Jeremy Quastel and Horng-Tzer Yau 1

DIFFUSION IN DISORDERED MEDIA

J. QUASTEL AND H. T. YAU^{1}

September 17, 1996

1Department of Mathematics, University of California, Davis and Courant Institute, New York University. The second author was partially supported by NSF grant 9101196, Sloan Foundation Fellowship, and David and Lucile Packard Foundation Fellowship

Abstract

We study the transport properties of a large system of interacting particles moving on an integer lattice in the presence of a random field. In this article we describe the problem and our results, and give an outline of our method. We give variational formulas for the transport coefficients. The details of the proof of the hydrodynamic limit can be found in [Q-Y].

0.1 Introduction.

At low temperatures and small concentration of impurities, transport in doped semiconductors takes place by hopping of electrons from occupied to unoccupied states localized near randomly distributed traps of varying strength. If we neglect the electron-electron interactions except to say that not more than one electron can occupy a site, then we have a sytem of particles with random jump rates with jumps to occupied sites excluded. Besides disordered semiconductors such models are used to descibe diffusion in hot solids through the vacancy mechanism, and diffusion of hydrogen in various metal hydrides.

Of great physical interest are the transport coefficients of such a sytem. The main results are classical homogenization, which ignores the interaction between particles, and various approximations of the interactions.

In this article we present variational formulas for the bulk diffusion and self diffusion coefficients. These formulas are in some sense equivalent to the principal of minimal entropy production. Formally they are equivalent to the corresponding Green-Kubo formulas. Unlike the Green-Kubo formulas however they do not require explicit knowledge of the correlation functions. And these are formulas which we can establish rigorously, directly from the microscopic dynamics.

0.2 The Model.

Let α : Z^{d} → R be a stationary random field which is bounded and satisfies some mild mixing conditions (see []). For example, α could be periodic, or the α(x) could be independent identically distributed random variables. We have a system of particles moving on Z^{d} in this field. They are moving as excluded continuous time random walks. A particle at x attempts to jump to a nearest neighbor site y at rate 1 + exp(α(y) − α(x)). The interaction is through the exclusion rule: Attempted jumps to occupied sites are suppressed.

The most convenient way to describe the dynamics is through the generator L of the associated Markov semigroup. If η ∈ {0, 1}^{Z^{d}} represents the configuration of particles then

Lf(η) = ∑_{x∼y} a_{xy}(η)∇_{xy}f(η).

Here ∇_{xy}f(η) = f(η^{xy}) − f(η) where

η^{xy}(z) = \{ \begin{matrix}η(z), & z ≠ x, y; \\ η(y) & z = x; \\ η(x) & z = y, \end{matrix}

represents the exchange of occupation numbers at x and y and

a_{xy}(η) = 1 + exp ((α(y) − α(x))(η(x) − η(y)))

is the rate at which this exchange takes place. The sum is over all nearest neighbor sites x and y.

The crucial property of the rates we have chosen is that the dynamics is stochastically reversible with respect to the family of random Bernoulli measures, ν_{λ}, such that

ν_{λ}(η(x) = 1) = \frac{e^{α(x)+λ}}{1 + e^{α(x)+λ}} .

The parameter λ is called the chemical potential. The effect of varying λ is to change the density m of particles. To determine λ in terms of m one solves the equation,

E [\frac{e^{α(0)+λ}}{1 + e^{α(0)+λ}} ] = m

where E is the expectation with respect to the random field.

We could choose other dynamics which are reversible with respect to the ν_{λ}. One possibility is

a_{xy}(η) = e^{−α(x)}η(x)(1 − η(y)) + e^{−α(y)}η(y)(1 − η(x))

in which a particle at x attempts to jump to each nearest neighbor site at rate e^{−α(x)}. We will refer to this model as the symmetric model and the earlier model as the asymmetric model. The method we will describe works just as well for the symmetric model. The reason we consider the asymmetric model is that the Dirichlet form given by

D(f) = ⟨f, −Lf⟩ = ∑_{x∼y} ⟨a_{xy} (∇_{xy}f)^{2}⟩

takes in that case the particularly simple form

D(f) = ∑_{x∼y} ⟨(∇_{xy}f)^{2}⟩ .

For technical reasons it is convenient to consider the system on T_{ϵ}^{d} = ϵZ^{d}/Z^{d}, a box with L = ϵ^{−1} lattice sites to a side, with periodic boundary conditions. ϵ is the small parameter in our problem. We will speed up the process by ϵ^{−2} to compensate for the lattice spacing of ϵ.

We are interested primarily in studying mass transport in the system. To this end we introduce the empirical density field which is the following measure on T^{d}, the d-dimensional torus,

μ_{ϵ} = ϵ^{d} ∑_{x∈T_{ϵ}^{d}} η(x)δ_{x}

where δ_{θ} gives mass one to θ ∈ T^{d}.

For each ϵ and each realization of the field we have a process P_{ϵ}^{noneq} corresponding to the randomly evolving empirical density field. The initial distributions are chosen so that, as ϵ → 0,

μ_{ϵ}(0, dθ) ⇒ m^{0}(θ)dθ

in P_{ϵ}^{noneq}-probability, for some given smooth m^{0}(θ) on T^{d}. What we would like to show is that this property is preserved at time t > 0, but with a new m(t, θ) obtained from the initial m^{0}(θ) by solving a nonlinear diffusion equation. Our main result is

Theorem 1 (Hydrodynamic Limit) For almost every α,

P_{ϵ}^{noneq} ⇒ P

where P is the point mass on the deterministic evolving profile measure m(t, θ)dθ where m(t, θ) is the unique solution of the diffusion equation,

\frac{∂m}{∂t} = ∇D(m)∇m, m(0, θ) = m^{0}(θ),

and D(m) = σ(m)λ′(m) where σ is a symmetric d×d matrix with associated quadratic form given by the variational formula

(l, σ(m)l) = inf_{g} E [⟨ ∑_{|v|=1} a_{0v} (l · v∇_{0v}η − ∇_{0v} ∑_{x} τ^{x}g)^{2}⟩_{m}]

We will now desribe all the terms in this rather formidable expression. Here l ∈ R^{d} is arbitrary. The infimum is taken over all local g(η, α). The shift is given by τ^{x}g(η, α) = g(τ^{x}η, τ^{x}α). The sum is over all shifts of g. On the face of it this makes no sense, but since g is local, ∇_{0v}τ^{x}g = 0 except for finitely many x. The expectation ⟨⟩_{m} is over the infinite product measure ν_{λ(m)}, and the expectation E is over the random field, α.

The diffusion coefficient D(m) is called the bulk diffusion coefficient of the system. Later we will give another characterization of it. Since they also deal with bulk transport the same coefficient appears when we study the fluctuation theory or the large deviations. In particular let 0 ≤ m ≤ 1 and consider the equilibrium process P_{ϵ}^{eq} obtained by prescribing ν_{λ}(m) as the initial distribution of the process and let Q_{ϵ}^{eq} be the distribution fo the corresponding evolving fluctuation fields,

ζ_{ϵ} = ϵ^{1/2} ∑_{x∈T_{ϵ}^{d}} (η(x) − m) δ_{x}.

Theorem 1 (Equilibrium Fluctuations) Q_{e}^{eq} converges weakly in D([0, T] → H_{−1}) to the generalized Ornstein-Uhlenbeck process characterized by the stochastic partial differential equation

dζ(t) = D(m)Δζdt + d∇W (t)

where ∇W is the Gaussian random field with covariance

E [⟨∇W (s), J⟩ ⟨∇W (t), G⟩] = δ(s − t) ⟨J, G⟩ .

Now we describe the large deviations from the hydrodynamic limit. Define the rate function I : D [[0, T]; M(T^{d})] → R I _{+} as follows: I(μ(·, ·)) = ∞ unless μ(t, dθ) = m(t, θ)dθ for each t ∈ [0, T] with

∫_{0}^{T} ∫_{T^{d}} ∇λ(m) · σ(m)∇λ(m)dθdt < ∞

in which case

I(μ(·, ·)) = ∫_{T^{d}} h(m(0, θ))dθ + ∫_{0}^{T} ∫_{T^{d}} F · σF dθdt

where h is the convex conjugate of

Φ(λ) = E [log ⟨e^{λη(0)}⟩_{m}]

and

∇ · F = \frac{∂m}{∂t} − ∇ · D(m)∇m.

Theorem 1 (Large Deviations) For almost every α, P_{ϵ}^{eq} satisfies the large deviations principle with rate function I. In particular, I is lower semicontinous, preimages of bounded sets are precompact, and for all closed sets C and open sets U in D([0, T]; M_{1}(T^{d})),

lim sup_{ϵ→0} ϵ^{d} log P_{ϵ}^{eq}(C) ≤ − inf_{μ(·,·)∈C} I(μ(·, ·))

and

lim inf_{ϵ→0} ϵ^{d} log P_{ϵ}^{eq}(U) ≥ − inf_{μ(·,·)∈U} I(μ(·, ·)).

The one conspicuous absence in this list of results is the nonequilibrium fluctuations, which have only been obtained for systems of gradient type [].

There is another transport coefficient floating around which we have not considered. This is the self-diffusion coefficient. Suppose that we have a system of particles performing our dynamics on Z^{d}. As initial distribution we choose ν_{λ(m)} so that the process is in equilibrium and the average density is m. Initially we place a tagged particle at the origin and follow its trajectory x(t) as the system evolves. It can be shown that even under this conditioning the rest of the system as seen from the tagged particle remains in equilibrium.

Theorem 1 (Self-Diffusion) Under diffusive scaling the tagged particle process converges in the Skorohod space to a Brownian motion with covariance D_{self}(m). The associated quadratic form is given by the variational formula,

(l, D_{self}(m)l) = inf_{f(η,α)} E[\frac{e^{α(0)}}{Z} ⟨ ∑_{|v|=1} a_{0v}(1 − η(v)) (l · v + τ^{−v}f − f)^{2}

+ ∑_{x∼y,x,y≠0} a_{xy} (∇_{xy}f)^{2}⟩_{m}]

where Z = E[e^{α(0)}] is a normalization. The expectation is over the infinite product measure ν_{λ(m)} and the field α.

The relation between these transport coefficients is that D_{self}(m) < D(m) unless m = 0 in which case D_{self}(0) = D(0) = a is the limiting covariance of a free particle in the field. This is a classical homogenization problem, and it is well known how to obtain the variational formula,

a = E [e^{α(0)}]^{−1} inf_{U(α)} E[ ∑_{|v|=1} (e^{α(0)} + e^{α(v)}) (l · v + τ^{−v}U − U)^{2}].

In one dimension in the case of periodic field, or independent random variables this can be computed explicitly. The solution for our asymmetric model is

a = 2 (E[e^{α(0)}]E [(e^{α(0)} + e^{α(1)})^{−1}])^{−1} .

One can also show rigorously that lim_{m→0} D(m) = a.

The only case in which D(m) is known explicitly is the case of a period two field in one dimension. In that case it turns out that D(m) ≡ a. That this is in general not the case can be seen as follows. For the asymmetric model there is a particle-hole duality. The dynamics of holes is the same as that of particles, except that they see the field −α instead of α. This has a number of consequences. If the field is statistically invariant under the transformation α ↦ −α then D(m) = D(1 − m). Another consequence is that as m → 1, D(m) → a(−α) where a(α) is given by (). One can check that in general a(α) ≠ a(−α). Therefore D(m) is non-constant. In other words the diffusion equation we get is nonlinear. A similar argument works for the symmetric model even though the model does not have such a nice particle-hole duality.

The particular case of the symmetric model in which α(x) are independent Bernoulli random variables with P (α(x) = 0) = 1 − p and P (α(x) = M) = p for some large M and small p is known as the random trap model. Monte Carlo simulations of the one dimensional random trap model by Kehr et. al. show a diffusion coefficient of the following shape,

It might seem paradoxical that the presence of other particles enhances the diffusivity. We will now give a heuristic explanation of this phenomenon. Consider the second class particle whose dynamics are given as follows. It jumps with the standard jump rates unless another particle attemps to jump to where it is. Such jumps are not suppressed, instead the two particles exchange positions. The reason for this definition is that if one couples two systems, one with exactly one more particle than the other, then one sees that the discrepancy follows these dynamics. One expects that under diffusion scaling the trajectory of the second class particle converges to a Brownian motion with covariance D(m). This is just linear response theory. Now one can see why the curve is the shape it is. When there are other particles around the effect of the traps on the second class particle is very different. It sits in a trap only until a first class particle comes to take it out. So even a small density of particles is enough to ameliorate the effect of the traps. On the other hand, first class particles stuck in traps tend to act as barriers. At high density this considerably slows down the second class particle. In dimensions other than one, however, this effect should be much less pronounced, and we conjecture that the curve has a maximum at m = 1.

0.3 Gradient and Nongradient Systems.

We will now outline the proof of the hydrodynamic limit. Suppose that we have a system of the type described in (). The evolution of the empirical density field is described by the following set of stochastic integral equations.

∫_{T^{d}} φ(T, θ)μ_{ϵ}(T, dθ)− ∫_{T^{d}} φ(0, θ)μ_{ϵ}(0, dθ)

= ∫_{0}^{T} ∫_{T^{d}} \frac{∂φ}{∂t}(t, θ)μ_{ϵ}(t, dθ)

− ∫_{0}^{T} ϵ^{d−2} ∑_{x∼y} (φ(t, ϵy) − φ(t, ϵx)) w_{xy}(η)dt

+ ∫_{0}^{T} ϵ^{d} ∑_{x∼y} (φ(t, ϵy) − φ(t, ϵx))∇_{xy}ηdM_{xy}.

where

w_{xy}(η) = a_{xy}(η)∇_{xy}η

is the current and M_{xy} are independent Poisson sawtooth martingales running at rates ϵ^{−2}a_{xy}. Certainly

ϵ^{−1} (φ(ϵy) − φ(ϵx)) = (y − x) · ∇φ(ϵx) + o(1).

A martingale calulation gives

E[(∫_{0}^{T} ϵ^{d+1} ∑_{x∼y} ∇φ∇ηdM_{xy})^{2}] = E[∫_{0}^{T} ϵ^{2d} ∑_{x∼y} a_{xy} (∇φ∇η)^{2} dt] = O(ϵ^{d}).

So the main work is to show that in some sense,

∫_{0}^{T} ϵ ∑_{x∼y}(x − y) · ∇φ(t, ϵx)ϵ^{−1}w_{xy}(η)dt → ∫_{0}^{T} ∫_{S} ∇φ(θ)D(m(θ))∇mdθdt.

Then we are left with the weak formulation of the nonlinear diffusion equation, and uniqueness follows by standard results in partial differential equations.

Historically, the first attempt to establish () was to assume that the system satisfies the so called gradient condition, namely that the current can be expressed as the gradient of a local function. In particular, there exist a function h({η(x)}_{x∈Λ_{K}}) depending on the variables in a box Λ_{K} of side length K, so that

w_{x,y} = τ^{y}h − τ^{x}h.

Then one can perform a second summation by parts

∫_{0}^{T} ϵ ∑_{x∼y}(x − y) · ∇φ(t, ϵx)ϵ^{−1}w_{xy}(η)dt = ∫_{0}^{T} ϵ ∑_{x} Δφ(t, ϵx)h(η)dt + o(1).

What one expects is that for any local function h and smooth test function φ,

∫_{0}^{T} ϵ ∑_{x} φ(t, ϵx)τ^{x}hdt → ∫_{0}^{T} ∫_{T^{d}} φ(t, θ)ĥ(m(t, θ))dθdt

where

ĥ(m) = ⟨h⟩_{ν_{λ(m)}} .

The resulting nonlinear diffusion equation is

\frac{∂m}{∂t} = Δĥ.

We will now give some of the ideas behind this. There is a very basic convexity inequality called the entropy inequality which says that for any function f,

E^{P} [f] ≤ log E^{Q}[exp f] + H(P; Q)

where H(P; Q) is defined as follows: H(P; Q) = ∞ unless P << Q in which case

H(P; Q) = E^{P} [log (dP/dQ)] .

If one takes P = P_{ϵ}^{noneq} and Q = P_{ϵ}^{eq} then the relative entropy all comes from the initial distribution.

H(P_{ϵ}^{noneq}; P_{ϵ}^{eq}) ≤ ϵ^{−d} log 2.

If we let

V_{K} = |Av_{y∈Λ_{K}}(τ^{y}h − ĥ(m̄_{K}))|

and

f = ∫_{0}^{T} ϵ^{d} ∑_{x} τ^{x}V_{K}dt

then one sees from the entropy inequality that it is enough to show that for each γ > 0,

lim sup_{K→∞} lim sup_{ϵ→0} ϵ^{d} log E^{P_{ϵ}^{eq}} [exp γ ∫_{0}^{T} ∑ τ^{x}V_{K}dt] ≤ 0.

One recognizes the expectation as the Feynman-Kac formula for the solution of

\frac{∂u}{∂t} = (L_{ϵ} + γ ∑ τ^{x}V_{K}) u.

Therefore we can estimate () in terms of the principal eigenvalue of L_{ϵ} + γ ∑ τ^{x}V_{K}, which is given by the Raleigh-Ritz formula as

sup_{∥f∥_{2}=1} {∑ ⟨τ^{x}V_{K}, f^{2}⟩ − ϵ^{−2} ∑_{x∼y} ⟨(∇_{xy}f)^{2}⟩} .

By convexity this is bounded above by

∑_{x} sup_{∥f∥_{2}=1} \{⟨V_{K}, f^{2}⟩ − ϵ^{−2} ∑_{{x∼y \atop x,y∈Λ_{K}}} ⟨(∇_{xy}f)^{2}⟩\} ,

where f is now a function only of variables in Λ_{K}. The ϵ^{d} normalizes the sum and we are left with ϵ^{−2}K^{−(d+2)} times the principal eigenvalue λ_{K,ϵ} of

K^{2}L_{K} + ϵ^{2}K^{d+2}V_{K}

where L_{K} is the generator restricted to Λ_{K}. If we write down the Rayleigh-Schrödinger series for λ_{K,ϵ} we find

ϵ^{−2}K^{−(d+2)}λ_{K,ϵ} = ϵ^{−2}K^{−(d+2)}{0 + ϵ^{2}K^{d+2} ⟨V_{K}⟩ + ⋯} → ⟨V_{K}⟩

as ϵ → 0. Since ⟨V_{K}⟩ → 0 as K → ∞ we are done.

Of course this is not really the proof. A detailed proof can be found in [Spohn].

The problem with the gradient condition is that it is neither natural nor ubiquitous. Since the result was obtained by simple averaging there is no net effect of fluctuations on the diffusion coefficent. So instead of solving the real problem we have simply avoided it.

For nongradient systems we are stuck at (). One cannot simply average w_{xy} because there is a huge factor ϵ^{−1} sitting in front of it. This is not a question of how to make the averaging rigorous. If you do the averaging you get the wrong answer.

Varadhan’s idea for the nongradient systems is as follows. Suppose that g is a local function. Then, since

∫_{0}^{T} ϵ^{d−1} ∑_{x} L_{ϵ}τ^{x}g = ϵ^{d−1} ∑_{x} τ^{x}g|_{0}^{T} − ∫_{0}^{T} ϵ^{d+1} ∑_{x∼y} a_{xy}∇_{xy}g dM_{xy}

we can see that the term Lg which we call a fluctuation term does not affect the hydrodynamics in the sense that

∫_{0}^{T} ϵ^{d} ∑_{x} φ(ϵx)ϵ^{−1} (w_{xy} − Lτ^{x}g) dt

and

∫_{0}^{T} ϵ^{d} ∑_{x} φ(ϵx)ϵ^{−1}w_{xy}dt

will have the same limit. Now we can hope that although w_{xy} is not exactly a gradient, it is well approximated by a fluctuation term plus a gradient term. In particular let

V = w_{01} − Lg − â(m̄_{K}) (τh − h) .

By the same argument as before we are led to the problem of showing that

lim sup_{K→∞} lim sup_{ϵ→0} ϵ^{−2}K^{−(d+2)}λ_{K,ϵ} ≤ 0

where λ_{K,ϵ} is the principal eigenvalue of

K^{2}L_{K} + ϵK^{d+2}V_{K}.

Writing down the Rayleigh-Schrödinger series again we get

lim sup_{K→∞} lim sup_{ϵ→0} ϵ^{−2}K^{−(d+2)}{0+ϵK^{d+2} ⟨V_{K}⟩+ϵ^{2}K^{2(d+2)} ⟨V_{K}, (−K^{2}L_{K})^{−1}V_{K}⟩+⋯}.

Since ⟨V_{K}⟩ = 0, the problem is reduced to showing that there is a sequence g_{K} so that

lim sup_{K→∞} K^{d+2} ⟨V_{K}, (−K^{2}L_{K})^{−1}V_{K}⟩ = 0.

This is still a rather formidable problem, however there is a fairly general argument which has been pushed through for several systems. We will not go into it here except to mention that the two main difficult steps are:

1. Establish that the system on Λ_{K} has a spectral gap no smaller than const.K^{−2}.

2. Integration by parts. One needs an estimate of the form,

⟨f(τh − h)⟩ ≤ const.K^{−d/2}\sqrt{∑_{x∼y,y∈Λ_{K}} ⟨(∇_{xy}f)^{2}⟩ .}

Note that the nonlinear diffusion one gets is

\frac{∂m}{∂t} = \frac{1}{2}∇â(m)∇ĥ(m).

0.4 The Method of Long Jumps.

The random field model we are considering turns out to be of nongradient type. However there is an added difficulty. Since the equilibria ν_{λ} are not translation invariant, no gradient τh−h can have mean zero, and therefore () cannot possibly be correct. It does not even make sense. It does not help to subtract something to make it mean zero (unless the dimension is sufficiently large). The problem is that the fluctuations in the gradient of the density field arising from the random field are large. One can only make sense of these gradients over sufficiently large distances where stochastic fluctuations are reduced by the central limit theorem.

The correct object to replace the gradient term turns out to be the following. Divide the box Λ_{K} into two halves Λ_{K}^{1} and Λ_{K}^{2}. For any x and y, let

w_{xy} = (1 + exp((α(y) − α(x))(η(x) − η(y))) η(y) − η(x) )

and

W_{K} = \frac{1}{|Λ_{K}^{1}|} ∑_{x∈Λ_{K}^{1}} \frac{1}{|Λ_{K}^{2}|} ∑_{x∈Λ_{K}^{2}} w_{xy}

It turns out that we can find g_{K} so that if

V_{l,K} = Av_{l}τ^{x} (w_{01} − Lg_{K} − \frac{σ(m)}{m(1 − m)} \frac{W_{K}}{K} )

then

lim sup_{l→∞} lim sup_{K→∞} l^{d}K^{d+2} ⟨V_{l,K}, (−K^{2}L_{K})^{−1}V_{l,K}⟩ ≤ 0.

the proof is essentially the same as in []. In fact it is easier because the problem of integration by parts is considerably finessed: Note first of all that

⟨w_{xy}f⟩ = ⟨∇_{xy}η∇_{xy}f⟩

As the first step in the proof of the spectral gap we show that

⟨(∇_{xy}f)^{2}⟩ ≤ C(y − x) ∑_{i=1}^{|y−x|} ⟨(∇_{e_{i}}f)^{2}⟩

where e_{1}, ..., e_{|y−x|} is a path from x to y. By Schwarz’s inequality therefore

⟨w_{xy}f⟩^{2} ≤ 2m(1 − m)C(y − x) ∑_{i=1}^{|y−x|} ⟨(∇_{e_{i}}f)^{2}⟩

This has another consequence which helps explain where W_{K} comes from. As part of the proof of () we show that

lim sup_{K→∞} K^{d+2}E [⟨\frac{W_{K}}{K} , (−K^{2}L_{K})^{−1} \frac{W_{K}}{K} ⟩] = \frac{m^{2}(1 − m)^{2}}{σ(m)}

By () we get immediately that C = m^{2}(1−m)^{2}/σ(m) is the optimal constant in the inequality

E [⟨(η(K) − η(0)) (f(η^{0K}) − f(η))⟩_{m}^{2}] ≤ CK ∑_{i=0}^{K} E [⟨(∇_{ii+1}f)^{2}⟩_{m}] .

Here the notation is meant to suggest we are on some canonical path from 0 to K.

At this point the reader may be wondering how much we have really improved the situation. We have shown how to replace

∫_{0}^{T} ϵ^{d} ∑_{x} φ(ϵx)ϵ^{−1}w_{xx+v}dt

by

∫_{0}^{T} ϵ^{d} ∑_{x} φ(ϵx)ϵ^{−1}τ^{x} \frac{σ(m)}{m(1 − m)} \frac{W_{K}}{K} dt

for large K, but it is not clear what the limit of the latter object is. The main observation is that simple averaging now gives the correct answer. If we calculate the average of W_{K} with respect to ν_{λ(m_{1})} on Λ_{K}^{1} and ν_{λ(m_{2})} on Λ_{2_{K}} we get ⟨W_{K}⟩_{m_{1},m_{2}} = F (m_{1}, m_{2}) where

F (m_{1}, m_{2}) = (e^{λ(m_{2})−λ(m_{1})}− 1) m̂_{1}(1 − m̂_{2}) + (1 − e^{λ(m_{1})−λ(m_{2})}) m̂_{2}(1 − m̂_{1}).

Here

m̂_{i} = \frac{1}{|Λ_{K}^{i}|} ∑_{x∈Λ_{K}^{i}} \frac{e^{α(x)+λ(m_{i})}}{1 + e^{α(x)+λ(m_{i})}}

By the mixing conditions, m̂_{i} = m_{i} + O(K^{−d/2}). Expanding, we get

F (m_{1}, m_{2}) = λ′(m)m(1 − m)(m_{2} − m_{1}) + O(m_{2} − m_{1})^{2} + o(K^{−d/2}).

So, if we can replace W_{K} by F we get

∫_{0}^{T} ϵ^{d} ∑_{x} φ(ϵx)τ^{x}σ(m)λ′(m)ϵ^{−1}K^{−1}((m_{2} − m_{1})+O(m_{2} − m_{1})^{2})dt

+o(ϵ^{−1}K^{−(d+2)/2})

Since ϵ^{−1}K^{−1} (m_{2} − m_{1}) ∼ v · ∇m, this converges to

∫_{0}^{T} ∫_{T^{d}} φ(t, θ)σ(m(t, θ))λ′(m(t, θ)) v ·∇m(t, θ)dθdt

as long as K = O(ϵ^{−2/(d+2)}). The result is a weak formulation of the diffusion equation (). To make it all work we therefore need the following result.

Lemma 1 Let

H = −K^{2}L_{K} + ϵK^{d+2}V.

If K^{2}L_{K} has spectral gap of order one then the Rayleigh-Schrödinger series for the ground state energy converges for K < c_{0}ϵ^{−2/(d+2)} provided V satisfies

⟨u, V u⟩ ϵK^{−d/2}D_{K}(u)^{1/2}

for all ∥u∥_{2} = 1.

We now indicate how we replace W_{K} by F. Since −K^{2}L_{K} has spectral gap of order one we can estimate

⟨(W_{K} − F) , (−K^{2}L_{K})^{−1} (W_{K} − F)⟩ ≤ C ⟨(W_{K} − F)^{2}⟩ .

The expectations here are in the canonical ensemble. In other words we have fixed the number of particles so that the density is m. The next lemma is a direct consequence of the local limit theorem.

Lemma 1 Let f be a function of the form K^{−1} ∑_{x=1}^{K} c_{x}η(x) where c_{x} is some uniformly bounded sequence. Then

E [f^{2} | η(1) + ⋯ + η(K) = Km] = ⟨f⟩_{m}^{2} + K^{−1}R + O(K^{−2})

where

R =

Applying the lemma, first on Λ_{K}^{1} with the variables in Λ_{K}^{2} frozen, then on _{K}^{2} we find that

⟨(W_{K} − F)^{2}⟩ = O(K^{?}).

Bibliography

[1] Richards, P. M., , Phys. Rev. B 16 (1977).

[2] Gartner, P. and Pitis, R., , Phys. Rev. B 45 (1992).

[3] Kehr, K. W., Paetzold, O. and Wichmann, T., Collective diffusion of lattice gases on linear chain with site-energy disorder, to appear in Phys. Lett. A.

[4] Kipnis, C. and Varadhan, S. R. S.,Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion, Comm. Math. Phys. 106.

[5] Quastel, J. Diffusion of color in the simple exclusion process, Comm. Pure Appl. Math. XLV, 623-679 (1992).

[6] Guo, M. Z., Papanicolaou, G. C., Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys. 118 (1988) 31-53.

[7] Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions II, Proc. Tanaguchi Symp., 1990, Kyoto.

[8] Papanicolaou, G. C., Varadhan, S. R. S. Boundary value problems with rapidly oscillating coefficeints, In: Random Fields, ed., by J. Fritz, J.L.Lebowitz, D. Szasz, pp. 835-853. North-Holland, Amsterdam.

[9] Wick, W. D. Hydrodynamic limit of non-gradient interacting particle process, J. Stat. Phys. 54 (1989) 873-892.

[10] Fritz, J. Hydrodynamics in a symmetric random medium, Comm. Math. Phys. 125, (1989) 13-25.

[11] Lu, S. L., Yau, H. T. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 399-433.

[12] Brak, R., Elliott, R. J. Correlated random walks with random hopping rates, Journal of Physics - Condensed Matter, 1989 Dec 25, V1 N51: 10299-10319.

[13] Brak, R., Elliott, R. J. Correlated tracer diffusion in a disordered medium, Materials Science and Engineering B - Solid State Materials for Advanced Technology, 1989 Jul, V3 N1-2: 159-162.

[14] Donsker, M., Varadhan, S.R.S. Large deviations from a hydrodynamic scaling limit, Comm. Pure Appl. Math. 42, 243-270, 1989.

[15] Quastel, J. Large deviations from a hydrodynamic scaling limit for a nongradient system, preprint.

[16] Kirkpatrick, S. Classical transport in disordered media: Scaling and effective-medium theories, Phys. Rev. Lett. 27 (1971) 1722.

[17] Ambegaokar, V., Halperin, B.I., Langer, J.S. Hopping conductivity in disordered systems, Phys. Rev. B 4 (1971) 2612.

[18] Miller, A., Abrahams, E. Impurity conduction at low concentrations, Phys. Rev. 120 (1960) 745.

[19] Spohn, H. Large scale dynamics of interacting particles, Springer-Verlag (1991).