Particle Model for the Reservoirs in the Simple Symmetric Exclusion Process

In this paper, we will study the long time behavior of the simple symmetric exclusion process in the “channel” ΛN=[1,N]∩N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda _N=[1,N]\cap \mathbb {N}$$\end{document} with reservoirs at the boundaries. These reservoirs are also systems of particles which can be exchanged with the particles in the channel. The size M of each reservoir is much larger than the one of the channel, i.e. M=N1+α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=N^{1+\alpha }$$\end{document} for a fixed number α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document}. Based on the size of the channel and the holding time at each reservoir, we will investigate some types of rescaling time.