Hydrodynamic limit for the Ginzburg-Landau interface model with non-convex potential
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.
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
with a symmetric function . The Langevin equation associated with is given by
where in the drift term is defined by
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
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
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
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
where stands for the hypercube in with center and side length , that is,
On we consider the dynamics governed by the following stochastic differential equations (SDEs)
with the boundary condition
with some and initial data , where for and , or more generally for and . The height variable in (2.2) is defined by
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 :
The function has the following representation:
where functions are symmetric functions and satisfy
There exist constants such that
There exists a constant such that
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,
For , the gradient is defined by
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
We denote equipped with the norm . We introduce the dynamics governed by the SDEs
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
We define the finite volume Gibbs measure on by
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
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
where is Lebesgue measure on , is the normalizing constant and is defined by for with and . We denote the law of by .
For each there exists a unique extremal such that
Furthermore, it can be obtained as the weak limit of the periodic Gibbs as .
Under Assumption 2.2, the sequence defined by
has a limit. We thus define the (normalized) surface tension surface tension by
Moreover, we can show the following thermodynamic identities between the surface tension and ergodic Gibbs measures:
which will be shown in Section 4.2. They play an essential role in the derivation of the hydrodynamic limit.
The surface tension is and is Lipschitz continuous. Furthermore, is strictly convex in the following sense: there exist two constants satisfying
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 .
We introduce a positive parameter corresponding to the inverse temperature, that is, the potential takes the form
where the symmetric functions satisfy
for some and for some . Then for , (independent of !) of the form
2.4. Main Result
The main result in this paper is the following:
The function has a compact support in .
The sequence of initial data for (2.1) satisfies
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)
where . Here, the function is the surface tension. More precisely, for every ,
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
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.
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.
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
Let be the solution of SDEs (2.4) with initial distribution . Introducing by
where is an arbitrary chain connecting to , we then obtain that solves the SDEs
Our calculation will be based on the energy estimate for introduced above.
Let and . For a deterministic with
with a martingale by Itô’s formula. Performing summation-by-parts, we get
We thus have
where and are defined by
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
and this implies that
We therefore obtain that for every there exists such that
holds for every .
We shall next calculate and its expectation. From Assumption 2.1, can be calculated as follows:
Using Schwarz’s inequality, we obtain the following estimate for the second term :
for arbitrary . If holds, we then have
by taking . Note that the estimate (3.5) trivially holds when . Summarizing above and taking expectation, we obtain
Here, we have used
which follows from the definition of and . From the relationship , the stationarity of and the definition of , we have
Since is translation invariant, we also have
with a constant . Applying above, we finally conclude
We next calculate the expected value of . Putting by
from the definition of . We shall thus calculate instead of . Using Schwarz’s inequality, we obtain
for an arbitrary , where and are define by
For , since is Lipschitz continuous, there exists a constant such that
by using the translation invariance of . For , let us use a similar argument to the proof of (2.12) in . Taking , we have
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
holds for every . Let us give a bound for the term . Using Itô’s formula, we obtain
and therefore we get