We study the hydrodynamic limit for a periodic 1-dimensional exclusion process with a dynamical constraint, which prevents a particle at site x from jumping to sitex ± 1 unless site x ∓ 1 is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than 1 2 , or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density 1 2 , the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.
In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in [6]. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of super-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in [8,9], in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.
We consider the dynamics of lattices which have constrained constitutive units flexible in only their mutual orientations. A continuum description is derived through which it is shown that the models have zero shear velocity, free-particle-like internal rotational modes and volume decreasing linearly with temperature. The relevance of models to a range of problems is pointed out.
ABSTRACT. We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter ρ ∈ (0,1). The rate of passage of particles to the right (resp. left) is 1 2 + a 2n γ (resp. 1 2 − a 2n γ ) except at the bond of vertices {−1,0} where the rate to the right (resp. left) is given bywe show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if γ > 1 2 , while for γ = 1 2 it is an energy solution of the stochastic Burgers equation. For γ ≥ β = 1, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For γ ≥ β > 1, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.
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