Bulk diffusivity of lattice gases close to criticality

We consider lattice gases where particles jump at random times constrained by hard core exclusion (simple exclusion process with speed change). The conventional theory of critical slowing down predicts that close to a critical point the bulk di(cid:11)usivity vanishes as the inverse compressibility. We con(cid:12)rm this claim by proving a strictly positive lower bound for the conductivity.


Introduction
Close to a critical point dynamical processes become sluggish.Such a critical slowing down can be infered already from the most primitive approximation.E.g. if the order parameter m is supposed to satisfy m = ?V 0 (m) with V (m) = am 2 + bm 4 ; b > 0; then as a !0 + the relaxation to the equilibrium point m = 0 becomes slow.A more demanding problem is to extract such a slow behavior out of a microscopic model with many degrees of freedom.Now, the conventional theory 1,2] asserts that dynamical processes slow down because close to criticality certain thermodynamic susceptibilities diverge.No extra complications are supposed to arise from the dynamics itself.In fact, the conventional theory turns out to permanent address: Theoretische Physik, Theresienstr.37, 80333 M unchen, Germany y Research partially supported by U.S. National Science Foundation grant 9101196, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship.
be wrong for an Ising model with spin ip dynamics, at least below the upper critical dimension.There is then an independent dynamical scaling exponent governing the slow decay at T c 3]: On the other hand for a conserved eld a renormalization group calculation supports the conventional theory 3,4].There has been considerable numerical e ort to verify these predictions, cf.5], e.g.. Since for lattice gases close to criticality there is slowing down on top of a conservation law, Monte Carlo simulations are plagued by substantial numerical uncertainties.To our knowledge, the most extensive Monte Carlo computation 6] obtains results at least consistent with the conventional theory.
Our paper is organized as follows: In Section 2 we brie y recall the de nition of lattice gases, explain in more detail the claims of the conventional theory, and de ne the bulk di usivity.In Section 3 we establish a lower bound on the bulk di usivity, the upper bound in (1.1) being trivial.

Conventional Theory and Bulk Di usivity
We consider a lattice gas on a simple hypercubic lattice Z I d : Although the bound to be established holds in fair generality (see remarks below), for the sake of concreteness and notational simplicity we restrict ourselves to the nearest neighbor case.As standard, the occupation variables are denoted by (x); (x) = 0; 1 with x 2 Z I d ; and a whole particle con guration is denoted by : Z I d !f0; 1g: The dynamics is governed by the exchange rates c(x; y; ): We require c(x; y; ) = 0 unless jx ?yj = 1 and to be nondegenerate in the sense 0 < c ? c(x; y; ) c + < 1 (2:1) for jx ?yj = 1.The rates c(x; y) are assumed to depend on only through f (z) : jx ?zj < R 0 ; jy ?zj < R 0 g; 1 R 0 < 1, to be translation invariant, namely c(x; y; ) = c(x + a; y + a; a ) with a the shift by a, and to be invariant under lattice rotations.The generator for the dynamics is then given by Lf( ) = X <x;y> > 0 is an attractive and < 0 a repulsive lattice gas.For a bond (x; y) let xy H( ) = H( xy ) ?H( ) be the energy di erence.We impose then the condition c(x; y; ) = c(x; y; xy )e ?xy H( ) : (2:4) In particular, Eq. (2.4) implies that the set of canonical Gibbs measures with the nearest neighbor potential J fx;yg = ?(x) (y) is invariant and reversible under L 8].We denote a translation invariant (not necessarily extreme) canonical Gibbs measure by and its expectations by < > , where the subscript labels the average density, < (x) > = .The stochastic process t ; t 2 R I; with initial measure is space-time stationary.Expectations with respect to this process are denoted by E .The Gibbs measures will play a prominent role and it might be useful to rst describe their phase diagram (for d 2) as presented in Figure 1.
We plot the average density against (not to scale).In region I there is a unique Gibbs measure.In region II, we have two extremal Gibbs measures which transform into each other by a unit shift.Their typical con gurations have a checkerboard pattern.The shaded regions III and IV correspond to mixtures of the Gibbs measures living at the boundary points of the density interval with xed.( c ; 1  2 ) is the critical point in the attractive case.For ? t < ?c ; there is a line of critical points which terminates in two critical endpoints 9,10].Only some parts of the phase diagram have been established mathematically 11,12,13].We de ne the static compressibility by (2:5) for in region I and II.For = c , ( ) diverges as ! 1 2 : Similarly, for ?t < < ?c and d = 2; 3, ( ) diverges as approaches the line of critical points.
Our real interest is the dynamics.The most basic property is the spreading of a density disturbance in equilibrium.As a quantitative measure one adopts the bulk di usivity, D. Since our lattice gas is isotropic by assumption, D is a scalar and can be de ned through the normalized second moment as x 2 E ( t (x) 0 (0)) ? 2 ] ?1 : (2:6) If satis es an exponential mixing condition, then the limit (2.6) exists and is given through the variational formula 14] < c(0; e j ) (e 1 e j )( (0) ?(e j )) + D 0e j X x2Z I d x G] 2 > : (2:7) The in mum is over all local functions G.Here e j is the unit vector along the positive jaxis and the exchange operator is de ned by D xy f( ) = f( xy ) ? f( ): Note that the sum over Z I d contains only a nite number of non-zero terms.Physically has the meaning of a conductivity, i.e., if the exchange rates are slightly biased along the 1-axis, then a steady state current is induced which turns out to be proportional to 14].From now on, we regard Eq. (2.7) as the de nition of .
Taking G = 0 in (2.(2:11) The scaling form (2.11) is somewhat remote from rigorous analysis.Our goal here is more modest.Theorem 1.Let be a box with periodic boundary conditions.Let ; be the Gibbs measure Z ?1 exp P <x;y>;x;y2 (x) (y)], constrained to the set f 2 f0; 1g : j j] = P x2 (x)g; 0 < < 1, where ] denotes the integer part and j j the number of sites in .
(2) The bound d ? is fairly explicit, cf.Eqs.(3.1), (3.10), but not too precise.It basically re ects that at low temperature the time scale for di usion increases as exp 2dj j]: (3) ( ) = as !0 and ( ) = 1 ?for ! 1. c( ) has the same limit behavior.Therefore D( ) tends to a non-zero value as !0; !1: (4) The proof of Theorem 1 goes through whenever the potential for the Gibbs measure and the exchange rates are of nite range.
Addendum Although somewhat o the main track, the reader might be curious to know how D( ) vanishes as !c according to current knowledge.To facilitate comparison with the literature we use standard symbols for the critical exponents.In d = 2 most of them come from the Onsager solution.Above the upper critical dimension one has mean eld values.In between one has a variety of approximate methods.Three cases have to be distinguished.(1) = c , c = 1 2 : The order parameter eld and the conserved eld agree.This is known as Model B in critical dynamics.The scaling form for the structure function is written as jkj ?2+ exp ?jkj z jtj] with z = 4 ?= 2 + (2 ?).(ii) ?t < < ?c ; c = c ( ): The order parameter eld is the staggered density which di ers from the conserved eld.This is known as Model C in critical dynamics.The scaling form for the structure function is written as jkj ?= exp ?jkj z jtj] with z = 2 + = .(iii) = ?t ; = t : This is a tricritical point.The notation is as in (ii) with index t: ad (i): One has D( ) = j ? 1  2 j ?1 with ? 1 = 14 for d = 2; ? 1 = 3:8 for d = 3 , and ? 1 = 2 for d 4: Also = 1 4 for d = 2, = 0:03 for d = 3 , and = 0 for d 4: ad (ii): c ( ) is parabolic close to = 1 2 : One has D( ) = j ?c j =(1? ) with D vanishing as an inverse logarithm in d = 2; =(1 ? ) = 0:12 for d = 3; and D bounded for d 4: At = ?c , D remains bounded.The structure function diverges logarithmically, for d = 2 diverges as jkj ?0:17 for d = 3, and remains bounded for d 4; cf.9].ad (iii): One has D( ) = j ?t j t =(1? t ) with t =(1 ?t ) = 8 for d = 2 and t =(1 ?t ) = 1 for d 3; cf.9].

A Lower Bound
We prove Theorem 1.Let = 0; 2`] d Z I d with `> 0 and integer.We make out of a torus by considering (: : : ; 2`; : : :) and (: : : ; 0; : : :) as nearest neighbors (= periodic boundary conditions).Let `> R 0 : Then c(x; y) is de ned for every bond on the torus .Let < > denote expectation with respect to the Gibbs measure Z ?1 e ?H on the torus constrained to the set f 2 f0; 1g : P x2 (x) = j j]g: We write x = (x 1 ; x ? ) where x 1 2 Z I is along the 1-axis and x ? 2 Z I d?1 is orthogonal to the 1-axis.

7 )
yields the upper bound c( )d + : (2:8) with d + = c + =2 and c( ) =< ( (0) ?(e 1 )) 2 > .The conventional theory postulates that remains strictly positive close to a critical point.By (2.6) this would imply that D( ) must vanish as ( ) ?1 for !c : At rst sight this claim looks rather innocent.However along with it goes a prediction on the scaling close to c as we explain now.We de ne the structure in region I and II.The conventional theory argues that for small k and large t b S(k; t) = b S(k; 0) exp ?k 2 jtj= b S(k; 0)]: (2:10) Away from criticality = lim k!0 b S(k; 0) and we recover the standard di usive spreading of a density disturbance with di usion coe cient D = = : However, at criticality b S(k; 0) = c 0 jkj ?2+ for k !0: If does not vanish, then at the critical point b S(k; t) = c 0 jkj ?2+ exp ?( =c 0 )jkj 4? jtj]:

(3: 3 )
If either y 1 = 0 or y n = `; then the corresponding summands in (3.3) have to be ommitted.Note that terms on the right hand side of (3.3) are normalized by the jump length just as (