Exact spectral solution of two interacting run-and-tumble particles on a ring lattice

TL;DR

This paper provides an exact spectral solution for a model of two interacting run-and-tumble particles on a ring lattice, revealing complex eigenvalue behavior and dynamical transitions influenced by the ratio of reversal to jump rates.

Contribution

It introduces an exact analytical solution for the spectral properties of interacting run-and-tumble particles, including eigenvalues, eigenvectors, and the dependence on key parameters.

Findings

Spectrum exhibits real bands for large reversal rates

Eigenvalues coalesce at exceptional points leading to non-diagonalizable matrices

Dynamical transitions manifest as non-analytic minima in relaxation times

Abstract

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, $\omega$. For both one and two particles, the spectrum consists of separate real bands for large $\omega$, which mix and become complex-valued for small $\omega$. At exceptional values of $\omega$, 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 $\omega$ (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.