Mean Field Limits of a Class of Conservative Systems with Position-Dependent Transition Rates

In this paper, we are concerned with a class of conservative systems including asymmetric exclusion processes and zero-range processes as examples, where some particles are initially placed on N positions. A particle jumps from a position to another at a rate depending on the coordinates of these two positions and the number of particles on these two positions. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation which is consistent with a mean field analysis. Furthermore, in the case where the number of particles on all positions are bounded by \(\mathcal {K}<+\infty \), we show that the fluctuation of our model is driven by a generalized Ornstein–Uhlenbeck process. A crucial step in the proofs of our main results is to show that the number of particles on different positions are approximately independent by utilizing a graphical method.