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Exact solution of the zero-range process: fundamental diagram of the corresponding exclusion process | | Masahiro Kanai
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10 Jan 2007 | | Subject: | Statistical Mechanics | | Abstract: | In this paper, we propose a general way of computing expectation values in the zero-range process, using an exact form of the partition function. As an example, we provide the fundamental diagram (the flux-density plot) of the asymmetric exclusion process corresponding to the zero-range process. It was previously pointed out that the zero-range process can be mapped onto an exclusion process, which is suitable for modelling traffic flow. The fundamental diagram clearly shows statistical properties of collective phenomena in many-particle systems, such as a traffic jam in traffic-flow models. The zero-range process is a stochastic process defined originally on a one-dimensional periodic lattice, where particles hop from site to site with a zero-range interaction. Each site of the lattice may contain an integer number of particles, and the particles hop to the next site with a hop rate which depends on the particle number at the departure site. It is remarkable that the zero-range process has an exactly-solvable steady state with a particle flow. We express the partition function for the steady state by the Lauricella hypergeometric function, and thereby have two exact fundamental diagrams each for the parallel and random update rules. Meanwhile, from the viewpoint of equilibrium statistical mechanics, we work within the canonical ensemble but the result obtained is certainly in agreement with previous works done in the grand canonical ensemble. | | Source: | arXiv, cond-mat/0701190 | | Services: | Forum | Review | PDF | Favorites | |
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