Navier-Stokes equations for a stochastic particle systems ... - CiteSeerX

Navier-Stokes equations for a stochastic particle systems on the lattice by R. Esposito1, R. Marra2 and H.T.Yau3

Abstract:

We introduce a class of stochastic models of particles on the cubic lattice Zd with velocities and study the hydrodynamical limit on the diusive space-time scale. Assuming special initial conditions corresponding to the incompressible regime, we prove that in dimension d  3 there is a law of large numbers for the empirical density and the rescaled empirical velocity eld. Moreover the limit elds satisfy the corresponding incompressible Navier-Stokes equations, with viscosity matrices characterized by a variational formula, formally equivalent to the Green-Kubo formula.

1. Introduction.

One of the main open problems in nonequilibrium statistical physics is the derivation of the hydrodynamical equations of uid from the microscopic Hamiltonian dynamics. The main uid equations are the Navier-Stokes equations and the Euler equations. The Euler equations represent the conservation of macroscopic mass, energy and momentum and have an obvious hyperbolic scaling invariance

x ! x; t ! t:

(1:1)

The Navier-Stokes equations are more complicated and have no obvious scaling. They are given by correcting the Euler equations with viscous terms described by second order 1 Dipartimento di Matematica Pura ed Applicata, Universita di L'Aquila, Coppito, AQ. Italy. Partially

supported by GNFM-CNR and MURST 2 Dipartimento di Fisica, Universita di Roma Tor Vergata, Via della Ricerca Scienti ca, 00133 Roma Italy. Partially supported by GNFM-CNR, INFN and MURST 3 Courant Institute, 251 Mercer Street, New York NY 10012, USA. Partially supported by U. S. National Science Foundation grant 9403462 and David and Lucile Packard Foundation Fellowship.

1

derivatives of conserved quantities such as energy, momentum and mass. In the incompressible regime, the Navier-Stokes equations become div u = 0; @t u + u
ru + rp = 
u

(1:2)

with u the velocity eld, p the pressure and  > 0 the kinematic viscosity. Since Euler and Navier-Stokes equations involve macroscopic quantities, derivations of these equations from microscopic Hamiltonian dynamics are understood in the sense of law of large number, as the number of particles tends to in nity. We x the space and time scales by choosing the typical interparticle distance as the unit length scale and the mean free time of particles as unit time scale. Suppose the range of molecular interactions are of the order of the typical interparticle distance. Then each particle typically interacts with at most nite number of nearby particles. Let " denote the ratio between the microscopic and macroscopic length units. In order that particles travel a unit macroscopic length in a unit macroscopic time, the macroscopic time scale is chosen to be "?1 t. The derivation of Euler equation can be stated as proving that the Euler equations are exact in the scaling limit x ! "?1 x; t ! "?1 t. This scaling is usually called the Euler scale or hyperbolic scaling. We now sketch Morrey's idea [1] for the derivation of the Euler equations. Since the Hamiltonian dynamics is conservative, the time derivatives of conserved quantities are given by microscopic conservation laws characterized by the corresponding microscopic currents. The currents involve microscopic interactions of particles and are not functions of local conserved quantities. To obtain closed equations, one has to represent these currents as functions of the local conserved quantities through some law of large numbers. If we assume that the system is locally in equilibrium and described by Gibbs states characterized by local conserved quantities, from the law of large numbers for Gibbs states these currents can be replaced by their expectations with respect to Gibbs states depending on local conserved quantities. Hence we obtain closed equations. The basic ingredient in the previous heuristic derivation is a strong ergodic theorem for Hamiltonian systems. Though this ergodic theorem is a basic problem in the theory of Hamiltonian systems and is very easy to state, without further assumption, it is nevertheless still an open question. If certain random perturbations are added to the Hamiltonian dynamics, or if certain assumptions are made, it can be rigorously proved [2]. Alternatively, if instead of Hamiltonian dynamics one starts from the Boltzmann equation, very detailed derivations of Euler equation are available ([3,4,5]). Of course the Boltzmann equation is not a microscopic equation and, because the Boltzmann equation describes the low density regime, the state equation obtained in this case is just the one of the perfect gases. It should also be noted that all these derivations are valid only up to the times that the Euler equations have smooth solutions. The similar program for the Navier-Stokes equations is much harder to carry out. First of all, the Navier-Stokes equations have no obvious scaling and thus they cannot be a scaling limit. Furthermore, although the basic equations in the classical physics are hyperbolic and completely reversible, the Navier-Stokes equations are irreversible due to the viscosity. 2

The rst diculty is more of technical nature and there are several possible solutions. One of them is to consider the following incompressible limit. The diusive eects, like the viscosity and the heat conduction are hard to detect, because they are small corrections to the Euler equations. They become relevant on the diusive scale, i.e.

x ! "?1 x; t ! "?2 t: (1:3) The Navier-Stokes equations are not invariant under such scaling, so the prescription of the diusive scaling (1.3) cannot be used in full generality. In order to restore the scaling invariance one needs to scale also the velocity eld u as u ! "u; (1:4) and, consequently, the pressure as p ! p + "2 p with p an arbitrary constant. In other words, we consider a regime where macroscopic velocities are very small compared with the sound speed. To summarize, the incompressible Navier-Stokes equations (1.2) can be understood as the scaling limit under the the rules given by (1.3) and (1.4). This limit is called the incompressible limit. It is used in [6,7] to derive the incompressible NavierStokes equations (1.2) from the Boltzmann equation and in [8] to derive the viscous Burgers equation as diusive limit of the simple exclusion process. The second diculty is a deep conceptual question and relates to the origin of diusivity in classical physics. Up to now, there is no good mathematical understanding for a derivation of diusive behavior from reversible systems. A heuristic explanation, however, can be given. We follow [9] where a formal derivation of the equations (1.2) from the Hamiltonian systems are given. From Morrey's heuristic argument or from the recent work of [2], the many body Liouville equation can be approximately solved using ansatz of local Gibbs states, i.e., Gibbs states with slowly varying chemical potentials. Let us strengthen the local equilibrium assumption by demanding that the Liouville equation can be solved asymptotically in orders of ". Then locally the solution of the Liouville equation is given by a Gibbs state with an " order correction. Therefore, the currents should be replaced by their expectations with respect to this state. The leading order contribution to the currents will be the same as in the Euler equation case; the " order correction will give rise to the viscosity. The diculties in carrying out this approach rigorously are obvious. First of all, there is no rigorous proof that local Gibbs states give even the leading order solution of Liouville equation unless some noise are added to the Liouville equation. Furthermore, it is in general hard to quantify the meaning of the asymptotic expansion involving an in nite number of variables. One way to make sense of this asymptotic expansion is through the use of relative entropy [2,10]. But lack of analytic tools for general Hamiltonian systems force us to consider lattice gases models. The models we consider are de ned as follows. Let V be a nite set of vectors in Rd of xed length  > 0, invariant under re ections and exchanges of the coordinate axes. It will represent the set of the possible velocities of the particles. For each v 2 V we consider a species of particles moving on the lattice Zd according to a simple exclusion process (SEP), with jump intensities such that the resulting drift is v. This means that a particle of the species v in a site x 2 Zd jumps with exponential law to one of the neighboring 3

sites x + e 2 Zd, jej = 1, with intensity pe (v), if x + e is not already occupied by a particle with velocity v. We denote by (x; v) = 0; 1 the occupation number of particles of the species v in x. The intensities pe (v) are chosen so that the drift of the process for the v-particles is v. The jumps of each particle are independent of the particles with dierent velocities. They are also independent of the particles with the same velocity, as far as the exclusion condition is satis ed. The changes of velocity are due to collisions. Particles collide when are in the same site and have suitable velocities. The essential condition on the collisions is that they conserve the number of particles involved in the collision and their total momentum, and nothing else. In the next section we de ne two models, called Model I and Model II, by giving the set V and the collision rules. Our method can be extended to a wider class of models and also to more general lattices. We take the set V for Model I to be the smallest possible by including only the 2d unit vectors parallel to the coordinate directions; for the Model II the set V includes several vectors with non-coordinate directions to make the model as isotropic as possible. In both case the jump intensity is chosen of the form

pe (v) =  + 21 v
e:

(1:5)

with  big enough to make positive all the jump intensities . Note that we allow a particle with velocity v to jump also in \wrong" directions, as the one orthogonal to v. This is convenient to prove ergodicity, but the assumption that such a jump happens with intensity bigger than =2 is unnecessary: any positive intensity would work. As for the collisions, we consider binary collisions, which exchange the velocities of any pairs of particles at the same site into another pairs of velocities having the same total momentum chosen at random among all the possible ones (if any), provided that the exclusion condition is not violated. The evolution of our model will be described in terms of a generator L which will be the sum of the exclusion part Lex and the collision part Lc. Lattice gas models for the hydrodynamics have been widely considered before. We refer to [11] and references quoted therein for a comprehensive review. The aim of these works was to construct simple models for hydrodynamical equations which could be carefully and fastly simulated on the computer. Hence the eort is to make them as deterministic as possible. The numerical results provided a remarkably good agreement with hydrodynamical equations. The macroscopic behaviors were also analyzed theoretically, under the assumption of propagation of chaos. This produced a type of Boltzmann equation for the models, although no explicit low density condition was introduced and no proofs were given. After adding a small amount of stochasticity, the Euler regime was studied in [12] using the approach of [2]. In [13,14] the Euler and Navier-Stokes regimes were considered for models with discrete space, time and velocities, and with stochastic free motion and deterministic or stochastic collisions. However the kinetic limit was used as an intermediate step. In the present paper we deal directly with the hydrodynamic limit and no intermediate step or low density assumption is needed. As discussed before, our analysis is con ned to the incompressible regime. The incompressibility assumption corresponds to choose the chemical potentials of the local equilibrium suitably small i.e. of order ". Following the heuristic derivation of viscosity in 4

the previous paragraph, we have to compute the next order correction term of this state, which is of order "2 . This is the key element of this paper. The viscosity will be determined through this correction term. This analysis allows us to prove our main result, the validity of the law of large numbers for the empirical density and the rescaled empirical velocity eld as the scaling parameter " goes to 0. In fact, Theorem 2.1 states their weak convergence in probability to deterministic elds solving the incompressible Navier-Stokes equation with the viscosity determined according to the previous argument. A very short sketch of the idea of the proof is the following. Recall the dynamics has two parts: the exclusion part Lex and the collision part Lc. Without the collisions our process is just the asymmetric simple exclusion process. The parallel problem for the asymmetric simple exclusion processes has been studied in [8] using the spectral gap and the logarithmic Sobolev inequality. Unfortunately, these estimates are hard to prove for the present models. We use an alternative approach. Let vx denote the \local density" of particles with velocity v 2 V . We rst use the spectral gap and the logarithmic Sobolev inequality of the exclusion part of the dynamics, Lex, to reduce the local relevant parameters to vx; v 2 V . On the other hand, only certain linear combinations of vx; v 2 V representing local mass and momentum are conserved by the collisions. We then use the collision part to eliminate the irrelevant modes. Details will be presented in Section 4 and Section 6. Finally we comment on the history of the methods used in this paper. The rst mathematical rigorous proof of hydrodynamical limit for reversible nongradient models is the pioneering work of [15]. Though the Green-Kubo formula has widely been used to derive the hydrodynamic equation and certain mathematical rigorness had been achieved before (see, for example, [16]), this work represents the rst completely mathematical rigorous proof of Green-Kubo formula and it gives a variational formula which is much more stable to work with. The key assumptions of this work are the reversibility of the process and the spectral gap of the dynamics. The spectral gap, though dicult, is more of technical nature. It is proved in several models by [17,18,19] and a general approach is outlined in [18]. A class of nonreversible models is treated by [20] but the nal hydrodynamical equation is required to be a nonlinear heat equation without drift. The restriction on the drift is removed in [8] provided the space dimension d > 2 and the logarithmic Sobolev inequality can be proved. A general argument to obtain the logarithmic Sobolev inequality is given in [21]. The restriction on the space dimension d > 2 in [8] is optimal and one expects completely dierent behavior for d  2. In this case, the scaling is not diusive and the hydrodynamical equation probably is nonlocal. There is no mathematical results concerning this phenomenon up to now. A similar phenomenon is expected also in the real hydrodynamics for d  2, due to the long time tails which cause the divergence of the transport coecients [22]. We conclude with a few remarks on the viscosity matrix. The viscosity matrix we obtain does not reduce to the contribution due to the stochasticity of the model, corresponding to the symmetric part of the generator. Actually, we prove that the contribution due to the \deterministic motion", i.e. to the anti-symmetric part of the generator, is nonnegative and nonvanishing. For the real incompressible Navier-Stokes uid all the eigenvalues coincide, 5

but we cannot prove that this is true for Model II. We do not know if this is due to the lack of our analytic tools or due to the nonisotropy of the underlying cubic lattice. Our analysis can be extended to lattices with other geometries, but we do not formulate an abstract framework for all possible lattices.

2. The models and the main theorem. Let L 2 Zd be the cubic sublattice f?L; : : : ; Lgd with periodic boundary conditions. Let L = "?1. We denote by ei, i = 1; : : : ; d, the coordinate vectors with component (ei )j = i;j Moreover, we put E = fe = ei for some i = 1; : : : ; dg. Riv will denote the re ection of the vector v 2 Rd w.r.t. the plane i orthogonal to ei , i.e. the vector with coordinates 

vi if i = j (Ri v)j = ? vj otherwise. Denote by  be any element of J , the permutation group of the set f1; : : : ; dg, and let v = (v(1); v(2); : : : ; v(d)). Let V be a nite subset of Rd representing the possible velocities and N be its cardinality. Assume that V is symmetric w.r.t. re ections and permutations: Ri V = V i = 1; : : : ; d; V = V ;  2 J For our purposes it will be sucient to consider velocities of xed modulus, so we also assume jvj =  for any v 2 V On each site of the lattice at most one particle for each velocity is allowed, with no exclusion rule between dierent velocities. A con guration of particles on the lattice is denoted by  = fx; x 2 Lg where x = f(x; v); v 2 Vg and (x; v) = 0; 1; x 2 Zd; v 2 V is the number of particles in x with velocity v. The generator of the dynamics is de ned by Lf = Lexf + Lcf (2:1) Lex is the generator of the exclusion process for particles with several colors (velocities), without exclusion between dierent colors X X   p(y; y + e; w; ) f (y;y+e;w ) ? f () (2:2) Lexf () = w2V y2Z d e2E

where

8 <  (z; v )

if v 6= w or z 6= x; y x;y;w(z; v) = : (x; w) if v = w and z = y (y; w) if v = w and z = x 6

The jump intensity is given by

p(x; y; w; ) = (x; w)p(x ? y; w); p(x ? y; w) = [ + (y ? x)
w=2]; v 2 V ;

(2:3)

for some  > =2. The constant  has to be chosen large enough so that the jump rates are nonnegative. The collision operator is de ned as follows: let 



Q = q = (v; w; v0 ; w0 ) 2 V 4 : v + w = v0 + w0 : We call the quadruples q 2 Q \collisions". Q is the set of all the possible collisions. The

rst two arguments of q, v and w are the incoming velocities, while v0 , w0 are the outgoing velocities of the collision q. We de ne

Lc f () =

X X

y2Z d q2Q



f (y;q ) ? f () :

(2:4)

Here, for q = (v; w; v0 ; w0 ) we put

p~(y; q; ) = (y; v)(y; w)(1 ? (y; v0 ))(1 ? (y; w0 )) and

(2:5)

8 <  (z; u)

if z 6= y or u 2= q or p~(y; q; ) = 0 0 if p~(y; q; ) = 1 and z = y and (u = v or u = w) : 1 if p~(y; q; ) = 1 and z = y and (u = v0 or u = w0 ) Note that if  does not contain in y two particles with the incoming velocities v and w, or there are particles with outgoing velocities v0 and w0 , y;q = . Putting (Qqy f )()    f (y;q ) ? f () , we rewrite (2.4) as

y;q (z; u) =

Lcf () =

XX

(Qqy f )()

y q2Q

(2:6)

Speci c choices of the the set V will be given later on in this section. For the moment, we simply note that the collision operator conserves the total mass and the total momentum. More precisely, the quantities X (2:7) I0 (x) := (x; v); v2V

and, for  = 1; : : : ; d:

I (x) :=

X

v2V

(v
e)(x; v)

7

(2:8)





are conserved in a collision, i.e., Lc I(x) = 0 for all x 2 L. The choices of V will ensure that these are the only conserved quantities during a collision. Since jumps conserve the total number of particles for each color, we have X

L

x



I (x) = 0;  = 0; : : : ; d:

As a consequence, the following measure is invariant for L

dL;r;n = Z ?1

L;r;n

Y

x2L

expfrI0 (x) +

d X =1

nI (x)g

(2:9)

with n the d-dimensional vector fn;  = 1; : : : dg. All the measures we will consider in the sequel are absolutely continuous w.r.t. the global equilibrium measure r corresponding to n = 0, namely the one de ned as

r () = Zr?1

Y

x2L

expfrI0 (x)g;

r 2 R:

(2:10)



When possible we also omit the subscript r and denote by
the expectation w.r.t. r . The generator L is not symmetric w.r.t. this measure, because the generator of the exclusion part is not symmetric. However, one can check immediately that

E r [f Lc g] = E r [gLcf ]; so Lc is symmetric w.r.t r . The formal adjoint w.r.t  is given by

L = (Lex ) + Lc with

(Lex ) f () =

X X

w2V y2Z d



p (y; y + e; w; ) f (y;y+e;w ) ? f ()



(2:11)

e2E

and

p (x; y; w; ) = (y; w)[ + (y ? x)
w=2] = p(y; x; ?w; ):

(2:12)

There are d + 1 locally conserved quantities I (2.8) for this dynamics and the corre ;  = 0; : : : ; d are de ned by sponding currents wx; 

d X



L I(x) =

=1

8

 r? wx;

(2:13)

with

r? g(x)  g(x) ? g(x ? e );

r g(x)  g(x + e ) ? g(x);   Note that the collision operator plays no role in (2.13) because Lc I(x) = 0. Similarly ;

we de ne wx; as

d

  X L I(x) = r?w;

=1

(2:14)

x;

Since the current depends only on the exclusion part of the process and the exclusion dynamics is de ned for each color separately, we can compute the currents using formulae for the asymmetric simple exclusion. More precisely, we can compute the current of the asymmetric simple exclusion process by d X ex L (x; v) = r?e wx; (v); =1

d

X  (v ): L (x; v) = r? wx;

=1

 (v ) are given explicitly by The currents wx; (v) and wx; (a) (v ); wx; (v) = 21 (p(e ; v) + p(?e ; v))r (x; v) + wx;  (v ) = 1 (p(e ; v ) + p(?e ; v ))r  (x; v ) ? w(a) (v ); wx; x; 2

(2:15)

(a) (v ) = (p(e ; v ) ? p(?e ; v ))b (v ); wx; x;

with

h i bx; (v) = (x + e ; v)(x; v) ? 21 (x + e ; v) + (x; v)

(2:16):

Here p(e ; v) is the jump rate of the color v in the direction e . With the convention (2.3), p(e ; v) ? p(?e ; v) = e
v and 21 (p(e ; v) + p(?e ; v)) =  0 and the momentum current Using these equations, we obtain the mass current wx;  as wx; 0 = wx;

 = wx;

X

v2V

X

v2V

wx; (v) = r I0(x) +

X

v2V

(e
v)wx; (v) = r I +

Similarly, ;0 = r I0 (x ) ? wx;

X

v2V

(e
v)bx; (v)

X

v2V

(e
v)(e
v)bx; (v); ; = 1; : : : ; d:

(e
v)bx; (v)

; = r I ? X (e
v )(e
v )bx; (v ); ; = 1; : : : ; d: wx; v2V

9

(2:17)

(2:18)

It is convenient to introduce also the symmetric and antisymmetric part of the currents ;  + w;  wx; x; ; w(a); = wx; ? wx; : x; x; 2 2 We will need to compute averages w.r.t. distributions of the form

w(s); =

n;L = Z ?1

Y

x2L

expfn0(x)I0 (x ) +

d X =1

n (x)I (x)g:

(2:19)

(2:20)

De ne the velocity distribution at x as

f (x; v; n) = E n [(x; v)] =

en0 (x)+n(x)
v ; 1 + en0 (x)+n(x)
v

 w.r.t. a measure of this form are with n = (n1 ; : : : ; nd ). The average of the currents wx; given by 0 ]= E n [wx;

=

X

v2V

X

v2V

v2V

(e
v)E n [bx; (v)] + r E n [I0]

  (v
e ) f (x; v; n)f (x + e ; v; n) ? 21 (f (x; v; n) + f (x + e ; v; n)) + r E n [I0];

 ]= E n [wx;

=

X

X

v2V

(e
v)(e
v)E n [bx; (v)] + r E n [I]

  (e
v)(e
v) f (x; v; n)f (x + e ; v; n) ? 21 (f (x; v; n) + f (x + e ; v; n)

+ r E n [I]:

(2:21)

Let us give now a heuristic derivation of the hydrodynamic equations. We start with the Euler scale. We choose as initial state for the system the local equilibrium measure (2.20) with chemical potentials n~("x; 0) slowly varying in space. Then we look at the system on the Euler time scale, i.e. at times of order "?1 , that corresponds to replace the generator L with "?1L. Then the hydrodynamic equations are recovered from the conservation laws d d "d X J ("x)E ft r [I ( )] = "d?1 X X  ] J ("x)r? E ftr [wx;  x dt x x =1

where ft r denotes the distribution of the process at time t. 10

(2:22)

Assuming that ft is still well approximated by a local equilibrium with chemical potentials n~("x; t), by (2.21) and (2.22), in the limit " ! 0 we get the Euler equations d @ + X @t =1 @z ja = 0;

(2:23)

d X

@ u + @  = 0; @t  =1 z where @z denotes the partial derivative w.r.t. the macroscopic coordinate z ,

("x; t) = E n~ [I0(x)] =

X

f (x; v; n~ ):

(2:24)

(v
e )f (x; v; n~)

(2:25)

v2V

is the mass density,

u("x; t) = E n~ [I(x)] =

X

v2V

is the momentum density,

j("x; t) = is the mass current and

X

v2V

(v
e )(f (x; v; n~) ? f (x; v; n~)2 )

h

i

 ("x; t) = E n~ (I (x) ? E n~ [I(x)])(I (x) ? E n~ [I (x)]) =

X

v2V

(e
v)(e
v)[f (x; v; n~) ? f (x; v; n~)2 ];

(2:26)

is the stress tensor. A more explicit expression for , u, j and  is obtained from the symmetry of V in the exchange v ! ?v. We have: e2~n0 + en~ 0 cosh(~n
v) ; ~ 2~n n~ v2V 1 + e 0 + 2e 0 cosh (n
v ) n~ 0 sinh(~n
v ) X e ; u = (v
e) 1 + e2~n0 + 2en~ 0 cosh ~(n
v) v2V n~ 0 X n
v)(1 ? e2~n0 ) ; j = (v
e) e 2~sinh(~ (1 + e n0 + 2en~0 cosh ~(n
v))2 v2V n~ 0 X n
v)(1 + e2~n0 ) + 4e2~n0 :  = (v
e )(v
e ) 2e cosh(~ (1 + e2~n0 + 2en~0 cosh ~(n
v))2 v2V

=

X

11

Next, we consider the incompressible limit; we follow the same strategy. We choose as initial state for the system the local equilibrium measure (2.20) with chemical potentials n~ ("x; t) given by n~ 0 = r + "2 0("x); n~  = " ("x); (2:27) such that div  :=

d X

=1

@z  = 0:

The the conservation laws are the same as (2.22) except a change of scaling is needed, i.e. we replace L by "?2 L: d @ "d X J ("x)E fl r [I ( )] = "d?2 X X  ]: J ("x)r? E fl r [wx;  x @t x x =1

(2:28)

Assume that a local equilibrium measure with chemical potentials like in (2.27) describes the system also at time t. With this choice of the local equilibrium, we have r (2:29)  = N + O("2 );  = 1 +e er ; E n~ [(x; v) + (x; ?v)] = 2 + O("2 ) (2:30) Moreover, the assumption on V imply that X

X

v2V

v2V

(e
v)(e
v) = K; ;

(e
v)(e
v)(
v)2 = ; A2 + ; B2 + C

where A; B; C and K are constants independent of . Therefore, u = "t() + O("3 ); (2:31) with t() = K(1 ? ); (2:32) We have used the following identities: exp[a + "b + "2 c] =  + "(1 ? )b + "2 (1 ? )h2c + (1 ? 2)b2 i + O("3 ) 1 + exp[a + "b + "2c] 2 n exp[a + "b + "2 c] ? exp[a + "b + "2 c] o2 1 + exp[a + "b + "2c] 1 + exp[a + "b + "2 c] h i 2 = (1 ? ) + "(1 ? )(1 ? 2)b + " (1 ? ) 2(1 ? 2)c + (1 ? 6(1 ? ))b2 + O("3 ); 2 (2:33) 12

where  = ea=(1 + ea). From this identities and (2.27), h i 2 f (x; v; n) =  + "(1 ? )(
v) + "2 (1 ? ) 20 + (1 ? 2)(
v)2 + O("3 ) f (x; v; n)2 ? f (x; v; n) = ?(1 ? ) ? "(1 ? )(1 ? 2)(
v) i h 2 " 2 ? 2 (1 ? ) 2(1 ? 2)0 + (1 ? 6(1 ? ))(
v) + O("3 ): The equations (2.29)-(2.31) are direct consequences of the last identities. From (2.21) and (2.34) one can check that

"d?2

d XX x =1

0 ]J ("x) = "d r? E n~ [wx

because

d XX x =1

n

(2:34)

o

J ("x) (2 ? 1)t()@z  + @z2 z  + O("2 ); (2:35)

r? [("(x + e )) + ("x)] = 2"@z ("x) + O("3 ):

From (2.29) the average of the l.h.s. of (2.28) with  = 0 is O("2 ). Moreover, the last term on the right side of (2.35) is of order "2 . Hence we have to assume div  = 0 to assure the continuity equation. Also, the leading order correction to (2.35) is of order "2, hence it is consistent with the assumption that the density correction is of order "2 (2.27). We remark that, if there is a rst order correction to the density, which can be achieved by adding a term "1 ("x; t) in the rst equation of (2.27), then the leading contribution to the last term in (2.35) is

"d+1

d XX

x =1

@z2 z J ("x)1

where 1 is the corresponding rst order correction to the density (i.e.,  = 0 + "1 ). Note that this term appears because of the stochastic nature of our dynamics (more precisely, because we use asymmetric simple exclusions to replace the deterministic dynamics). The averages of the momentum currents in (2.28) can be computed as d

d

d

=1

=1

=1

X  ] = ?t()(1 ? 2) X @z 0 + t() X @ 2  "?3 E n~ r? [wx  z z

? 2 (1 ? )(1 ? 6(1 ? ))

d XX

(e
v)(e
v)@z (
v)2 + O("):

(2:36)

v2V =1

Recall that V is invariant under the re ection w.r.t each coordinate axis. Hence, by (2.36), the equations for  are of the form d d X @  = ? 1 @ p ? Bh() @ 2 ? Ch() X 2  ;  @  +  @  z z z     z z  @t t() t() t() =1 =1

13

(2:37)

where

p(; ) = t()(1 ? 2)0 + Ah()2 ; h() = 2 (1 ? )(1 ? 6(1 ? )): We obtain the usual Navier-Stokes equation (up to a scale factor if the coecient B = 0. An example of V such that this is true will be given in the Model II below. This heuristic derivation is not entirely correct. The diusion coecient appearing in (2.37) comes completely from the stochastic uctuation, namely, the uctuation associated with the asymmetric simple exclusion. Therefore, if (2.37) were correct, there would be no contribution to the viscosity from the \deterministic part" of the dynamics. We shall prove that (2.37) still holds but the diusion coecient is not given by . The true diusion coecient is always bigger than  and this accounts for the viscosity from the \deterministic part" of the dynamics. We now choose the space V . First notice that, because of the symmetry properties of V , the constants A, B, C and D have the more explicit expressions

A=

X

v2V

v12v22 = C=2; B =

X

v2V

[v14 ? 3v12v22 ]; K =

X

v2V

v12:

(2:38)

Model I

The simplest choice is the following: Let

V = E: With this choice, the only possible collisions are those q = (v; w; v; w0 ) such that v + w = 0 and v0 + w0 = 0. For this model, one can easily check that

A = C = 0; B = 2; K = 2

(2:39)

Model II

Let d = 3 and  denote any element of J , the permutation group of f1; 2; 3g. Let

V = fv : v = (1; 1; $) for some  2 J g where $ is the positive solution of

$4 ? 6$2 ? 1 = 0:

V is invariant under re ections w.r.t. any coordinate plane and permutations of coordinate axes. Moreover, with the above choice of $, one gets:

B=0 14

and

A = 8(1 + 2$2 ); C = 16(1 + 2$2 ); K = 8(2 + $2 ) (2:40) The proof that the only collision invariants are total mass and total momentum is part of the ergodic theorem proven in next section. Now we are ready to state our convergence result. Let ft the distribution density, w.r.t. r , of the process on the diusive time scale, namely the solution of the forward equation @ f = "?2 L f : t @t t We consider the initial datum ft=0 =

0 h

(2:41)

given by

?1 0 = Z" exp "

d X X x2L =1

 ("x)I (x)

i

(2:42)

with  smooth periodic functions. Recall the de nition of the speci c relative entropy of two densities f and g w.r.t. r :

 s(f j g) = "d f log fg

(2:43)

and let

s(f )  s(f j 1): The rest of this paper is devoted to prove the following Theorem 2.1. Assume d  3 and V as in Model I or Model II. Let ft the density w.r.t. the measure r , de ned in (2.10), at time t. Then there are d  d diagonal matrices D ,  = 1; : : : ; d such that D  Ds =  I (as matrices) and the following holds: given a smooth solution p(x; t); u(x; t) of the incompressible Navier-Stokes equations div u = 0;

(2:44)

d d X  @ u + h() B@ u2 + C X  @z @z u ; = ?@z p + D u @ u  z z   ;  2 @t t() ; =1 =1

(2:45)

we de ne the following density t w.r.t. r h

t = Zt?1 exp "

d X X x2L =1

i

("x; t)I (x) ;  = t()?1 u:

Then:

15

(2:46)

1)

lim "?2 s(ft j t) = 0:

"!0

(2:47)

2) De ning the empirical elds " for  = 0; : : : ; d, as

0"(x; t) = "d " (x; t) = "d?1

X

x2L

(z ? "x)I0 (x(t))

(2:48)

X

(z ? "x)I (x(t));  = 1; : : : ; d; (2:49) x2L we have that, for  = 1; : : : ; d, " converges weakly in probability to u, solution of the "

Navier-Stokes equation, and 0 converges to the constant N. 3) The matrices D  Ds are diagonal and have the form  =  ; [D1 ; + D2 (1 ? ; )] D ;

Moreover

D 6= Ds;  = 1; : : : ; d:

The proof of this theorem is given in the Sect.3 and it is based on the Lemmas and Theorems of Sections 4{6. Remarks. (i) The matrix Ds is the diusion matrix which appears in the limiting equations when one consider only the symmetric part of the generator Ls = 1=2(L + L ). The true diusion matrices D are bigger than Ds, as stated in the last part of Theorem 2.1 because of the contribution of the deterministic motion. (ii) The second part of the theorem is a consequence of the relative entropy estimate by means of the following entropy inequality. If f and g are normalized densities w.r.t. d, then   (2:50) E f [X ]  ?1"?d s(f j g) + ?1 log E g expf X g for any positive and for any random variable X . See [23, 10] and lemma 2.2 below for details. (iii) The factor t() is due to the relation between the momentum density u and its conjugate chemical potential  at the rst order as in (2.31). Here u = "u + O("2 ). (iv) We need the speci c form of V only in the proof of the ergodic theorem, Theorem 3.3 in next section. The rest of the proof is valid under quite general conditions. Therefore Theorem 2.1 holds for any model in d  3, with V a nite set of vectors of given length, invariant under re ections and permutation of axes, for which the ergodic Theorem 3.3 is valid. To conclude, we recall a Lemma from [8] 16

Lemma 2.2. Let f be a density satisfying ?2 s(f j ) = 0 lim " "!0

where is a density of a product measure of the type (2.46). Then for any J and any bounded local function F one has

lim [E f ? E ]["d?1 "!0

X

x

J ("x)x F ] = 0:

If instead of "?2 s(f j ) ! 0 one has "?2 s(f j )  const. then the following bound holds

j[E f ? E ]["d?1

X

x

J ("x)xF ]j  const:

In particular, can be the equilibrium measure r . Finally, a direct computation shows that

s( ) = "dE r [ log ]  const."2 P

so that the previous lemma assures that j[E ? E r ]["d?1 x J ("x)x F ]j  const:

3. The relative entropy estimate

The strategy of the proof of Theorem 2.1 is similar to the one used in [8]. The rst step is to choose a local equilibrium density, ~ t, (and suitable corrections) to compare to the density ft of the actual time evolution. This density will be chosen in such a way that its parameters to the rst order satisfy the hydrodynamic equations, while the second order terms take into account the contribution of the fast modes. More precisely, we put 

~ t = Z~t?1 exp " + "2

 X

x2L

3 X X

(  !^ )("x; t)I (x)

x2L =1

(0  !^ )("x; t)I0 (x) + ()



:

(3:1)

Here Z~t is the normalization,  is the convolution product on Zd, and  (z; t) , !^ and  are chosen as follows. (3.A) Let ` and k = `?2=d be integers, ` = `d+2 and suppose that k is divided in disjoint cubes of size (2` + 1), with centers  2 (2` + 1)Zd , jj  k. Let `1 = ` ? `1=d2 and 17

S consider the cubes `1; and ~ k = jjk `1; is the region k without corridors of width 2`1=d2 . De ne !^ and ! to be the normalized characteristic functions

!^ (x) = j~ k j?11l (x 2 ~ k ); !(x) = (2k + 1)?d1l (x 2 k ): (3.B) The chemical potential ,  = 1; : : : ; d, are solutions of div  

d X =1

@z  = 0 d X

d X

@  =? 1 @ p?  @ 2  2 +C + D B@   @   z z z    z z  @t t() t() ; =1 =1 h() n

o

(3:2)

 given as in Theorem 2.1, with D

p(; 0 ;  ) = t()(1 ? 2)0 + Ah()2 ; t() = K(1 ? ); h() = 2 (1 ? )(1 ? 6(1 ? )): P

P

(3.C) Choice of : () = ? x2L d; =1 @  ("x; t)(^!  y F ) where F are local functions satisfying the conditions in De nition 4.2 of the next section. The choice of !^ is for the convenience that no boundary terms arise in the multiscale analysis; one can simply use ! but a few steps are needed to bound the boundary terms. The multi-scale analysis is an important tool in proving the Theorem 4.6 of [8]. Its extension to the present setup will be used extensively here. For the convenience of later reference, recall for our models the currents are given by ; = w(s)  ? w(a)  ; wx; x; x;

and

(a)  = wx; (a) 0 = wx;

where

X

v2V

(s)  = r I wx; 

(3:3)

(e
v)(e
v)bx; (v);

X

v2V+

(e
v)bx; (v)

h i 1 bx; (v) = (x + e ; v)(x; v) ? 2 (x + e ; v) + (x; v) 18

(3:4)

Furthermore, the density, velocity, stress tensor and the pressure are related by r  = N + O("2 );  = 1 +e er ; u = "t() + O("3 )

?
 = t() + "2p(; ) ; + "2h()



B2 

 ; + C

(3:5)



:

We return to the proof Theorem 2.1. The density ~ t dier from t only at the second ~ is small. order in ". The following Lemma shows that the dierence of s(f j ) and s(f j ) Lemma 3.1. Suppose that f is a probability measure on L . Let

 = Z ?1 exp 0 = Z 0?1 exp

d h X X x2L =1

d ? h X X

i

" S("x; t)I (x)

" S("x; t)I (x

x2L =1

) + "2 t

 ("x; t)x F


i

two densities w.r.t. r where S and t are smooth functions of (z; t) and F local functions of the con guration such that E [F] = 0. If "?2 s(f j) ! 0, then h

i

lim "?2 s(f j) ? [s(f j0 ) = 0 "!0

(3:6)

The proof is an obvious extension of the one given in [8]. From this remark, to prove Theorem 2.1 we shall only prove lim "?2 s(ft j ~ t ) = 0:

(3:7)

"!0

To state the main Theorem of this section, we need the following de nition. De nition 3:2. Let G be a local function such that E r [G] = 0. For any vector m = fmgd=0 we de ne the \variance" V`(G; m) by

X

V`(G; m) = (2`1 + 1)?d [

jxj`1

(x G ? `(G))](?Ls;` )?1 [

X

jxj`1



(x G ? `(G))] `;m (3:8)

where `1 = ` ? `1=d2 and `;m is the canonical Gibbs state of (2` +1)d sites with parameters such that the mean of I,   0, on the block ` are equal to m, namely,

`;m = ?1

`;m

d Y =0

(I;` ? m)` :

19

Here ` denotes the counting measure on the con gurations in `, `;m is the normalization and `(G) = E r [GjI`+] where E r [GjI`+] is the conditional expectation given the averages

I;` = (2` +1 1)d

X

jyj`

I (y )

 = 0; : : : ; d

(3:9)

The generator Ls;` is the symmetric part of the generator restricted to `, in the sense that the sums in (2.2) and (2.4) are restricted to the sites and bonds contained in `. We de ne also the \variance" of G by

V (G; r) = lim sup E r [V`(G; I` )]: `!1

In order to make above de nition meaningful, one needs to give sense to L?s;`1 . This is consequence of the nite volume ergodicity of the generator, summarized in the next Theorem 3.3. We will use the following notation: given a con guration  on a box ` of size `, we put X N() = I (x);  = 0; : : : ; d: x2`

Moreover, for any choice of m , we denote by `;m the set of con gurations  in ` such that N() = m(2` + 1)d for  = 0; : : : ; d. Note that there are many choices of the m 's such that `;m is empty. We consider only the non trivial choices. We have the following Theorem 3.3. The process generated by Ls;` on `;m endowed with the measure `;m is ergodic. Proof: Since the process is a nite Markov chain, it is sucient to show that it is transitive, in the sense that, given any couple of con gurations  and  in `;m, it is possible to nd a sequence of jumps and collisions which transform  into . We rst consider Model I. Let

N ? () = 

X

2`

[(x; e ) + (x; ?e )];  = 1; : : : ; d;

d X =1

N? () = N0()

Step 1: If ;  2 `;m , then N? () ? N? () is even. Proof of step 1: If N? () ? N? () were odd, then there is at least x 2 ` such that (x; ") ? (x; " ) = 1 and (x; ?" ) ? (x; ?" ) = 0 or viceversa. Hence [(x; " ) ? (x; ?e)] ? [(x; " ) ? (x; ?e )] = 1. But N() = N(), so for each such x there is

20

an y 2 ` such that [(y; ") ? (y; ?e )] ? [(y; " ) ? (y; ?e )] = 1, to compensate the discrepancy of momenta in x. But this means that N? () ? N? () is not odd. Step 2: Let N? () ? N? () = 2k. If k > 0 for some  = 1; : : : ; d, then there is a sequence of jumps and collisions which transform  into . Proof of step 2: To simplify the argument, let d = 3. We have N0() = N0 (), hence P3 =1 k = 0. Without loss of generality we can assume k1 < 0 and k2  0, k3  0. Therefore, with a suitable sequence of jumps, we can transform  into a con guration 1 with N (1) = N () and N? (1) = N? () such that there are at least ?k1 sites x1 ; : : :; x?k1 where there is no particle both with velocity e1 and with velocity ?e1. In the same way, one can construct a con guration 2 such that N(2) = N () and N? (2) = N?() such that there are k2 sites y1; : : : ; yk2 without particles with velocity e2 and ?e2 and k3 sites yk2 +1; : : : ; yk2 +k3 without particles with velocities e3 and ?e3 . Since k2 + k3 = ?k1, there is no loss of generality assuming that the two sets of point coincide, because otherwise extra jumps can be done to reduce to this situation. Then we are in position to make a collision in each of the sites xi , transforming 1 into 2. Step 3: If k = 0 for each  = 1; : : : ; 3 then a sequence of jumps is sucient to transform  into . P P Proof of step 3: In fact, if k = 0, x2` (x; v) = x2` (x; v) for each v 2 V . Therefore jumps are enough to transform  into . As for Model II, we sketch the proof. De ne    i there exist a collision q 2 Q and a site y 2 L such that Qqy  =  . This de nes a equivalence relation and we say  and  are connected if they are in the same equivalence class. If  is a con guration in 0, we will also denote by  the subset of V such that (v) = 1. Lemma 3.4 For any two con gurations  and  at 0,  and  are connected i I +() = I +( ). Proof: Step 1 We claim rst the same result holds on the space W = f(1; 1)g. This can be checked easily. Step 2 Next, we claim Lemma 3.4 holds if we restrict ourselves to the subspace V2(3) = f(1; 1; $)g. This can be checked easily too. We give a brief sketch. From the previous claim, the projection of  and  onto the space W are connected. Let 3 = the number of fv 2  with v3 = $g. Since the momenta of  and  are the same, 3 = 3. Hence we only have to prove the following: Suppose fu; vg   with u 6= v. Let  = ( n fu; vg) [ f(u1; u2; v3 ); (v1 ; v2 ; u3)g. Then  and  are connected. But u + v = (u1 ; u2; v3) + (v1 ; v2 ; u3), hence we have proved the claim. This proposition is obvious and will be used repeatedly in the following proof. It basically means that the third component can be changed freely in V2(3). Step 3 Suppose  and  are two con gurations in V2(3) with the same momentum but not the same number of particles. Suppose I0 () > I0 ( ). We claim that  is connected to  [ A for some A satisfying that A \  =  and A is re exive, i.e., if v 2 A then ?v 2 A. We rst claim that I0() ? I0( ) = 2n for some n 2 N since otherwise I3() 6= I3( ). We 21

now follows previous arguments by rst considering the projection onto the space W and then adjusting the third components. In more details, we rst prove our claim in the two dimensional space W . This can be checked directly. We can then use the remark from the step 2 to adjust the third components. This proves the claim. Step 4 Finally, the general case. Let (i) = fv 2  : vi = $g. Clearly, I ((i)) = I( (i)) for  > 0. Hence (i) and  (i) dier only for pairs of opposite vectors. Since any pair of opposite vectors can be brought into any dierent pair of opposite vectors by a collision, it is not hard to check  and  are connected. We now consider the general cases: Lemma 3.5 For any two con gurations  and  on `,  and  are connected i I +() = I +( ). Proof. The proof follows almost exactly the same arguments as in the previous lemma except some explanations are needed for the use of the simple exclusion. Let us illustrate it by proving the step 1 carefully. We are now in the setting of two dimensional con gurations  and  with the same momentum and number of particles. We now bring particles with opposite velocities to the same sites as much as possible. We can then decompose  as union of two con gurations pairs, consisting of pairs of opposite vectors at the same sites, and the rest, singles. Since we have only four directions and no pairs are allowed, we can assume, without loss of generality, with obvious changes to cover the dierent situations, that singles contains  particles in the direction (1; 1) and particles in the direction (1; ?1) and nothing else. Similarly, we can decompose  in the same way. Since (1; 1) and (1; ?1) are linearly independent, we can check that  singles contains exactly the same number of particles in the directions (1; 1) and (1; ?1) and nothing else. Hence pairs and  pairs contains the same number of pairs because the total number of particles of  and  are the same. We can connect pairs and  pairs by simple exclusions and collisions and we can also connect singles and  singles by simple exclusions. This concludes the proof of step 1. Step 2 is obvious. We now prove the claim in step 3. First we check if instead of having the same number of particles and momentum in step 1,  and  only have the same momentum (note they are now con gurations on W ). From previous arguments we still have that singles and  singles have the same number of particles. Hence  and  can be brought into con gurations 0 and  0 such that 0 n  0 consists of pairs (assuming I0()  I0( )). With this comment, we can prove step 3 easily. Finally, step 4 is by now straightforward. The main Theorem of this section is Theorem 3.6 Let ~ t be de ned as above, k = `"?2=d and ft as in (2.41). Then, for d  3, and for any > 0 ?2 s(fT j ~T) lim lim " `!1 "!0

 C `lim lim !1 "!0

Z

0

T

"?2 s(ft j ~ t)dt + CT

22

d X ; =1

V (H ; r) + const.

(3:10)

Here C is a positive constant, V is de ned in (3.8) and

H

d X ( a ) @ I (x ) ? L xF Hx; = wx; ? D~  

=1

 H0 ; :

where

(3:11)

= D ? ; : D~  

We need the following lemmas. The rst one is a large deviation bound; the other two are bounds on the entropy and the Dirichlet form. Lemma 3.7. Let be the density given by (2.46) and let, for  = 0; : : : ; d,

mx;k = E [I;k (x)]; I;k (x) = (2k + 1)?d

X

jx?yjk

I(x)

 = E [I;k (x); I;k (x)]  E [(I;k (x) ? m )2 ] x;k x;k where E [A; B] = E [AB] ? E [A]E [B]. Then, for k = `"?2=d and for any q < q0 with q0 a small xed constant,

lim lim "d?2 log E `!1 "!0



n X

exp q

x

  (I;k (x) ? mx;k (v))2 ? x;k

o

=0

(3:12)

In general, if Gx = Gx(y0 ;


; yd ) is a family of bounded smooth functions such that x Gx (mx;k ) = @G @y y=mx;k = 0;  = 0; : : : ; d;



then for any smooth function J , there is q0 such that for any q < q0

lim lim "d?2 log E `!1 "!0



n X

exp q

x



J ("x) Gx(I;k (x)) ? E [Gx(I;k (x))]

o

= 0: (3:13)

This proof of Lemma 3.7 is similar to the one for the corresponding lemma in [8]. We omit the details. Lemma 3.8 The density ft satis es the following bounds: (i) d s(f )  ?const:"?2+dD (pf ); L t dt t 23

where DL (g) = (DexL + Dc L )(g) is the Dirichlet form. Here

Dex (g) = L

with and Dc L (g) is given by

d X XX x2L v2V =1

Z

 [rvx;g()]2dr ();

(3:14)

rvx;g() = g(x;x+e;v ) ? g() Dc L (g) =

X XZ

x2L q2Q

[(Qqx g)()]2 dr ():

(ii)] For any t  0 , "?2 s(ft )  const: (iii) For any local function F and any J,

j[E ft ? E r ]["d?1

(i)

X

x

J ("x)x F ]j  const:

Lemma 3.9. The relative entropy s(ft j ~ t ) satis es the bound

d s(f j ~ )  "d Z f ~ ?1("?2 L ? @ ) ~ d : t t dt t t @t t r (ii) There is a constant ct independent of ft such that ?2 [ d s(ft j ~ t ) ? "d lim " "!0 dt

(iii)

Z

Z

@ ) log ~ d ] ? c  0: ft ("?2 L ? @t t r t Z

@ ~ d : @t t r Lemma 3.8 is obtained by adapting arguments from [23]. The rst equation of Lemma 3.9 is proven in [2, 10]. The last equation is obvious. The second equation diers from the rst one only because the term L ~ t is replaced by L log ~ t. From Lemma 2.2, their dierence is small. Details were given in [8]. From now on we use the notation \const:" for any constant independent of ft and of the con guration (but may depend on the initial state and the parameters ). We shall not compute these constants at each step but will use equation (iii) of Lemma 3.9 to determine the nal constant.

L ~ t d = 0 =

24

By Lemma 3.9 to estimate the derivative of the relative entropy it is enough to compute the quantity I  "d?2 E ft [("?2 L ? @t@ ) log ~ tdr ]: We rst compute "d?4L log ~ t :

"d?4 L log ~ t ="d?4(Lex (log ~ t ? "2) + "d?2 L 

d X X d ? 3  =" (  !^ )("x; t)r? wx; x ; =1 d XX 0 + "d?2 L  = A1 + A2 + A3 d ? 2 (0  !^ )("x; t)r? wx +" x =1

(3:15)

The term A2 is easier than A1 because of the factor "d?2 . We rst sum by parts. The dierence operator is thus moved to act on 0  !^ . Recall the expression of the current given by (3.3). We can separate the current into the symmetric part and the asymmetric part. Let A2 = B1 + B2 with B1 (B2 resp.) related to the symmetric (asymmetric resp.) part. Hence

P

d XX d ? 2 r? r 0[^!  I0]x B1 = " x =1 d XX (a) 0 ) d ? 2 r? 0("x; t)(^!  wx B2 =" x =1

where [^!  I]x = y !^ (x ? y)I (y ). From the Taylor expansion we can approximate r? r by "2 @z @z with a negligible lower order term. Since our density ft is very close to the equilibrium, we can compute B1 to great accuracy. Because E r [B1] = 0, we have from (ii) and (iii) of Lemma 3.4

jE f (B1)j = jE f (B1 ) ? E r [B1]j  ": Hence B1 is negligible. We now examine the dicult term A1 . d X X d ? 3  ]x A1 = ? " (r )("x; t)[^!  wy; x ; =1 d d X X X X ? d ? 3 d ? 3 (a) ] = ?" r r  (^!  I)x ? " r [^!  wx; x x ; =1 x ; =1

25

As remarked after (3.15), we can replace "?2r? r by @z @z up to a negligible error term. To deal with the currents appearing in A1 and in B2, we need to look at their expectations w.r.t. suitable grandcanonical measures modeled according to empirical averages of the conserved quantities. To this end, for any Y = (Y0; : : : ; Yd), let 1;Y be the grandcanonical measure such that

E 1;Y [I ] = Y;  = 0;


; d

(3:16)

The measure 1;Y is uniquely de ned because it is required to be an invariant measure of the dynamics (exclusion + collision). Explicitly, 1;w is the product measure with marginal at one point point given by d

X 1;Y () = Z ?1 expf n(Y )I ()gr ();

r;n

=0



(3:17)

for  = f(v); v 2 Vg. The chemical potentials n(Y ) are suitably chosen to give the correct averages (3.16). The subscript 1 is used because we will need the grandcanonical measures for the symmetric simple exclusion which will be introduced later on and carries a subscript 2. Given an integer k > 0 for any con guration  in L , we consider the local empirical averages I;k (x),  = 0; 1; : : : ; d, de ned in (3.9). The measure 1;Ik+(x) is de ned according to (3.17), with Y = I;k (x). n(Y ) = n(Ik(x)),  = 0; : : : ; d are the corresponding chemical potentials; here Ik+(x) = fI;k (x);  = 0; : : : ; dg. Moreover we put

f (v; n(Y ) ) :=

en(0Y ) +n(Y )
v : (Y ) 1 + en0 +n(Y )
v

(3:18)

With above notations, we have, for Y = I;k (x) X I0;k (x) = E 1;Ik(x) [I0] = f (v; n(Y ) );

v2V

X I;k (x) = E 1;Ik(x) [I] = (v
e)f (v; n(Y ) )

(3:19)

v2V

We can now de ne  = (^!  w(a)  )x ? E 1;Ik (x) [(^!  w(a) )x ]; gx; y; y; (a) 0 ) ? E 1;Ik (x) [(^ 0 = (^ (a) 0 ) ] ;  = 1;


; d gx; !  wy; !  wy; x x

26

(3:20)

From previous de nitions we have, n

(a)  ) ] = X (v
e )(v
e ) [f (v; n(Y ) )]2 ? f (v; n(Y ) ) E 1;Y [(^!  wy; x  v2V : = ? (Y ) n

o

o

X (a) 0 ) ] = E 1;Y [(^!  wy; (v
e ) [f (v; n(Y ))]2 ? f (v; n(Y )) x v2V

(3:21)

: = ?0 (Y )

The equations (3.21) are not explicit in terms of the conserved quantities. To obtain an explicit expression one has to solve (3.19). For our purpose, some approximation will be sucient. This will be carried out later on in this section. Summarizing, we have d d X X X X 2 d ? 3 d ? 1 (r )("x; t) (Ik+(x)) @z z  [^!  I ]x ? " A1 = " x ; =1 x ; =1 d X X + o(1): (r )("x; t)gx; + "d?3 x ; =1

Let

+ ; : =D ~ ; D;

(3:22)

d X X d ? 1 (@z @z  )("x; t)[^!  I ]x A1 = " D;  x ; ; =1 d d   i h X X d ? 3 ~ +" (@z  )("x; t) gx ? D; !^  r I (y ) x

=1 x;; =1 d X X d ? 3 ?" (r )("x; t) (Ik+(x)) + o(1): x ; =1

(3:23)

We can rewrite

Similarly, B2 = C3 + B3 where

C3 = "d?1

X

B3 = "d?2

X

x;

x;

0 @z 0gx;

r00 (Ik+(x))

27

From (3.15), (3.23) we rearrange the remaining terms as

A1 + A2 + A3 = C1 + C2 + C3 + B3 + o(1) where d d h     i X X X  d ? 3 ~ (@z  )("x; t) gx ? D; !^  r I (y ) x ? !^  L (y F ) x ; C1 = " x ; =1

=1

(3:24)

and

d i X + d ? 3  d ? 1   D; (@z @z  )("x; t)I ;k (x) ? " (r )("x; t) (Ik (x)) C2 = " x ; =1

=1 d h X X

(3:25)

We also need to compute

? "d?2E f f @t@ log ~ tg

d @  d?1 X X f ;k (x) + "d X @ 0 ("x; t)I0;k (x) + const. = ?E "  ( "x:t ) I  x =1 @t x @t

Using an argument similar to the one explained in bounding B1, one can prove that the second term on the right side is negligible up to a constant. Summarizing these computations, we have

I = E f [C1 + C3 + C4] + const: + o(1)

(3:26)

X

(3:27)

where

C4 = "d?2

and ?x(Y ) = "

d n X =1 d X

? "?2

=1

x

?x(Ik+(x))

d  X @ @z @z  ("x; t)Y ? @t  + D ; =1

r ("x; t) (Y ) ? "?1 r0("x; t)0 (Y )

o

The terms E f [C1] and E f [C3] are dealt with by using the following theorem. 28

(3:28):

Theorem 3.10. Suppose h is a local function, J is a smooth function and is a local Gibbs state de ned as in (3.1). For any > 0 (

lim lim "d?2 `!1 "!0

Z

f

Z

X

x2L

J ("x)f(^!  h)(x) ? E

1;I+ (x) k



[h] d )

p

? const. ?1 J (z)2 dzV (h; r) ? "d?4DL ( f ) ? ?1 "?2s(f j )  0 ;

(3:29)

provided  is small enough. Here ` = `d+2, k = `"?2=d and V is de ned in (3.8).

Corollary 3.11. For any > 0, any f and any Gibbs state such that "?2 s(f j ) is bounded,

lim lim "d?2 `!1 "!0

Z X

x2L

Z

n

J ("x) !^  (y h ? E

1;I+(x) k

o

[h] (x)fdr

d?4 ?1  21 J 2(z)dz V (h; r) + "lim lim "?2 s(f j ): !0 " DL ( f ) +  `lim !1 "!0 p

In particular, if ft is the density in Theorem 2.1 then for any > 0,

lim lim `!1 "!0

Z

T 0

dt

n

"d?2

Z X n

x2L

J ("x)^!  (y g ? E

 ?1CTV (h; r) + ?1 `lim lim !1 "!0

Z

0

T

1;I+(x) k

o

[g]) (x)ft dr

o

dt"?2s(ft j ~ t) + const.

(3:30)

where C is some constant depending on J and . The proofs of both the theorem and its corollary are postponed to the last section. From Corollary 3.11 we can bound E f [C1] by a variance term and the Dirichlet form of the density f . Because we are interested in a time integrated inequality in Theorem 3.6, the Dirichlet form will be integrated in time and we can use (3.30). Hence

E f [C1]  const. ?1

d X ; =1

V (H ; r) + ?1"?2 s(ft j ~ t) + const. + const: + o(1)

where the o(1) denotes an error negligible after time integration. Similarly, we can bound C3. This time we have an extra " factor and the variance term vanishes in the limit " ! 0. 29

Summarizing,

I  const. ?1

d X ; =1

V (H ; r) + ?1"?2 s(ft j ~ t) + const. + E f [C4] + const: + o(1) (3:31)

The constant is determined by the condition i h X E ~ t "d?2 ?x(Ik+(x)) + const. = o(1)

x

By Lemma 2.2 one can replace ~ t by t. Let m x;k = E t [I;k (x)],  = 0; : : : ; d. From the central limit theorem for the product measure t, m x;k = mx + O(k?d), with mx = E t [I (x)] . Hence the constant in (3.31) equals to h i X X E ~ t "d?2 ?x(Ik+ (x)) + o(1) = "d?2 E t [?x(mx )] + o(1)

x

x

Using the entropy inequality (2.50) with = "2?d we get

I "?2 s(ft j ~ t) + const. ?1 h

+"d?2 log E t exp

n X

x

d X ; =1

V (H ; r) + ?1 "?2 s(ft j ~ t) + const.

?x(Ik+(x)) ? ?x(mx )

oi

+ o(1)

(3:32)

We will use the large deviations Lemma 3.2 to bound last term in (3.32). To apply it, we have to check that the function ?x satis es the following condition

@ ?x @Y Y =mx = o(");  = 0;


; d

(3:33)

To check this we have to evaluate derivatives of  (Y ), de ned in (3.21). To this end we compute  (Y + Y ) ?  (Y ) to the rst order in Y . We use the notation Y = (Y0 ; Y ). From (3.18) and (2.33) (and a simple computation), to the rst order in Y we have

f (v; n + n) ? f (v; n + n)2?(f (v; n) ? f (v; n)2 ) h i = (n0 + n
v) f (v; n)(1 ? f (v; n))(1 ? 2f (v; n)) 30

with n = n(Y +Y ) ? n(Y ). Hence

 (Y + Y ) ?  (Y ) n o X 2 2 = (v
e )(v
e ) f (v; n + n) ? f (v; n + n) ? (f (v; n) ? f (v; n) ) v2V

=

X

v2V

(3:34)

(v
e )(v
e )[n0 + (n
v)]f (v; n)(1 ? f (v; n))(1 ? 2f (v; n)) + O(n2 )

Recall (2.34). Hence for Y = m we have

 (Y + Y ) ?  (Y ) X = [n0 + (n
v)](v
e )(v
e )f (v; n)(1 ? f (v; n))(1 ? 2f (v; n)) v2V

= [S + O("2 )]n0 ; + "T

X

v2V

(v
e )(v
e )(
v)(n
v)

(3:35)

= Sn0; + 2"h() B n; + C n +2  n + O("2 ) h

i

where S and T are constants depending on  and can be computed explicitly. The analogous computation for 0 is much simpler because an extra factor " appears before 0 . We have

0 (Y + Y ) ? 0 (Y ) =

X

v2V

[n0 + (n
v)](v
e )f (v; n)(1 ? f (v; n))(1 ? 2f (v; n)) + o(1)

= t()(1 ? 2)n + o(1): Moreover,

(3:36)

Y = t()n + o(1); Y0 = t()n0 + o(1) Therefore, up to order o(") we have d  @  (z; t) + X @ ?x  = " ? (@z @z )(z; t)D @Y Y =mx @t  ; =1

d X   1 h (  ) 2 ? t() @z p(z; t) ? t() B@z  (z; t) + C  (z; t)@z  (z; t) + o("); =1

d X @ ?x @Y0 Y =mx = const." =1 @z (z; t) + o(")

31

(3:37)

The vanishingness of the right sides of (3.37) is exactly the incompressible Navier-Stokes equations. Therefore, the last term in (3.32) vanishes in the limit as ` ! 1. Applying Lemma 3.9 integrated on time we get Z

T

(const. + ?1 )"?2 s(ft j ~ t)dt+ lim lim "?2s(fT j ~ T )  `lim lim `!1 "!0 !1 "!0 0 d CT X ; r) + const. ) V ( H 

; =1



The proof of Theorem 3.4 is then achieved by the use of Gronwall lemma. We shall prove that there exists a positive diusion matrix D such that inf F

d X ; =1

V (H ; r) = 0;

(3:38)

with H de ned in (3.11). Hence Theorem 2.1 follows from Theorem 3.4. The rest of the paper is devoted to prove that (3.38) holds with F in a suitable class of local functions.

4. The structure theorem. In this section we de ne the spaces of functions we will work with, and characterize their elements. We use several results of [8] for the simple exclusion. We rst introduce some notations. Since the reference density is xed, we shall use the symbol E  instead of E r . Fixed any x 2 L consider the following alternative description of the con gurations in x. Recall that #(V ) = N . We introduce N 2 numbers c(v) with  = ?N + d + 1; : : :; 0; 1; : : : ; d and v 2 V . We put

I(x ) =

X

v2V

c(v)(x; v):

(4:1)

To be consistent with the previous de nition of the conserved quantities I(x) for  = 0; : : : ; d, we put c0(v) = a; c(v) = be
v;  = 1; : : : ; d with the constants a and b speci ed by the normalization conditions below. The constants c(v) for  < 0 are chosen so that the mapping x ! fI(x)g be invertible and the orthogonality conditions below are satis ed. De ne r

 e ~(
; v) = (
; v) ? ;  := (
; v) = 1 + er : 32

For any f and g functions of x de ne the scalar product (f; g) := E  [f (x ); g(x)] = E  [f (x )g(x)] ? E [f (x )]E  [g(x)]: Then we have the orthogonality relations: ((
; v); (
; v0 )) = v;v0 (1 ? ) We also require the conditions (I (x); I (x )) = ; ; ; = ?N + d + 1; : : : ; 0; 1; : : : ; d:

(4:2)

For Model I for example we have N = 2d. Let

I(x) = [2(1 ? )]?1=2

X

v2V

I0(x) = [2d(1 ? )]?1=2

(e
v)(x; v);  = 1; : : : ; d

X

v2V

(x; v)

and de ne I(x),  = ?d + 1;


; ?1 by

I :=

?1 X =?d+1

A [(x; e ) + (x; ?e )];

where A are chosen so that

E [I(x ); I (x)] = ; ; ; = ?d + 1;


; d

(4:3)

The relation (4.3) for  = 0;


; d holds because of the de nition of I (x),  = 0;


; d. It remains to choose A so that (4.3) holds for  < 0. This can be done easily for d  2. In particular, for d = 3 one can choose

A1 = [2(1 ? )]?1=2 (1; ?1; 0); A2 = [8(1 ? )]?1=2 (1; 1; ?2) In general, we will use the notation I ? ; I + to mean the sets fI ;  < 0g, fI ;   0g and I~ will denote the centered variable I~ = I ? E [I]: (4:4) We now introduce two dierent spaces G ex and G~ex which are suitable to describe the exclusion part only. The rst one is constructed specifying the local densities of particles of each color, while the second one involves in nite volume grandcanonical averages. The space G~ex was introduced in [8]; the space G ex was introduced in [24]. 33

De nition 4.1 Let G ex = fg local function of  : E [g j Ik ] = 0 for some k > 0 and all values of Ik g: Here E [
j Ik ] denotes the expectation conditioned on xed values of I;k = Avx2k I(x); for  = ?N + d + 1; : : : ; d: The conditional expectation in principle depends on the measure . Suppose  is a measure on  given by

() = Z ?1

;p

Y

d Y

x2 =?N +d+1

exp[pI (x)]:

(4:5)

When it is conditioned to Ik , it is the same as the uniform distribution of (x; v) on k conditioned on Ik . Therefore the choice of the measure  is not important as long as it is in the class of measure satisfying (4.5). De nition 4.2 Let G~ex = fg local function of  satisfying (4.6)g

E [g] = 0;

X

x

E  [g; I(x )] = 0; for  = ?N + d + 1; : : : ; d:

(4:6)

Let ^ be the measure such that E ^ [I] = m for  = ?N +d+1; : : : ; d and g^(m) = E ^ [g]. Then the second condition is equivalent to

@ g^(m) @m m=m = 0 for  = ?N + d + 1; : : : ; d;

(4:7)

where m is the vector fm;  = ?N + d + 1; : : : ; dg and m   = E r [I],  = ?N + (d + 1); : : : ; d, are the values corresponding to the equilibrium measure r . We introduce in G ex the following bilinear form and the associated seminorm, based on the symmetric simple exclusion generator

 g; h ex= 14 V ex(g + h) ? V ex(g ? h) ; k g k2?1;ex= g; g ex; ?



(4:8)

with V ex(g) de ned similarly to V (g; r) in De nition 3.1 of the previous section, but with the generator Ls replaced by Lex s . More precisely, let

V`ex(g; m) = (2`1 + 1)?d

X

jxj`1

34

?1 xg(?Lex s;`)

X

jxj`1

x g



`;m

with `1 = ` ? `1=d2 . Here `;m is the canonical Gibbs state of (2` +1)d sites with I;` = m,  = ?N + d + 1; : : : ; d. Explicitly

`;m = Z ?1

`;m

d Y =?N +d+1

(I;` ? m )`:

Here ` denotes the counting measure on the con gurations in ` and Z`;m is the normalization. With these notations,

V ex(g; r) = lim sup E r [V`ex(g; I`)]: `!1

We omit the label r because it is xed through this section. Let d XX ex a; r I(0 ); a; G0 = f  =1

2 Rg = f

d XX  =1

a; r I~(0 ); a; 2 Rg

P

(4:9)

P

be the space of the gradients. We used the shorthand  instead of d=?N +(d+1). From [8], we have the following theorem. Theorem 4.3 The bilinear form 
;
ex is a scalar product and it can be extended to G~ex. Let Gex be the closure of G ex under the norm (4.8). One has ?




ex Gex = G~ex = Lex s G ex  G0

(4:10)

Previous de nitions are natural when one disregards the collisions, so that all the I 's are conserved. With collisions one needs the analogous of de nitions 4:1 and 4:2 involving only the conserved quantities I , for  = 0; : : : d. De nition 4.4 Let

G = fg local function of  : E [g j Ik+] = 0 for some k > 0 and for all values of Ik ;   0g: Here E [
j Ik+] means expectation conditioned to xed values of Avx2k I(x) for  = 0; : : :; d. De nition 4.5 Let

G~ = fg local function of  satisfying (4.11)g; E  [g] = 0;

X

x

E [g; I(x)] = 0;  = 0; : : : ; d: 35

(4:11)

Let  be the product measure such that E  [I ] = m for   0 and g^(m) = E m [g]. Then as before the second condition is equivalent to

@ g^(m) @m m=m = 0;  = 0; : : : ; d:

(4:12)

where m is the vector fm;  = 0; : : : ; dg and m   = E r [I],  = 0; : : : ; d, are, as before, the values corresponding to the equilibrium measure r . Clearly, for any function g,

g?

@ g^(m) I~ ( ) 2 G~: m=m  0 0 @m X

Let

(4:13)

k
k2?1= V (
)

with V de ned in de nition (3.1). Up to this point we have not proved that k g k?1< 1 for all g 2 G . The only information we have is that for all g 2 G ex (recall G ex  G )

k g k?1k g k?1;ex : Let

G0 = f

d d X X =0 =1

(4:14)

a; r I~(0); a; 2 Rg

be the space of the gradients of the conserved quantities. The main result of this section is the following theorem. Theorem 4.6 For all g 2 G~, one has k g k?1< 1. Furthermore, let G denote the closure of G under k
k?1 seminorm. Then G~ can be decomposed as

G = G~ = Ls G  G0 := H;

(4:15)

where the closure are taken w.r.t. k
k?1. Furthermore, let

 g; h = 41 k g + h k?1 + k g ? h k?1 : 



(4:16)

Then 
;
 is an inner product and G equipped with this inner product is a Hilbert space. Remark 1 Though the seminorm k
k?1 has not been proved to be nite outside the space G ex, one can easily check that it is nite on the space

H0  L s G  G 0 36

Indeed, it is not hard to check that Ls G and G0 are orthogonal w.r.t. 
;
 and 
;
 de ne an inner product such that H0 is a pre-Hilbert space. Hence the space Ls G  G0 = H is well de ned. This and some more properties of 
;
 will be proved in Theorem 5.1. Note that in H one has LsG \ G0 = f0g: Remark 2 We emphasize that the boundedness of k
k?1 on G~ is a consequence of the theorem. Proof. We outline the proof rst. The detailed proofs of Lemmas will be presented at the end of this section. Step 1: Let X G? = f a I~(0 ); a 2 Rg: (4:17)