Exact spectral solution of two interacting run-and-tumble particles on a ring lattice
Exact spectral solution of two interacting run-and-tumble particles on a ring lattice
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DOI：
10.1088/1742-5468/aaf631
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发表时间：
2018-10
期刊：
影响因子：
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通讯作者：
E. Mallmin;R. Blythe;M. Evans
中科院分区：
文献类型：
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作者：
E. Mallmin;R. Blythe;M. Evans
相互作用随机游走模型的精确解，如一维晶格气体，提供了对非平衡现象起源的精确洞察。在这里，我们研究了一个模型的运行和翻滚的粒子在一个环形晶格通过硬核排斥相互作用。我们提出了一个和两个粒子使用生成函数技术的精确解。对于两个粒子，本征向量和本征值明确表示使用两个参数想起Bethe根，其数值由多项式方程，我们推导。该谱以复杂的方式依赖于方向反转率与晶格跳跃率之比。对于单粒子和双粒子，大粒子的光谱由独立的真实的谱带组成，小粒子的谱带混合并变成复值谱带。在的特殊值处，两个或多个特征值合并，使得马尔可夫矩阵不可对角化。这种复杂的参数依赖性的后果是出现的动态过渡：非解析最小值的最长弛豫时间的函数（对于一个给定的晶格尺寸）。例外点在理论上和实验上是相关的，例如开放量子系统和多通道散射。我们建议，这种现象应该是一个普遍存在的功能，以及经典的非平衡模型，并在此背景下的物理观测值的相关性。
Exact solutions of interacting random walk models, such as 1D lattice gases, offer precise insight into the origin of nonequilibrium phenomena. Here, we study a model of run-and-tumble particles on a ring lattice interacting via hardcore exclusion. We present the exact solution for one and two particles using a generating function technique. For two particles, the eigenvectors and eigenvalues are explicitly expressed using two parameters reminiscent of Bethe roots, whose numerical values are determined by polynomial equations which we derive. The spectrum depends in a complicated way on the ratio of direction reversal rate to lattice jump rate, . For both one and two particles, the spectrum consists of separate real bands for large , which mix and become complex-valued for small . At exceptional values of , two or more eigenvalues coalesce such that the Markov matrix is non-diagonalizable. A consequence of this intricate parameter dependence is the appearance of dynamical transitions: non-analytic minima in the longest relaxation times as functions of (for a given lattice size). Exceptional points are theoretically and experimentally relevant in, e.g. open quantum systems and multichannel scattering. We propose that the phenomenon should be a ubiquitous feature of classical nonequilibrium models as well, and of relevance to physical observables in this context.