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" ${\Lambda }_N=[1,N]\cap N$ 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. $M=N^{1+\alpha }$ for a fixed number $\alpha >0$. Based on the size of the channel and the holding time at each reservoir, we will investigate some types of rescaling time.