In this talk I will consider the symmetric simple exclusion process with a slow bond. Its dynamics can be described as follows. At each site x we place a random clock Tx exponentially distributed with parameter 1 and we assume that clocks in different sites are independent. Initially we randomly distribute particles along the lattice and each time a clock rings, if there is a particle at the corresponding site, then it decides to jump to one of its nearest-neighbors. If there is no particle at that site, then nothing happens and the clocks restart. A particle jumps from a site x to x+1 and from x+1 to x at the same rate which is 1 for all sites, except for x = ?1 where it equals ?n^{??}, with ? > 0 and ? ? [0,?]. I will present scaling limits for this model at the level of hydrodynamics and fluctuations. In the hydrodynamics, for ? ? [0, 1), the density of particles evolves according to the periodic heat equation; if ? = 1, it evolves according to the heat equation with some Robin?s boundary conditions and if ? ? (1,?], it evolves according to the heat equation with Neumann?s boundary conditions. A similar phase transition is also present on the fluctuations of the density, the current and the tagged particle.
This is a joint work with Tertuliano Franco (UB - Brazil) and Adriana Neumann (UFRGS - Brazil). |