Integral equation formulation of run-and-tumble particles in a harmonic trap: the special status of a system in two-dimensions

Abstract

Statistical-mechanical models often exhibit a dimension-dependent solvability: in 1D, exact solutions are straightforward; in 2D, solutions are exact but require nontrivial derivations; and in 3D, closed-form solutions are typically unavailable. This logic is repeated for a simple model of self-propelled particles, run-and-tumble particles (RTP) in a harmonic trap, confirming the claim that the system in 2D enjoys special status. This study revisits the RTP-harmonic-trap model using an integral-equation formulation recently proposed in Ref. \cite{POF-Frydel-2024}. The formulation is based on reinterpreting RTP motion as a jump process. The key quantity of the formulation is a transition operator $G(x,x')$, representing the probability distribution of the jumps of an auxiliary system. The stationary distribution is then obtained from the integral equation $\rho(x) = \int dx' \, \rho(x') G(x,x')$. In 2D, we find that $G(x,x')$ is reversible. This implies the $\rho(x)$ satisfied the detailed balance condition, $\rho(x') G(x,x') = \rho(x) G(x',x)$, from which $\rho(x)$ can be obtained without need of an integral equation. The reversibility of $G(x,x')$ does not mean that RTP particles are in equilibrium. It only means that our specific interpretation of the RTP motion leads to an auxiliary system that is in equilibrium. The reversibility of the system in 2D is lost if the probability distribution of the waiting times (the times that determine the duration on the "run" stage of the RTP motion) deviates from an exponential distribution.