Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system
type: Item
instance of: scholarly article
title: Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system
zbMATH Open document ID: 0819.60096
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1995-04-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77502
review text: A one-dimensional generalized symmetric simple exclusion process is considered. At most two particles per site are permitted for this process. The state space of the process is \(\{0,1,2\}^{T_ N}\), \(T_ N = \{0,1,2,3, \dots\}\) for each integer \(N\). The process is defined as the Markov one generator of which is given by \[ (L_ Nf) (\eta) = \sum_{k=0}^{N-1} (L_{k, k+1} f)(\eta) + (L_ gf) (\eta) + (L_ df) (\eta). \] The elementary generators \(L_{k,k+1}\), \(L_ g\), \(L_ d\) are defined as follows \[ (L_{k,k+1}f) (\eta) = d_{k,k+1} (\eta) \bigl[ f(\eta^{k,k+1}) - f(\eta) \bigr] + g_{k,k+1} (\eta) \bigl[ f (\eta^{k+1,k}) - f(\eta) \bigr] \] with the rate functions \[ d_{k,k+1} (\eta) = \begin{cases} 1 \quad & \text{if } \quad \eta_ k \geq 1,\;\eta_{k+1} \leq 1,\\ 0 \quad & \text{otherwise}, \end{cases}\quad g_{k,k+1} (\eta)= \begin{cases} 1 & \text{if } \quad \eta_ k \leq 1,\eta_{k+1}\geq 1, \\ 0 & \text{otherwise}, \end{cases} \] \[ \eta^{k,i}_ j = \begin{cases} \eta_ j \quad & \text{if} \quad j \neq k,i, \\ \eta_{k-1} \quad & \text{if} \quad j = k,\\ \eta_{i+1} \quad & \text{ if } j=i.\end{cases} \] The generators \(L_ g\), \(L_ d\) are the creation-destruction operators taking into account the boundary conditions. Thus the system is open with the boundaries which are two infinite reservoirs of particles with different densities. The hydrodynamic limit \(N \to \infty\) is studied. It is proved that in the stationary state the nonrandom density of random field converges to the solution of a nonlinear elliptic equation. In particular, it is proved Fick's law of the transport current.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 740324
zbMATH Keywords: interacting particle system
zbMATH Keywords: hydrodynamic limit
author: Claude Kipnis
author: C. Landim
author: Stefano Olla
reviewed by: Yu. P. Virchenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 740324