In this talk, we study the spectral gap and mixing time of the simple exclusion process in a finite line segment, where we swap the contents of two sites $x, x+1$ at rate $c(x, x+1)$. Under some assumption on $c(x, x+1)_x$ and the number of particles, we identify the spectral gap, and prove that the total variation distance to equilibrium drops abruptly from 1 to 0 at a certain instant.
