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

TL;DR

This work derives the exact nonequilibrium stationary state of a $d$-dimensional run-and-tumble particle confined in an isotropic harmonic trap, reducing the problem to the one-dimensional marginal $p_X(x)$ by rotational invariance. The authors solve a generalized 1D RTP with arbitrary post-tumble velocity distribution $W(v)$ via a Kesten recursion and a Dirichlet-process (stick-breaking) representation, obtaining closed-form expressions for $p_X(x)$ and its moments; specializing to the projected velocity law yields the full stationary statistics in $d=1,2,3$. In $d=1$ and $d=2$, the radial density is a beta distribution and exhibits a persistence-driven shape transition at the turning surface $r_0=v_0/\mu$, while in $d=3$ the statistics remain tractable but do not reduce to a beta form, with a more intricate turning-surface behavior. Introducing thermal noise with diffusion coefficient $D>0$ shows that the stationary law is a Gaussian convolution of the $D=0$ result, regularizing singularities and governing the crossover between persistence- and diffusion-dominated regimes as $D$ varies; all predictions are validated by numerical simulations.

Abstract

We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in $d$ dimensions confined in an isotropic harmonic trap $V(\mathbf r)=μr^{2}/2$, 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 $r=v_0/μ$, 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.

Figures (11)

• Figure 1: Illustration of a trajectory of the generalized RTP process governed by the Langevin equation (\ref{['generalizedRTP']}). Left: A sample trajectory for $D=0$, where the inter-tumbling velocities ${\sf v}_i$ are drawn from an arbitrary distribution $W(v)$. Right: The corresponding trajectory in the presence of white noise ($D>0$). In this case, the stationary distribution is given by the convolution of the $D=0$ distribution with the Gaussian distribution $\mathcal{N}(0,D/\mu)$ -- see Section \ref{['DonSection']}.
• Figure 2: Illustration of the iterative steps of a stick-breaking process, in which a stick of unit length is successively broken into infinitely many pieces from left to right. At step 1, a fraction $Y_1=\overline{U}_1$ is removed, leaving a segment of length $1 - \overline{U}_1$. At step 2, a fraction $\overline{U}_2$ of the remaining stick is broken off, producing a new piece of length $Y_2=\overline{U}_2(1 - \overline{U}_1)$ and leaving a residual segment of length $(1 - \overline{U}_2)(1 - \overline{U}_1)$. The procedure continues indefinitely, yielding pieces of lengths $Y_n$. Each $\overline{U}_n \in (0,1)$ is drawn independently from a distribution $\mathrm{Beta}(1,\alpha)$.
• Figure 3: Stationary probability density $p_X(x)$ of a single component of the RTP in dimensions $d=1$ (left), $d=2$ (center), and $d=3$ (right), for different values of the activity parameter $\alpha = \gamma/\mu$. In $d=1$ and $d=2$, the distribution undergoes a shape transition at $\alpha=2$ and $\alpha=1/2$, respectively, signaled by the emergence or disappearance of singularities at the turning points $\pm v_0/\mu$. In contrast, in $d=3$ the distribution remains finite and vanishes algebraically at the boundaries for all $\alpha$. Solid lines are exact analytical results, while symbols correspond to numerical simulations of the RTP dynamics. Parameters are $\mu = 1$ and $v_0 = 1$.
• Figure 4: Stationary radial probability density $p_R(r)$ for an RTP confined in a harmonic trap, shown for dimensions $d = 1$ (left), $d = 2$ (center), and $d = 3$ (right), and for different values of the activity parameter $\alpha = \gamma/\mu$. In all panels, we set $\mu = 1$ and $v_0 = 1$. For small $\alpha$ (high persistence), the probability mass accumulates near the turning radius $r = v_0/\mu$. At the critical value of $\alpha$ (dimension-dependent), the distribution has finite mass there. For large $\alpha$ (frequent tumbling), the distribution contracts toward the origin and approaches a Maxwell distribution. Solid lines are exact analytical predictions, while markers correspond to numerical simulations.
• Figure 5: Stationary joint distribution $P(x,y)$ of the RTP in $d=2$, shown for increasing values of the activity parameter $\alpha$. For $\alpha<1$, the probability is concentrated near the turning circle $r=v_0/\mu$. At the critical point $\alpha=1$, the density becomes uniform along the circle. For $\alpha>1$, the distribution progressively contracts toward the origin, approaching a Gaussian form in the passive limit. Parameters are $\mu = 1$ and $v_0 =1$.