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 coordinates of these two positions and numbers 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 numbers of particles on all positions are bounded by $\mathcal{K}<+\mathrm{\infty }$, we show that the fluctuation of our model is driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of our main results is to show that numbers of particles on different positions are approximately independent by utilizing a graphical method.