Exact Stationary State of a $d$-dimensional Run-and-Tumble Particle in a Harmonic Potential

TL;DR

This paper derives the exact nonequilibrium steady state of a run-and-tumble particle in a harmonic trap across multiple dimensions, revealing shape transitions, effects of thermal noise, and providing explicit distributions and moments.

Contribution

It provides the first exact analytical solutions for the stationary distributions of RTPs in harmonic potentials in multiple dimensions, including effects of thermal noise and velocity distributions.

Findings

Radial distribution simplifies to a beta distribution in 1D and 2D.

Explicit closed-form expressions for the 3D joint distribution.

Identification of a shape transition at the turning surface controlled by persistence.

Abstract

We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in $d$ dimensions confined in an isotropic harmonic trap $V(\mathbf r)=\mu r^{2}/2$, with $r=\|\mathbf r\|$. Rotational invariance reduces the problem to the stationary single-coordinate marginal $p_X(x)$, from which the radial distribution $p_R(r)$ and the full joint stationary density follow by explicit integral transforms. We first focus on a generalized trapped RTP in one dimension, where post-tumble velocities are drawn from an arbitrary distribution $W(v)$. Using a Kesten-type recursion, we represent its stationary position in terms of a stick-breaking (or Dirichlet) process, yielding closed-form expressions for its distribution and its moments. Specializing $W(v)$ to the projected velocity law of an isotropic RTP, we reconstruct $p_R(r)$ and the full joint distribution of all the coordinates in $d=1,2,3$. In $d=1$ and $d=2$, the radial law simplifies to a beta distribution, while in $d=3$, we derive closed-form expressions for $p_R(r)$ and the stationary joint distribution $P(x,y,z)$, which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface $r=v_0/\mu$, where $v_0$ is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient $D>0$, showing that the stationary law is a Gaussian convolution of the $D=0$ result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as $D \to 0$ and $D \to \infty$ respectively. All analytical predictions are systematically validated against numerical simulations.