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arXiv · 2401.12512

Mean field limits of a class of conservative systems with position-dependent transition rates

Xiaofeng Xue
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Abstract

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 $\mathcal{K}<+\infty$, 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.

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Xiaofeng Xue. 2024-01-23. Mean field limits of a class of conservative systems with position-dependent transition rates. https://arxiv.org/abs/2401.12512

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