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arXiv 2310.03343cond-mat.stat-mechmath.COmath.PR
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一种精确可解的非对称$K$-排斥过程

An exactly solvable asymmetric $K$-exclusion process

Arvind Ayyer, Samarth Misra

备注：
35 pages, 4 figures, simulations are also available at http://www.math.iisc.ac.in/~arvind/KASEP/, signification corrections, many new references
发表信息：
J. Phys. A: Math. Theor., Vol 57 no. 31, (2024) 315001, 24pp
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AI总结：

本研究提出精确可解的$(q, t)$ $K$-ASEP模型，推导其乘积形式稳态的显式回文多项式权重，分析密度与最概然粒子数，并构建对应二维排斥过程，为misanthrope过程二维推广提供示例。

摘要：

我们研究了具有$L$个格点的有限环上的相互作用粒子过程，每个格点最多容纳$K$个粒子，其中粒子以由$t$-变形整数和非对称参数$q$给出的速率跳迁至最近邻，这里$t>0$且$q \geq 0$为参数。我们将该模型称为$(q, t)$ $K$-ASEP，当$K=1$时它退化为环上常见的ASEP，当$t=q=1$时退化为Schütz与Sandow（《物理评论E》，1994）研究的模型。该模型是misanthrope过程的一个特例，因此其稳态不依赖于$q$且具有乘积形式，推广了ASEP的同类现象。此处值得关注的是，稳态权重由涉及$t$-二项式系数的显式公式给出，且是$t$的回文多项式。有趣的是，尽管$(q, t)$ $K$-ASEP不满足粒子-空穴对称性，但其稳态却满足。我们分析了密度，并计算了在不同$t$区间内稳态下单个格点的最概然粒子数。最后，我们构建了一个高度为$K$、周长为$L$的离散柱面上的二维排斥过程，该过程投影后得到$(q, t)$ $K$-ASEP，且其稳态分布同样具有乘积形式。我们认为该模型将为构建misanthrope过程的二维类似物提供一个说明性示例。模拟结果作为附件文件随附。

英文摘要：

We study an interacting particle process on a finite ring with $L$ sites with at most $K$ particles per site, in which particles hop to nearest neighbors with rates given in terms of $t$-deformed integers and asymmetry parameter $q$, where $t>0$ and $q \geq 0$ are parameters. This model, which we call the $(q, t)$~$K$-ASEP, reduces to the usual ASEP on the ring when $K = 1$ and to a model studied by Schütz and Sandow (\emph{Phys. Rev. E}, 1994) when $t = q = 1$. This is a special case of the misanthrope process and as a consequence, the steady state does not depend on $q$ and is of product form, generalizing the same phenomena for the ASEP. What is interesting here is the steady state weights are given by explicit formulas involving $t$-binomial coefficients, and are palindromic polynomials in $t$. Interestingly, although the $(q, t)$~$K$-ASEP does not satisfy particle-hole symmetry, its steady state does. We analyze the density and calculate the most probable number of particles at a site in the steady state in various regimes of $t$. Lastly, we construct a two-dimensional exclusion process on a discrete cylinder with height $K$ and circumference $L$ which projects to the $(q, t)$~$K$-ASEP and whose steady state distribution is also of product form. We believe this model will serve as an illustrative example in constructing two-dimensional analogues of misanthrope processes. Simulations are attached as ancillary files.

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