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

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.

💡 Research Summary

This paper presents an exact analytical solution for the nonequilibrium steady state of a run‑and‑tumble particle (RTP) confined in an isotropic harmonic potential V(r)=μr²/2 in arbitrary spatial dimension d. The authors exploit rotational invariance to reduce the full d‑dimensional problem to the stationary marginal distribution of a single Cartesian coordinate, p_X(x). Once p_X(x) is known, the radial distribution p_R(r) and the full joint density of all coordinates follow from explicit integral transforms.

The analysis begins with a generalized one‑dimensional RTP in which the post‑tumble velocity is drawn from an arbitrary distribution W(v) supported on a finite interval. By writing the position update as a Kesten‑type linear recursion, X_{n+1}= (γ/(γ+μ)) X_n + (μ/(γ+μ)) v_n, the authors recognize that the stationary position can be represented as an infinite stick‑breaking (Dirichlet‑process) construction. This representation yields a closed‑form expression for p_X(x) (Eqs. 16‑17) and for all moments ⟨X^n⟩ in terms of Bell polynomials and the moments of W(v) (Eq. 18).

Specializing to the physical RTP, the post‑tumble velocity component follows the projection of an isotropically oriented vector of fixed magnitude v₀. The projected distribution W_proj(v) is the arcsine law for d=2 and uniform for d=3. Substituting this W(v) into the general formulas gives the exact stationary statistics of each coordinate for any d≥2.

For d=1 and d=2 the radial distribution turns out to be a beta distribution: p_R(r) ∝ r^{2α−1} (1−(μr/v₀)²)^{α−1}, 0≤r≤v₀/μ, where α=γ/μ is the ratio of the tumbling rate to the trap relaxation rate. The density exhibits a shape transition at α=1: for α<1 (high persistence) the distribution diverges at the turning surface r=v₀/μ, whereas for α>1 it vanishes there, producing a unimodal profile centered at the origin.

In three dimensions the radial law does not reduce to a beta function. The authors derive a novel closed‑form expression (Eqs. 21‑23) involving the auxiliary functions A(r) and f(u). At the critical point α=1 the expression simplifies to a combination of elementary functions (arctanh, cosine, sine). As in lower dimensions, a persistence‑controlled transition occurs at α=1, but the three‑dimensional case displays richer phenomenology: an inner peak at the origin can coexist with an outer shell near r=v₀/μ, and the global maximum may jump discontinuously as parameters vary.

Thermal noise is incorporated by adding an independent Gaussian white noise term with diffusion coefficient D to the Langevin equation. Because the dynamics remain linear, the stationary distribution for D>0 is simply the convolution of the D=0 solution with a Gaussian of variance D/μ (Eq. 15). This convolution regularizes the turning‑point singularities, generates Gaussian tails beyond r=v₀/μ, and introduces a dimensionless temperature‑like parameter θ=2μD/v₀² (or equivalently θ=2α D/D_eff). As θ→0 the system recovers the pure active‑persistent regime; as θ→∞ it crosses over to a purely diffusive Boltzmann‑Gaussian profile. The authors map out the full θ‑α phase diagram, showing how the bimodal structure emerges in d=1,2 and how a coexistence region with inner and outer maxima appears in d=3.

Finally, the framework is extended to an N‑state RTP where the velocity can take N discrete values {v_i} with probabilities {p_i}. The stationary marginal p_X(x) becomes the distribution of a linear combination of a Dirichlet random vector, expressed as a multiple integral (Eq. 26). This result provides a tractable description of experimentally relevant models with a finite set of speed states.

All analytical predictions are validated against extensive numerical simulations, showing perfect agreement. The paper thus fills a long‑standing gap by delivering exact steady‑state distributions for active particles in harmonic traps in any dimension, clarifying the role of persistence, dimensionality, and thermal fluctuations, and offering a versatile toolbox for future theoretical and experimental studies of confined active matter.