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

T0 review · 0 major / 7 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The stationary state of a run-and-tumble particle in a harmonic trap is exactly solvable in any dimension, and in three dimensions takes a closed but non-beta form.

desk verdict The d=3 closed-form stationary densities are genuinely new, the Kesten/Dirichlet-process method is clean, and the paper deserves serious peer review. read the letter →

arxiv 2602.08436 v2 pith:KVOMSJH2 submitted 2026-02-09 cond-mat.stat-mech cond-mat.dis-nncond-mat.softmath.PR

Mathis Guéneau , Satya N. Majumdar , Grégory Schehr This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nncond-mat.softmath.PR
keywords run-and-tumbleparticleharmonictrapstationarydistributionDirichletprocessstick-breakingKestenrecursionshapetransitionthermalnoise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Run-and-tumble particles confined by a harmonic trap reach a nonequilibrium steady state that, despite decades of study, had no exact description beyond one and special two-dimensional cases. This paper closes that gap by showing that the full stationary distribution in any dimension is encoded in the marginal distribution of a single Cartesian coordinate, which itself follows from a generalized one-dimensional model with arbitrary post-tumble velocities. Solving that model through a Kesten recursion and its stick-breaking (Dirichlet-process) representation gives closed-form densities and moments for every dimension. Specializing to the projected velocities of an isotropic tumbling particle yields explicit radial laws: beta distributions in d=1 and d=2, and a genuinely different closed form in d=3. Adding thermal noise simply convolves these results with a Gaussian, regularizing the turning-point singularities and producing a finite-temperature shape transition.

What carries the argument

The central object is the Kesten recursion x_n = U_n x_{n-1} + V_n for the position just after each tumble, with U~Beta(α,1). Unrolling the recursion maps the stationary position to a weighted sum of the post-tumble velocities, whose random weights are exactly the stick-breaking weights of a Dirichlet process of concentration α and base measure W(v). This 'mean functional of a Dirichlet process' representation yields, via a known identity, closed-form densities and moments for arbitrary W(v). Combined with rotational invariance — the radial and joint densities are integral transforms of the single-coordinate marginal — it turns the d-dimensional stationary problem into a one-dimensional calc

What would settle it

Simulate a three-dimensional run-and-tumble particle in a harmonic trap with μ=1, v0=1, and α=0.5, and measure the stationary radial histogram near the turning surface. If p_R(r) does not scale as ε^{α-1} (with log corrections) as ε = 1 - r → 0⁺, the closed form (83) is wrong.

Watch

Extended reading notes

Core claim

The paper's central claim is that the stationary position of a d-dimensional run-and-tumble particle in an isotropic harmonic trap is fully determined by the one-coordinate marginal p_X(x), and that p_X(x) for any post-tumble velocity law W(v) is exactly the density of a mean functional of a Dirichlet process. This identification turns an intractable nonlocal Fokker-Planck equation into closed-form expressions: p_X(x) is given by an integral formula involving a simple function φ_α, and all moments are Bell polynomials in the moments of W(v). For the isotropic d-dimensional RTP the projected velocity W(v) is the Beta-type law of Eq. (7); the resulting radial distribution is a beta law in d=1

Load-bearing premise

The derivation assumes that the infinite stick-breaking sum representing the stationary position actually converges to the unique invariant law of the Kesten recursion for every post-tumble velocity distribution W(v); the paper invokes the representation but does not prove this ergodicity step for arbitrary W(v).

Editorial extensions

If this is right

  • In d=1 and d=2, the radial stationary law is exactly a beta distribution, with a persistence-controlled shape transition at the turning radius.
  • In d=3, the radial distribution is no longer beta but is given in closed form, and it still exhibits a shape transition at α=1.
  • Thermal noise D>0 leaves the steady state as a Gaussian convolution of the noise-free law, so all turning-point singularities are rounded and the distribution acquires Gaussian tails beyond r=v0/μ.
  • The Dirichlet-process representation yields exact stationary states for N-state run-and-tumble models (discrete velocity sets) with piecewise-continuous densities.
  • The moments of the stationary position are given in closed form for any dimension and any W(v) via Bell polynomials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same machinery should extend to heterogeneous run speeds: if v0 is itself drawn from a distribution at each tumble, the single-coordinate formula (16) remains unchanged, so the whole d-dimensional solution carries over with the modified projected velocity law.
  • Because the Dirichlet-process representation is linear in the velocities, the approach likely also applies to linear switching dynamics, such as Brownian motion with a stochastically switching trap stiffness; this may yield exact two-time or multi-particle correlations.
  • The d=3 non-beta character suggests that for all d≥3 the radial law is not a beta distribution, with a dimension-dependent family of closed forms; the d=2 beta law may be a special coincidence linked to the planar projection being arcsine.
  • The finite-D two-step transition in d=3, with a discontinuous jump of the global maximum, resembles a first-order transition and could be probed experimentally in optical-tweezer setups with artificial swimmers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. The paper derives the exact stationary state of a run-and-tumble particle (RTP) in an isotropic harmonic potential in d dimensions. It first solves a generalized one-dimensional RTP with arbitrary post-tumble velocity distribution W(v) via a Kesten recursion, representing the stationary position as an infinite stick-breaking sum equivalent to a mean functional of a Dirichlet process. Using the Cifarelli-Regazzini inversion, it obtains closed-form expressions for the single-coordinate density (Eq. 45) and moments (Eq. 47). Specializing W(v) to the projected velocity law of an isotropic RTP, the paper reconstructs the radial distribution p_R(r) and the full joint stationary density. In d=1 and d=2 the radial law is a beta distribution, while in d=3 the authors derive a new closed-form p_R(r) (Eq. 83) and joint density P(x,y,z) (Eq. 85) that are not beta. The paper also treats thermal noise D>0, showing that the stationary law is a Gaussian convolution of the D=0 law, and analyzes the resulting shape transitions, including a two-step/discontinuous maximum transition in d=3. All predictions are compared with numerical simulations.

Significance. This is a significant contribution. The d=3 closed-form stationary distribution closes a gap that had remained open despite several recent efforts (Refs. 39-41). The Dirichlet-process/stick-breaking route is elegant and likely transferable to other linear switching systems. The d=2 result correctly reproduces the independent beta law of Frydel, and the d=3 formulas are new and nontrivial. The method yields parameter-free exact expressions and explicit moments via Bell polynomials. The finite-D analysis, including the universal low-D scaling form near the turning surface and the first-order-like jump of the global maximum in d=3, provides concrete testable predictions. The only formal gap—the convergence of the Kesten recursion—is covered by the classical Kesten theorem because E[log U]=-1/alpha<0 and V_n is bounded; I do not regard this as a load-bearing flaw.

minor comments (7)
  1. [Sec. III B, Eq. (42)] The notation 'U_n = 1 - U_n, U_n ~ Beta(1, alpha)' is confusing because the same symbol is used for the original and transformed variables. Suggest using, e.g., V_n or U'_n for the transformed variable to avoid ambiguity.
  2. [Sec. IV C, Eq. (73)] The statement that p_X(x) in Eq. (73) 'has been checked numerically' to satisfy the integro-differential equation (74) is a consistency check, not a proof. Since the derivation of Eq. (73) from the general formula is independent, this is acceptable, but it should be labeled as a numerical consistency check rather than a verification of the main result.
  3. [Appendix C, Eq. (C5)] The Cifarelli-Regazzini identity is quoted without proof. This is acceptable because it is a published theorem (Ref. [75]), but the paper would be more self-contained if it stated the precise theorem and its applicability conditions (e.g., bounded f(v)=v/mu and finite base measure) in one sentence.
  4. [References [79] and [80]] References [79] and [80] appear to have identical titles. Please check whether they are distinct works or whether one citation is erroneous.
  5. [References [82] and [95]] References [82] and [95] are Wikipedia links. For a journal submission, consider replacing them with standard textbook or DLMF citations for Bell polynomials and Appell hypergeometric functions.
  6. [Abstract and Title] The title and abstract emphasize the 'd-dimensional' solution, but explicit closed-form radial and joint distributions are worked out for d=1,2,3. For d>3, the result is an exact integral representation of the single-coordinate marginal plus explicit moments. This is not a flaw, but the abstract could be more precise by saying 'closed forms in d=1,2,3 and an exact integral representation in general d'.
  7. [Fig. 8] The figure labels/caption appear to mix the symbols theta_c and mu_c. Please check that the axis labels and critical lines are consistently denoted (theta_c(alpha) for d=1,2).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the closed-form stationary laws are derived by solving the Kesten recursion and using the external Cifarelli-Regazzini theorem, benchmarked against independent d=2 results and simulations.

full rationale

The paper contains no fitted-parameter or self-referential step that would make a prediction equivalent to an input. The stationary law is derived by solving the Kesten recursion (Eq. (27)) exactly; the resulting infinite sum (Eq. (41)) is rewritten as a stick-breaking sum (Eq. (43)) whose weights are Beta(1,alpha), and the closed form (Eq. (45)) is obtained from the external Cifarelli-Regazzini theorem (Ref. [75], restated in Appendix C). The theorem is not proved in the paper, but it is an external mathematical result with conditions satisfied here; citing it is not circular. The d=2 beta law is checked against the independent result of Frydel [39], the d=3 expressions are new and compared to simulations, and the D>0 convolutional structure follows from linearity and independence of noises. Self-citations (e.g., Refs. [24,71,72,86,89,93,94]) are methodological or contextual; in particular the Kesten recursion and MGF are rederived in Sec. III A and Appendix B, so the derivation does not reduce to those citations. The only soft spot is that ergodicity/uniqueness of the Kesten recursion is invoked with a citation to Kesten [74] rather than proved, but this is an external theorem and the stated conditions (E[log U]<0, bounded V) hold. No circular step is therefore identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters fitted to data; model parameters μ, v_0, γ, D are physical inputs. The axioms are standard Kesten/Dirichlet theory plus isotropy and noise-independence assumptions. The paper introduces no new physical entities.

assumptions (4)
  • domain assumption Existence and uniqueness of the stationary law of the Kesten recursion X_n = U_n X_{n-1} + V_n for all α>0 and all bounded W(v).
    Invoked when passing from Eq. (39) to Eq. (41) and in the MGF solution; not proved in the paper. If multiple invariant measures existed, p_X would not be unique.
  • domain assumption Rotational invariance of the full process implies the single-coordinate marginal is sufficient: X = R n_1 with n_1 uniform on the unit sphere and independent of R (Eqs. A7–A9).
    Used throughout to derive p_R from p_X; requires isotropic tumbling and isotropic harmonic trap, and stationarity.
  • standard math Cifarelli–Regazzini theorem: the cumulative distribution of a mean of a Dirichlet process DP(α,W) is given by M(x)=1/π ∫ dt (x-t)^{α-1} φ_α(t).
    Basis of the central formula Eq. (45); imported from Ref. [75].
  • domain assumption When D>0, the active noise and thermal noise are independent, so z = x + y with y an OU process independent of x; hence p_Z = Gaussian * p_X (Eq. 89).
    Used in Sec. V for all finite-D results; requires independence of v(t) and η(t).

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Pith. "Pith review of Exact Stationary State of a $d$-dimensional Run-and-Tumble Particle in a Harmonic Potential." pith.science (2026). https://pith.science/paper/KVOMSJH2

@misc{pith2026260208436,
  author       = {Pith},
  title        = {Pith review of: Exact Stationary State of a $d$-dimensional Run-and-Tumble Particle in a Harmonic Potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KVOMSJH2}},
  note         = {Machine review of arXiv:2602.08436}
}
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.

Figures

Figures reproduced from arXiv: 2602.08436 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of a trajectory of the generalized RTP process governed by the Langevin equation ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the iterative steps of a stick-breaking process, in which a stick of unit length is successively broken into [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Stationary probability density [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Stationary radial probability density [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Stationary joint distribution [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Cross sections of the stationary density [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Stationary radial probability density [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Diagrams in the ( [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Universal low- [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of the analytical prediction Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]

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