Hydrodynamic limit for interface model with non-convex potential

# Hydrodynamic limit for the Ginzburg-Landau ∇ϕ interface model with non-convex potential

Jean-Dominique Deuschel and Takao Nishikawa and Yvon Vignaud J.-D.Deuschel: Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

T. Nishikawa: Department of Mathematics, College of Science and Technology, Nihon University, Tokyo, Japan

Y. Vignaud: Lycée Jean Jaurès, Argenteuil, France
###### Abstract.

Hydrodynamic limit for the Ginzburg-Landau interface model was established in  under the Dirichlet boundary conditions. This paper studies the similar problem, but with non-convex potentials. Because of the lack of strict convexity, a lot of difficulties arise, especially, on the identification of equilibrium states. We give a proof of the equivalence between the stationarity and the Gibbs property under quite general settings, and as its conclusion, we complete the identification of equilibrium states under the high temparature regime in . We also establish some uniform estimates for variances of extremal Gibbs measures under quite general settings.

## 1. Introduction

We consider the large scale hydrodynamic behavior of the the Ginzburg-Landau interface model. This is an effective interface model, describing the stochastic dynamic of the separation of two distinct phases.

The position of the interface is described by height variables measured from a fixed -dimensional discrete hyperplane . Here, we will take when we consider the system on a discretized torus with the periodic boundary condition, or when we consider the system on the domain with Dirichlet boundary condition. is a microscopic domain corresponding to a given macroscopic domain which is bounded and has a smooth boundary. See Section 2 for the precise definition.

The corresponding Hamiltonian on for given height variable is of the form

 H(ϕ)=12∑x,y∈Γ,|x−y|=1V(ϕ(x)−ϕ(y))+∑x∈Γ,y∈Zd∖Γ,|x−y|=1V(ϕ(x)−ϕ(y)),

with a symmetric function . The Langevin equation associated with is given by

 dϕt(x)=−Ux(ϕt)dt+dwt(x),x∈Γ,

where in the drift term is defined by

 Ux(ϕ):=∂H∂ϕ(x)(ϕ)≡∑y∈Zd;|x−y|=1V′(ϕ(x)−ϕ(y))

and is a family of independent copies of the one dimensional standard Brownian motion.

The aim of this paper investigate and identify the hydrodynamic limit of at diffusive scaling, that is, for time while for space. In the case of a strictly convex potential for which there exist two constants such that

 c−≤V′′(η)≤c+,η∈R. (1.1)

the hydrodynamic limit has been established for periodic lattice in  and for discretized domain with Dirichlet boundary conditions in . In particular, the corresponding macroscopic motion is identified as the solution of the nonlinear partial differential equation

 ∂h∂t=div{(∇σ)(∇h(t,θ)},θ∈D,t>0,

where the surface tension is defined via thermodynamic limit.

In these results, the condition (1.1) plays an essential role in the analysis for the stochastic dynamics , especially, in the identification of equilibrium states and the establishment of the strict convexity of . The our aim in this paper is to prove the hydrodynamic limit without the strict convexity assumption (1.1), see Assumptions 2.1, 2.2 and 2.3 for details.

Our motivation comes from recent results in  and  where both strict convexity of the surface tension and identification of the extremal gradient Gibbs measures hold, for non-convex potential at sufficiently high temperature.

In the case of the dynamics on the torus , the limit follows quite simply from additional estimates. However, for the dynamics on the discretized domain with the Dirichlet boundary condition, the derivation is much harder, since we can not use the relative entropy and entropy production. The main step then is to characterize the set of stationary measures for the gradient field associated with the infinite system of SDEs, which is essentially used in order to establish local equilibrium as in  without using the relative entropy and the entropy production.

In case of strictly convex , the structure of the translation invariant stationary measures is completely identified by , its proof relying the assumption (1.1). To complete our proof of the hydrodynamic limit in the non-convex case, we need to identify the class of translation invariant stationary measures as the class of Gibbs distributions.

This subject has been intensively studied in the literature, cf.  for stochastic Ising models,  for the diffusion process on the infinite dimensional torus ,  for the diffusion process on ,  for the diffusion process on the infinite product with a Riemannian manifold with positive curvature. In this paper we show the similar result, adapting the argument of  to gradient Gibbs distributions. The main challenge here is the lack of ellipticity of the gradient dynamic, see Section 3 and 5 for details.

An alternative derivation of the hydrodynamic limit for the Ginzburg-Landau model based on a two scale argument has been proposed by  and . Unlike our proof, relying on the assumption on the uniqueness of the extremal gradient Gibbs distribution, the two scale argument uses logarithmic Sobolev inequalities. However, this approach seems restricted to the one-dimensional case in , respectively strict convexity assumption for the potential (1.1) in .

Before closing this section, let us give briefly the organization of this paper. In Section 2, we formulate our problem more precisely, and state the main result. In Section 3, we present some properties of translation invariant stationary measures, especially, the relationship between stationarity and the Gibbs property, and some uniform estimates for their variances. Note that results in this section hold under the quite general Assumption 2.1. In Section 4, after establishing a priori bounds for stochastic dynamics and summarize properties of the surface tension, we derive the macroscopic equation from the stochastic dynamics. Here, we rely quite explicitly on the further Assumptions 2.2 and 2.3. In Section 5, we give a proof of Theorem 3.1, presented at Section 3.

## 2. Model and main result

### 2.1. Model

Let be a bounded domain in with a Lipschitz boundary. For convenience, let contain the origin of . Let be the discretized microscopic domain corresponding to in the sense that

 DN={x∈Zd;B(x/N,5/N)⊂D},

where stands for the hypercube in with center and side length , that is,

 B(α,l)=d∏i=1[αi−l/2,αi+l/2).

On we consider the dynamics governed by the following stochastic differential equations (SDEs)

 dϕt(x)=−Ux(ϕt)dt+√2dwt(x),x∈DN, (2.1)

with the boundary condition

 ϕt(x)=ψN(x),x∈Zd∖DN (2.2)

with some and initial data , where for and , or more generally for and . The height variable in (2.2) is defined by

 ψN(x)=Nd+1∫B(x/N,1/N)f(θ)dθ (2.3)

for every , where is a function belonging to . We note that the function describes the macroscopic boundary condition and the height variable describes the microscopic one.

We make the following assumption on the interaction potential :

###### Assumption 2.1.

The function has the following representation:

 V(η)=V0(η)+g(η),η∈R,

where functions are symmetric functions and satisfy

1. There exist constants such that

 c−≤V′′0(η)≤c+,η∈R.
2. There exists a constant such that

 |g′(η)|+|g′′(η)|≤Cg,η∈R.
###### Example 2.1.

If a function is symmetric and satisfies

 c≤V′′(η)≤c′,|x|≥M

for some and , then the function admits the decomposition as in Assumption 2.1. Indeed, we can take as follows:

 V0(x)=⎧⎨⎩12V′′(M)x2−12V′′(M)M2+V(M)+αM,|x|≤M,V(x)+α|x|,|x|>M,

with . Letting , that is,

 g(x)=⎧⎨⎩V(x)−V(M)−12V′′(M)x2+12V′′(M)M2−αM,|x|≤M,−α|x|,|x|>M,

we can easily see that and they fulfill conditions (1) and (2) in Assumption 2.1.

Further assumptions dealing with the strict convexity of the surface tension and the characterization of extremal gradient Gibbs measures are stated below, see Assumptions 2.2 and 2.3 for details.

We regard (2.1) as the model describing the motion of microscopic interfaces and introduce the macroscopic height variable as follows:

 hN(t,θ)=∑x∈ZdN−1ϕN2t(x)1B(x/N,1/N)(θ),θ∈Rd,

where being the solution of (2.1) with (2.2).

### 2.2. Notations

Before stating the detail of our main result, we need to introduce several notations. Note that we will follow the same manner as in  and .

Let be the set of all directed bonds in . We write and for . We denote the bond by again if it doesn’t cause any confusion. For every subset of , we denote the set of all directed bonds included and touching by and , respectively. That is,

 Λ∗ :={b∈(Zd)∗;xb∈Λ and yb∈Λ}, ¯¯¯¯¯¯Λ∗ :={b∈(Zd)∗;xb∈Λ or yb∈Λ}.

For , the gradient is defined by

 ∇ϕ(b):=ϕ(x)−ϕ(y),b=(x,y)∈(Zd)∗.

Now, let be the family of all gradient fields which satisfy the plaquette condition (2.1) in , i.e., . Let be the set of all such that

 |η|2r:=∑b∈(Zd)∗|η(b)|2e−2r|xb|<∞.

We denote equipped with the norm . We introduce the dynamics governed by the SDEs

 dηt(b)=−∇U⋅(ηt)(b)dt+√2d∇wt(b),b∈(Zd)∗, (2.4)

where is the family of independent one dimensional Brownian motions. Since the coefficients are Lipschitz continuous in , this equation has the unique strong solution in for every . Note that defined from the solution of the SDE (2.1) on satisfies (2.4) for and boundary conditions for when letting for .

Since we define Gibbs measures on by Dobrushin-Lanford-Ruelle (DLR, for short) equation, we the finite volume Gibbs measure in advance. For a finite set and fixed , we define the affine space by

 XΛ,ξ={η∈X;η(b)=ξ(b),b∈(Zd)∗∖¯¯¯¯¯¯Λ∗}.

We define the finite volume Gibbs measure on by

 μΛ,ξ(dη)=Z−1Λ,ξexp⎛⎜⎝−∑b∈¯¯¯¯¯Λ∗V(η(b))⎞⎟⎠dη¯¯¯¯¯Λ∗,ξ,

where is the Lebesgue measure on and is the normalizing constant.

Let be the set of all probability measures on and let be those satisfying for each . The measure is sometimes called tempered. Let be the family of translation invariant, tempered Gibbs measures introduced by , namely, the family of satisfying the Dobrushin-Lanford-Ruelle equation

 μ(⋅|F(Zd)∗∖¯¯¯¯¯Λ∗)=μΛ,ξ(⋅),μ-a.s. ξ, (2.5)

where is the -algebra generated by . Note that the dynamics given by (2.4) is reversible under . We denote the family of with ergodicity under spatial shifts by .

### 2.3. Assumptions on Gibbs measures and the surface tension

In order to derive the hydrodynamic limit, we will assume both uniqueness of the extremal gradient Gibbs distributions and strict convexity of the surface tension. These assumption are always satisfied under (1.1), cf. see  and , or for non-convex potential at sufficiently high temperature, cf.  and . On the other hand, at critical temperature, Biskup and Kotecký give an example of gradient Gibbs measures with two different extremal states, cf. . The derivation of the corresponding hydrodynamic limit in this case is very challenging open problem.

More precisely, let be the periodic lattice and be the set of all directed bonds in . With , we consider the finite volume Gibbs measure on by

 ~μN,u(d~η)=Z−1N,uexp⎛⎜⎝−12∑b∈Γ∗NV(~η(b)+ub)⎞⎟⎠d~η,

where is Lebesgue measure on , is the normalizing constant and is defined by for with and . We denote the law of by .

###### Assumption 2.2.

For each there exists a unique extremal such that

 Eμu[η(ei)]=ui.

Furthermore, it can be obtained as the weak limit of the periodic Gibbs as .

Under Assumption 2.2, the sequence defined by

 σN(u):=−|ΓN|−1(logZN,u−logZN,0),

has a limit. We thus define the (normalized) surface tension surface tension by

 σ(u)=limN→∞σN(u). (2.6)

Moreover, we can show the following thermodynamic identities between the surface tension and ergodic Gibbs measures:

 Eμu[V′(η(ei))]=∇σ(u),u∈Rd, (2.7) Eμu[d∑i=1η(ei)V′(η(ei))]=u⋅∇σ(u)+1,u∈Rd, (2.8)

which will be shown in Section 4.2. They play an essential role in the derivation of the hydrodynamic limit.

Further we need some technical assumption on the regularity of which are well known in the strictly convex case (1.1), cf.  or in the high temperature regime .

###### Assumption 2.3.

The surface tension is and is Lipschitz continuous. Furthermore, is strictly convex in the following sense: there exist two constants satisfying

 C1|u−v|2≤(u−v)⋅(∇σ(u)−∇σ(v))≤C2|u−v|2,u,v∈Rd. (2.9)
###### Remark 2.1.

Note that the convexity of the surface tension, alternatively defined in terms of fixed boundary conditions has been established in  under very general conditions. Moreover, the strict convexity (i.e. lower bound in (2.9) with ) is not essential for the hydrodynamic limit since an approximation of could be implemented as in .

The following example shows that our Assumptions  2.2 and 2.3 hold in the high temperature regime:

###### Example 2.2.

We introduce a positive parameter corresponding to the inverse temperature, that is, the potential takes the form

 V(η)=β(~V0(η)+~g(η)),

where the symmetric functions satisfy

 0

for some and for some . Then for , (independent of !) of the form

 β0=(c−)3q2d22q(c++d+)q+1∥g′′∥2qLq(R)

both Assumptions 2.2 and 2.3 are satisfied when , see  and its arXiv version (arXiv:0807.2621v1 [math.PR]).

### 2.4. Main Result

The main result in this paper is the following:

###### Theorem 2.1.

We assume Assumptions 2.1, 2.2 and 2.3. Furthermore, we assume that there exists satisfying the following:

1. The function has a compact support in .

2. The sequence of initial data for (2.1) satisfies

 limN→∞E∥hN(0)−h0∥2L2(D)=0, (2.10)

where is the macroscopic height variable corresponding to .

Then, for every , converges in as to which is the unique weak solution of the partial differential equation (PDE)

 (2.11)

where . Here, the function is the surface tension. More precisely, for every ,

 limN→∞E∥hN(t)−h(t)∥2L2(D)=0 (2.12)

holds.

## 3. Stationary measures and estimate for variance

In this section, we mainly discuss properties of stationary measures of (2.4) while working on the general assumption, Assumption 2.1. We believe that the results of this section are relevant beyond the derivation of the hydrodynamic limit.

### 3.1. Generator of (2.4) and stationary measures

We at first note that the infinitesimal generator of (2.4) is given by

 LZd=∑x∈ZdLx, (3.1)

where

 Lx=∑b,b′∈(Zd)∗:xb=xb′=x{4∂2∂η(b)∂η(b′)−2V′(η(b))∂∂η(b′)}.

To keep notation simple, we sometimes denote by if it doesn’t cause any confusion.

We can see that the Gibbs property implies reversibility under (2.4), and therefore stationarity, see Proposition 3.1 in  for details. We note that the same argument as in  is applicable in quite general setting, including ours. In Theorem 2.1 of , the equivalence of the Gibbs property and stationarity is shown using (1.1), here we show this result using another approach.

###### Theorem 3.1.

We assume Assumption 2.1. If is invariant under spatial shift and a stationary measure corresponding to , i.e.,

 ∫XLf(η)μ(dη)=0,f∈C2loc(X),

then is a Gibbs measure, i.e., (2.5) holds.

Since the proof of Theorem 3.1 is slightly long, we postpone the proof until the end of this paper, see Section 5.

### 3.2. Uniform bound for the variance for stationary measures

If the potential is a strictly convex function satisfying (1.1), we then get the uniform bound for the variance for Gibbs measures as a direct consequence of the Brascamp-Lieb inequality. See  for details. Our next result based on dynamical approach shows that the variance remains bounded in the tilt for general potentials under Assumption 2.1.

###### Theorem 3.2.

We assume Assumption 2.1. Let be the family of stationary measures for the gradient field (2.4) which are tempered, translation invariant and ergodic under spatial shift. The variance of under are bounded from above by a constant independent of , that is,

 supμ∈SextVarμ[η(b)]<∞,b∈(Zd)∗

holds.

###### Proof.

We shall show the desired bound by arranging the argument of the proof of Proposition 2.1 of . We fix and we define the vector by

 ui=Eμ[η((ei,0))],1≤i≤d.

Let be the solution of SDEs (2.4) with initial distribution . Introducing by

 ϕt(0)=∫t0U0(ηs)ds+√2wt(0)

and

 ϕt(x)=ϕt(0)+∑b∈C0,xηt(b),x∈Zd,

where is an arbitrary chain connecting to , we then obtain that solves the SDEs

 dϕt(x)=−Ux(ϕt)dt+√2dwt(x),x∈Zd.

Our calculation will be based on the energy estimate for introduced above.

Let and . For a deterministic with

 ψ(x)=u⋅x,x∈Zd,

we obtain

 d∑x∈Λ(ϕt(x)−ψ(x))2 =−2∑x∈Λ(ϕt(x)−ψ(x))Ux(ϕt)dt+2|Λ|dt+Mt

with a martingale by Itô’s formula. Performing summation-by-parts, we get

 ∑x∈Λ(ϕt(x)−ψ(x))Ux(ϕt) =12∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))V′(∇ϕt(b)) −∑b∈¯¯¯¯¯Λ∗;xb∈Λ∁(ϕt(xb)−ψ(xb))V′(∇ϕt(b)).

We thus have

 ∑x∈Λ(ϕT(x)−ψ(x))2=I0+I1(T)+I2(T)+2|Λ|T+MT, (3.2)

where and are defined by

 I0 =∑x∈Λ(ϕ0(x)−ψ(x))2, I1(T) =−∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))V′(∇ϕt(b))dt, I2(T) =2∫T0∑b∈¯¯¯¯¯Λ∗;xb∈Λ∁(ϕt(xb)−ψ(xb))V′(∇ϕt(b))dt.

From now on, we shall give bounds for expectations of and separately.

We at first give a estimate for the expectation of . Here, the same argument as the proof of (2.14) in  can be applied. That is, from ergodicity and temperedness of , we have

 lim|x|→∞1|x|2E[(ϕ0(x)−ψ(x))2]=0, (3.3)

and this implies that

 limℓ→∞ℓ−2|Λ|−1E[I0]=0.

We therefore obtain that for every there exists such that

 E[I0]≤ϵℓ2|Λ| (3.4)

holds for every .

We shall next calculate and its expectation. From Assumption 2.1, can be calculated as follows:

 I1(T) =−∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))(V′0(∇ϕt(b))−V′0(∇ψ(b)))dt −∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))g′(∇ϕt(b))dt +∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))(V′0(∇ψ(b)))dt ≤−c−∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))2dt +∥g′∥∞∫T0∑b∈¯¯¯¯¯Λ∗|∇ϕt(b)−∇ψ(b)|dt +∫T0∑b∈¯¯¯¯¯Λ∗(∇ϕt(b)−∇ψ(b))V′0(∇ψ(b))dt =:I1,1(T)+I1,2(T)+I1,3(T).

Using Schwarz’s inequality, we obtain the following estimate for the second term :

 I1,2(T)≤12λ∥g′∥∞∫T0∑b∈¯¯¯¯¯Λ∗|∇ϕt(b)−∇ψ(b)|2dt+12λ−1∥g′∥∞∣∣¯¯¯¯¯¯Λ∗∣∣T

for arbitrary . If holds, we then have

 I1,2(T)≤12c−∫T0∑b∈¯¯¯¯¯Λ∗|∇ϕt(b)−∇ψ(b)|2dt+12c−1−∥g′∥2∞∣∣¯¯¯¯¯¯Λ∗∣∣T (3.5)

by taking . Note that the estimate (3.5) trivially holds when . Summarizing above and taking expectation, we obtain

 E[I1(T)]≤−12c−∫T0∑b∈¯¯¯¯¯Λ∗E[(∇ϕt(b)−∇ψ(b))2]dt+12c−1−∥g′∥2∞∣∣¯¯¯¯¯¯Λ∗∣∣T.

Here, we have used

 E[I1,3(T)]=0,

which follows from the definition of and . From the relationship , the stationarity of and the definition of , we have

 E[(∇ϕt(b)−∇ψ(b))2]=Varμ[η(b)].

Since is translation invariant, we also have

 ∑b∈¯¯¯¯¯Λ∗Varμ[η(b)]≥κ|Λ|∑b:xb=0Varμ[η(b)]

with a constant . Applying above, we finally conclude

 E[I1(T)]≤−12c−κT|Λ|∑b:xb=0Varμ[η(b)]+12c−1−∥g′∥2∞|¯Λ∗|T. (3.6)

We next calculate the expected value of . Putting by

 ~I2(T)=2∫T0∑b∈¯¯¯¯¯Λ∗;xb∈Λ∁(ϕt(xb)−ψ(xb))(V′(∇ϕt(b))−V′(∇ψ(b)))dt,

we have

 E[I2(T)]=E[~I2(T)]

from the definition of . We shall thus calculate instead of . Using Schwarz’s inequality, we obtain

 E[~I2(T)] ≤γℓ−1|∂Λ∗ℓ|∫T0supy∈∂ΛE[(ϕt(y)−ψ(y))2]dt +γ−1ℓ∫T0∑b∈¯¯¯¯¯Λ∗;xb∈Λ∁E[(V′(∇ϕt(b))−V′(∇ψ(b)))2]dt =:F2,1(T)+F2,2(T) (3.7)

for an arbitrary , where and are define by

 ∂¯¯¯¯¯¯Λ∗={b∈¯¯¯¯¯¯Λ∗;xb∈Λ∁}, ∂Λ={xb;b∈∂¯¯¯¯¯¯Λ∗}.

For , since is Lipschitz continuous, there exists a constant such that

 F2,2 ≤Cγ−1ℓdT∑b:xb=0Varμ[η(b)] (3.8)

by using the translation invariance of . For , let us use a similar argument to the proof of (2.12) in . Taking , we have

 (ϕt(y)−ψ(y))2 ≤2(ϕt(y)−ψ(y)−1|Λ′|∑x∈Λ′(ϕt(x)−ψ(x)))2 +2(1|Λ′|∑x∈Λ′(ϕt(x)−ψ(x)))2 =:A1+A2

for every . For the term , the calculations runs quite parallel to the argument in  and we can obtain that for every there exists such that

 E[A1]≤ϵℓ2

holds for every . Let us give a bound for the term . Using Itô’s formula, we obtain

 1|Λ′|∑x∈Λ′(ϕt(x)−ψ(x)) =1|Λ′|∑x∈Λ′(ϕ0(x)−ψ(x)) −1|Λ′|∫t0∑x∈Λ′∑b∈(Zd)∗;xb=xV′(ηs(b))ds+1|Λ′|∑x∈Λ′wt(x)

and therefore we get

 A2 ≤4(1|Λ′|∑x∈Λ′(ϕ0(x)−ψ(x)))2 +4⎛⎜⎝1|Λ′|∫t0∑x∈Λ′∑b∈(Zd)∗;xb=xV′(ηs(b))ds⎞⎟⎠2+4(1|Λ′|