From mesoscale back to microscale: reconstruction schemes for coarse-grained stochastic lattice systems

Starting from a microscopic stochastic lattice spin system and the corresponding coarse-grained model we introduce a mathematical strategy to recover microscopic information given the coarse-grained data. We define “reconstructed” microscopic measures satisfying two conditions: (i) they are close in specific relative entropy to the initial microscopic equilibrium measure conditioned on the coarse-grained, data, and (ii) their sampling is computationally advantageous when compared to sampling directly from the conditioned microscopic equilibrium measure. By using different techniques we consider the cases of both short and long range microscopic models

Fourier law, phase transitions and the stationary Stefan problem

We study the one-dimensional stationary solutions of an integro-differential equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising systems with Kac potentials, \cite{GiacominLebowitz}. We construct stationary solutions with non zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied. We show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains however its validity in the thermodynamic limit where the limit profile is again monotone away from the interface

Exponential rate of convergence in current reservoirs

In this paper we consider a family of interacting particle systems on [−N, N] that arises as a natural model for current reservoirs and Fick’s law. We study the exponential rate of convergence to the stationary measure, which we prove to be of the order N−2

Spectral gap in stationary non-equilibrium processes

In this paper we study the spectral gap for a family of interacting particles systems on $[-N,N]$, proving that it is of the order $N^{-2}$. The system arises as a natural model for current reservoirs and Fick's law

Truncated correlations in the stirring process with births and deaths

We consider the stirring process in the interval \La_N:=[-N,N] of $\mathbb Z$ with births and deaths taking place in the intervals $I_+:=(N-K,N]$, $K>0$, and respectively $I_-:=[-N,-N+K)$. We prove bounds on the truncated moments uniform in $N$ which yield strong factorization properties

On a class of solvable stationary non equilibrium states for mass exchange models

We consider a family of models having an arbitrary positive amount of mass on each site and randomly exchanging an arbitrary amount of mass with nearest neighbor sites. We restrict to the case of diffusive models. We identify a class of reversible models for which the product invariant measure is known and the gradient condition is satisfied so that we can explicitly compute the transport coefficients associated to the diffusive hydrodynamic rescaling. Based on the Macroscopic Fluctuation Theory \cite{mft} we have that the large deviations rate functional for a stationary non equilibrium state can be computed solving a Hamilton-Jacobi equation depending only on the transport coefficients and the details of the boundary sources. Thus, we are able to identify a class of models having transport coefficients for which the Hamilton-Jacobi equation can indeed be solved. We give a complete characterization in the case of generalized zero range models and discuss several other cases. For the generalized zero range models we identify a class of discrete models that, modulo trivial extensions, coincides with the class discussed in \cite{FG} and a class of continuous dynamics that coincides with the class in \cite{FFG}. Along the discussion we obtain a complete characterization of reversible misanthrope processes solving the discrete equations in \cite{CC}.Comment: 32 pages, 1 figur

Free energy expansions for renormalized systems for colloids

We consider a binary system of small and large spheres of finite size in a continuous medium interacting via a non-negative potential. We work in the canonical ensemble and compute upper and lower bound for the free energy at finite and infinite volume by first integrating over the small spheres and then treating the effective system of the large ones which now interact via a multi-body potential. By exploiting the underlying structure of the original binary system we prove the convergence of the cluster expansion for the latter system and obtain a sufficient condition which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods directly to the binary system). Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction