Abstract. The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first -and often inadequate -approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.Mathematics Subject Classification. 65C05, 65C20, 82B20, 82B80, 82-08.

Abstract. We consider a system of particles confined in a box Λ ⊂ R d interacting via a tempered and stable pair potential. We prove the validity of the cluster expansion for the canonical partition function in the high temperature -low density regime. The convergence is uniform in the volume and in the thermodynamic limit it reproduces Mayer's virial expansion providing an alternative and more direct derivation which avoids the deep combinatorial issues present in the original proof.

Abstract. We consider the symmetric simple exclusion process in the interval [−N, N ] with additional birth and death processes respectively on (N −K, N ], K > 0, and [−N, −N + K). The exclusion is speeded up by a factor N 2 , births and deaths by a factor N . Assuming propagation of chaos (a property proved in a companion paper, [3]) we prove convergence in the limit N → ∞ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.

We consider the stirring process in the interval Lambda(N) : = [-N; N] of Z with births and deaths taking place in the intervals I+ : = ( N K; N], and respectively I- : = [-N; -N + K), 1 <= K < N. We prove bounds on the truncated moments uniform in N which yield strong factorization properties

We study the one-dimensional stationary solutions of an integrodifferential equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising systems with Kac potentials, [5]. 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.