ˜Γ( η, ξ ) = ( η = η , η , ..., η m − , η m = ξ ) from η to ξ is a sequence of configurations η j such that every twoconsecutive configurations satisfies ˜ D ( η j , η j +1 ) = 1.Consider a family of continuous functions ( ˜ u ix ) ≤ i ≤ d,x ∈ Z d where ˜ u ix : ^ H x,x + e i → R . Foran ordered pair ( η, ξ ) satisfying ˜ D ( η, ξ ) = 1, define a path integral I η,ξ ) by I η,ξ ) (˜ u ) := ( ˜ u ix ( η ) if η ( x ) − ξ ( x ) and η ( x + e i ) + 1 = ξ ( x + e i ) − ˜ u ix ( ξ ) if η ( x ) + 1 = ξ ( x ) and η ( x + e i ) − ξ ( x + e i ) . A path integral can be naturally extended to paths of any length as I η,ξ ) (˜ u ) := m − X j =0 I η j ,η j +1 ) (˜ u ) . A family of continuous functions ( ˜ u ix ) ≤ i ≤ d,x ∈ Z d is called an I -closed form if I η,ξ ) (˜ u ) = 0hold for all closed paths ˜Γ( η, ξ ).Now, we construct one-to-one map from the set of I -closed forms to the set of I -closed forms. For an I -closed form ( u ix ) ≤ i ≤ d,x ∈ Z d , define a family of continuous functions( ˜ u ix ) ≤ i ≤ d,x ∈ Z d as ˜ u ix ( η ) := 1 √ C + u ix ( η )Ψ x,x + e i , ( η ) + 1 √ C − u ix ( η )Ψ x,x + e i , − ( η )+ 1 √ C A u ix ( η )Ψ x,x + e i , − ( η ) + 1 √ C C u ix ( η )Ψ x,x + e i , ( η ) , then, ( ˜ u ix ) ≤ i ≤ d,x ∈ Z d is an I -closed form. On the other hand, for an I -closed form( ˜ v ix ) ≤ i ≤ d,x ∈ Z d , define a family of continuous functions ( v ix ) ≤ i ≤ d,x ∈ Z d as v ix ( η ) := p C + ˜ v ix ( η )Ψ x,x + e i , ( η ) + p C − ˜ v ix ( η )Ψ x,x + e i , − ( η )+ p C A ˜ v ix ( η )Ψ x,x + e i , − ( η ) + p C C ˜ v ix ( η )Ψ x,x + e i , ( η ) − p C E h ( ˜ v ix ( η x =1 ,x + e i = − ) + ˜ v ix ( η x =0 ,x + e i =0 )) i Ψ x,x + e i − , ( η ) , then, ( v ix ) ≤ i ≤ d,x ∈ Z d is an I -closed form.Let us introduce the notion of an I -germ of closed form and an I -germ of closedform. A family of continuous functions ( g i ) ≤ i ≤ d (resp. ˜ g i ) where g i : H ,e i → R is an I (resp. I )-germ of closed form if u ix := τ x g i is an I (resp. I )-closed form.Main theorem of this section is formulated as follows: Theorem 3.18.
For every I -germ of closed form g , there exists a sequence of L ( ν ρ ) -functions h n and constants ( c i ) ≤ i ≤ d such that g i = lim n →∞ ( c i d X j =1 ( U j ) i + ∇ ,e i Γ h n ) in L ( ν ρ ) for all ≤ i ≤ d.
24y the one-to-one correspondence between the I -germ of closed form and the I -germ of closed form and Cauchy-Schwartz inequality, in order to prove Theorem 3.18 wehave only to prove the next theorem: Theorem 3.19.
For every I -germ of closed form ˜ g , there exists a sequence of L ( ν ρ ) -functions h n and constants ( c i ) ≤ i ≤ d such that ˜ g i = lim n →∞ ( c i d X j =1 ( ˜ U j ) i + ˜ ∇ ,e i Γ h n ) in L ( ν ρ ) for all ≤ i ≤ d. where ˜ U i ( η ) = 1 { η (0) ≥ ,η ( e i ) ≤ } and ˜ ∇ ,e i f ( η ) = Ψ ,e i , ( η )( f ( η ,e i ) − f ( η )) + Ψ ,e i , − ( η )( f ( η ,e i ) − f ( η ))+ Ψ ,e i , − ( η )( f ( η ,e i =0 ) − f ( η )) + Ψ ,e i , ( η )( f ( η − ,e i =1 ) − f ( η )) . Applying the method in [3], we can deduce this theorem since the proof depends onlyon the spectral gap estimates, which is proved by Theorem 1.8.1, and the fact that ν ρ istranslation invariant. The strategy of the proof is the same as given for liquid-solid system in [2]. The main stepis to establish the local ergodic theorem.First, we consider a class of martingales associated with the empirical measure assimilar to Case 1. We also use the same notations: T , M H ( t ), π Nt , Q µ N and Q ∗ which aredefined in Section 3. In Case 2, we assume the gradient condition, C + + C − − C A − C E = 0,so the martingale M H ( t ) is rewritten as M H ( t ) = h π Nt , H i − h π N , H i − Z t N − d X x ∈ T dN (∆ N H )( xN ) τ x h ( η s ) ds where h is a cylinder function defined by h ( η ) = C + { η (0)=1 } − C − { η (0)= − } and ∆ N H represents the discrete Laplacian of H :(∆ N H )( xN ) = d X i =1 N [ H ( x + e i N ) + H ( x − e i N ) − H ( xN )] . Following the same argument in Section 3, it is easy to prove thatlim N →∞ P µ N [ sup ≤ t ≤ T | M H ( t ) | ≥ δ ] = 0for every δ >
0, the sequence { Q µ N , N ≥ } is weakly relatively compact and that everylimit points Q ∗ is concentrated on absolutely continuous paths π t ( du ) = π ( t, u ) du withdensity bounded by 1 and -1 from above and below respectively: − ≤ π ( t, u ) ≤ { Q µ N , N ≥ } are concentrated on absolutely continuous trajectories π ( t, du ) = π ( t, u ) du whose densities are weak solutions of the equation (2.5). For thispurpose, all we have to show is thatlim N →∞ E µ N [ Z t N − d X x ∈ T dN (∆ N H )( xN ) τ x h ( η s ) ds ] = E Q ∗ [ Z t Z T d (∆ H )( u ) P ( π ( s, u )) ds ]where P is the function defined in (2.6).First, we establish the local ergodic theorem which enables us to replace the samplemean of microscopic variables with their average under the equilibrium measure having amicroscopically defined sample density as its density-parameter. Let µ Nt ∈ P ( χ dN ) be theprobability distribution of η t on χ dN and let ˜ µ N be the space-time average of { µ Nt } ≤ t ≤ T defined by ˜ µ N = 1 T N d X x ∈ T dN Z T µ Nt · τ − x dt. Then, we have that
Proposition 4.1 (local ergodic theorem) . For every cylinder function f , lim K →∞ lim sup N →∞ E ˜ µ N [ | ¯ f ,K ( η ) − h f i η K (0) | ] = 0 where ¯ f ,K = K +1) d P | x |≤ K τ x f ( η ) .Proof. Let L be an operator on C defined by ( Lf )( η ) = P b ∈ ( Z d ) ∗ L b f ( η ), where ( Z d ) ∗ stands for the set of all directed bonds. Following the method of the proof of Theorem 4.1in [2], it is easy to show that { ˜ µ N } N is tight in P ( χ d ) and an arbitrary limit µ ∈ P ( χ d )satisfies µ ( Lf ) = 0 for every cylinder function f . Moreover, by definition, µ is invariantunder spatial translations. Therefore, we have that support ( µ ) ⊂ {− , } Z d ∪ { , } Z d . Itis known that the translation-invariant L -stationary measure on {− , } Z d or { , } Z d isa superposition of Bernoulli product measures. Then, the law of large numbers concludesthe proposition.Next, we need to prove that the sample density defined microscopically can be re-placed in the limit with the macroscopic one. We can use Young measures to complete itby the exactly same way as in [2]. Then, combining with these results, the main theoremwill be concluded. The details are omitted. In Case 3, since the hydrodynamic equation is the heat equation, the replacements whichare required in Case 2 are unnecessary. So, this is the easiest case to prove the hydrody-namic limit. 26irst, we consider a class of martingales associated with empirical measure again.Then, as in Case 2, the martingale M H ( t ) is rewritten as M H ( t ) = h π Nt , H i − h π N , H i − C E Z t h π Ns , ∆ N H i ds + C + − C − Z t N − d X x ∈ T dN { η s ( x )=0 } (∆ N H )( xN ) ds where ∆ N H represents the discrete Laplacian of H .Following the same argument in Section 3 and 4, all we have to show is that(5.1) lim N →∞ E µ N [ Z t N − d X x ∈ T dN { η s ( x )=0 } ds ] = 0 . With the notation defined in the last section, (5.1) is rewritten aslim N →∞ E ˜ µ N [1 { η (0)=0 } ] = 0 . Following the method of the proof of Theorem 4.1 in [2] again, it is easy to show that { ˜ µ N } N is tight in P ( χ d ) and an arbitrary limit µ ∈ P ( χ d ) satisfies µ ( Lf ) = 0 for everycylinder function f . Moreover, by definition, µ is invariant under spatial translations.Therefore, we have that support ( µ ) ⊂ {− , } Z d and it concludes the theorem. In this section, we fix the terminology of weak solutions of parabolic equations and presentthe uniqueness of such equations. Hereafter φ : R → R is a strictly increasing and Lipschitzcontinuous function. We consider the Cauchy problem:(6.1) ∂ t ρ ( t, u ) = ∆( φ ( ρ ( t, u ))) (cid:16) = d X i =1 ∂ ∂u i φ ( ρ ( t, u )) (cid:17) ρ (0 , · ) = ρ ( · ) , and define weak solutions of this Cauchy problem. Definition 6.1.
Fix a bounded initial profile ρ : T d → R . A measurable function ρ ≡ ρ ( t ) = ρ ( t, u ) ∈ C ([0 , T ] , M ( T d )) ∩ L ([0 , T ] × T d ) is a weak solution of the Cauchy problem(6.1) if for every function H : T d → R of class C ( T d ) and for every ≤ t ≤ T Z T d H ( u ) ρ ( t, u ) du = Z t ds Z T d φ ( ρ ( s, u ))∆ H ( u ) du + Z T d H ( u ) ρ ( u ) du. We prove the uniqueness of weak solutions in this class.
Theorem 6.1.
Fix a bounded measurable function ρ : T d → R . There exists at most oneweak solution of the parabolic equation (6.1).Proof. We can apply the proof of Theorem A.2.4.4 in [3] straightforwardly.27 cknowledgement
The author would like to thank Professor T. Funaki for helping her with valuable sugges-tions.
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