Tadahisa Funaki

Stochastic Interface Models

In these notes we try to review developments in the last decade of the theory on stochastic models for interfaces arising in two phase system, mostly on the so-called ⊸φ interface model. We are, in particular, interested in the scaling limits which pass from the microscopic models to macroscopic level. Such limit procedures are formulated as classical limit theorems in probability theory such as the law of large numbers, the central limit theorem and the large deviation principles.

The scaling limit for a stochastic PDE and the separation of phases

SummaryWe investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter ε (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldMε of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofMε.

A certain class of diffusion processes associated with nonlinear parabolic equations

SummaryWe introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation. Then we show the existence and uniqueness theorems for solutions to the martingale problem.

Fluctuations for ∇φ interface model on a wall

We consider [backward difference][phi] interface model on a hard wall. The hydrodynamic large-scale space-time limit for this model is discussed with periodic boundary by Funaki et al. (2000, preprint). This paper studies fluctuations of the height variables around the hydrodynamic limit in equilibrium in one dimension imposing Dirichlet boundary conditions. The fluctuation is non-Gaussian when the macroscopic interface is attached to the wall, while it is asymptotically Gaussian when the macroscopic interface stays away from the wall. Our basic method is the penalization. Namely, we substitute in the dynamics the reflection at the wall by strong drift for the interface when it goes down beyond the wall and show the fluctuation result for such massive [backward difference][phi] interface model. Then, this is applied to prove the fluctuation for the [backward difference][phi] interface model on the wall.

Stationary states of random Hamiltonian systems

SummaryWe investigate the ergodic properties of Hamiltonian systems subjected to local random, energy conserving perturbations. We prove for some cases, e.g. anharmonic crystals with random nearest neighbor exchanges (or independent random reflections) of velocities, that all translation invariant stationary states with finite entropy per unit volume are microcanonical Gibbs states. The results can be utilized in proving hydrodynamic behavior of such systems.

A stochastic partial differential equation with values in a manifold

Abstract We investigate a certain stochastic partial differential equation which is defined on the unit interval with periodic boundary condition and takes values in a manifold. Such equation has particularly two different applications. Namely, it determines the evolution law of an interacting constrained system of continuum distributed over the unit circle, while it defines a diffusive motion of loops on a manifold. We establish the existence and uniqueness results and then show the smoothness property of the solutions. Some examples are given in the final section.

Singular limit for stochastic reaction-diffusion equation and generation of random interfaces

Singular limit is investigated for reaction-diffusion equations with an additive noise in a bounded domain of ℝ2. The solution converges to one of the two stable phases {+1, −1} determined from the reaction term; accordingly a phase separation curve is generated in the limit. We shall derive a randomly perturbed motion by curvature for the dynamics of the phase separation curve.

Zero temperature limit for interacting Brownian particles. I. Motion of a single body

We consider a system of interacting Brownian particles in R d with a pairwise potential, which is radially symmetric, of finite range and attains a unique minimum when the distance of two particles becomes a > 0. The asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects. If the system is rigidly crystallized, namely if the particles are rigidly arranged in an equal distance a, the crystallization is kept under the evolution in macroscopic time scale. Then, assuming that the crystal has a definite limit shape under a macroscopic spatial scaling, the translational and rotational motions of such shape are characterized.

Zero temperature limit for interacting Brownian particles. II. Coagulation in one dimension

We study the zero temperature limit for interacting Brownian particles in one dimension with a pairwise potential which is of finite range and attains a unique minimum when the distance of two particles becomes a > 0. We say a chain is formed when the particles are arranged in an almost equal distance a. If a chain is formed at time 0, so is for positive time as the temperature of the system decreases to 0 and, under a suitable macroscopic space-time scaling, the center of mass of the chain performs the Brownian motion with the speed inversely proportional to the total mass. If there are two chains, they independently move until the time when they meet. Then, they immediately coalesce and continue the evolution as a single chain. This can be extended for finitely many chains.

Hydrodynamic Limit for Lattice Gas Reversible under Bernoulli Measures

The hydrodynamic limit for the class of lattice gases that are reversible under the Bernoulli measures is studied by estimating the relative entropy of the microscopic state of actual system with respect to a local equilibrium state (the method of H.T. Yau). The model discussed in this article is of non-gradient type and this forces us to introduce the local equilibrium state of second order approximation that is made according to the variational formula (an equivalent of the Green-Kubo formula) for the diffusion coefficient due to S.R.S. Varadhan. The estimation of the relative entropy is carried out by adapting the “gradient replacement” devised by Varadhan for the study of the Ginzburg-Landau model of non-gradient type. Because of the method adopted we do not need tightness argument nor two-block estimate, but do need to assume that the solution of the limiting nonlinear diffusion equation is smooth.