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REVIEW 3 major objections 3 minor 70 references

Steady state and relaxation dynamics of run and tumble particles in contact with a heat bath

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a run-and-tumble particle in a heat bath trapped by a V-shaped potential, the exact steady state is a sum of two exponential modes, and each mode relaxes on its own time scale.

desk verdict The exact steady-state double-exponential result is solid and new; the relaxation-time claims rest on an approximation that fails in the Brownian limit, so the time-dependent part needs a major caveat or rework. read the letter →

arxiv 2506.02645 v1 pith:6A7OKHPA submitted 2025-06-03 cond-mat.stat-mech cond-mat.soft

R. K. Singh , Oded Farago This is my paper
classification cond-mat.stat-mechcond-mat.soft PACS 05.40.-a05.10.Gg05.70.Ln
keywords run-and-tumbleparticleactivematterthermalnoisesteady-statedistributionrelaxationdynamicspiecewiselinearpotentialentropyproductionLaplace
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

This paper asks how a run-and-tumble particle—a swimmer that runs straight and randomly tumbles—relaxes when it is also in contact with a heat bath and confined by a piecewise linear potential $U(x)=b|x|$. The authors establish an exact steady-state result: the position distribution is not a single Boltzmann or Laplace form but a normalized sum of two exponential distributions, $P(x)=\frac{A_1}{\lambda_1}e^{-\lambda_1|x|}+\frac{A_2}{\lambda_2}e^{-\lambda_2|x|}$, with the inverse lengths $\lambda_i$ fixed by the active speed, tumbling rate, potential slope, and temperature. The approach to this steady state is organized around the same two modes: each mode first spreads as a thermal Gaussian, then relaxes on a time $\tau_i=(\lambda_i^2 T)^{-1}$, with the distant tail reached by relaxation fronts moving at speed $2T\lambda_i$. The steady-state entropy production rate is also expressible through the mode weights and the $\lambda_i$, so a full non-equilibrium steady state and its relaxation are captured analytically for this active system.

What carries the argument

The central object is the two-mode decomposition of the position distribution, built from the coupled Fokker-Planck equations for the right- and left-moving subpopulations. In sum-and-difference variables, the steady-state equations reduce to a linear first-order system with a $3\times 3$ matrix $\mathbf{M}_3$; the two negative eigenvalues of $\mathbf{M}_3$ supply the inverse lengths $\lambda_1,\lambda_2$ of the two Laplace modes in the steady state. The time-dependent problem uses the same construction, producing a $4\times 4$ matrix $\mathbf{M}_4$ and the quartic eigenvalue equation whose two negative roots are the Laplace-space inverse lengths $\lambda_i(s)$. The argument is carried by the interpolation ansatz $\lambda_i(s)\approx\sqrt{s/T+\lambda_i^2}$ together with the coefficient ansatz $B_i(s)=A_i/s+1/(4T)$, which match the Brownian limits at small and large times and allow the Laplace transform to be inverted explicitly into Gaussian decays plus integrals that saturate to the steady state.

What would settle it

Compute the two negative roots of the quartic equation (18) numerically for parameter sets outside the three tested ones, for example very small or very large temperature $T$, and compare $\lambda_i(s)$ with $\sqrt{s/T+\lambda_i^2}$ over the full range of $s$; any substantial deviation at intermediate $s$ would invalidate $\tau_i=(\lambda_i^2 T)^{-1}$ and $v_i^*=2T\lambda_i$. A complementary experimental check is to measure the local relaxation time $t(x)$ in Langevin simulations for such parameters and test whether the large-$|x|$ slope equals $1/v_i^*$.

Watch

Extended reading notes

Core claim

The central claim is that the one-dimensional position distribution of a run-and-tumble particle with telegraphic active noise and white thermal noise, in the V-shaped potential $U(x)=b|x|$, splits into two dynamically distinct modes. At steady state this is exact: $P(x)=\frac{A_1}{\lambda_1}e^{-\lambda_1|x|}+\frac{A_2}{\lambda_2}e^{-\lambda_2|x|}$, where $\lambda_1,\lambda_2$ are the absolute values of the two negative eigenvalues of the matrix $\mathbf{M}_3$ that governs the steady Fokker-Planck system, and $A_1,A_2$ are fixed by symmetry at the origin and by normalization. In Laplace space the time-dependent solution has the same two-mode structure, with $s$-dependent inverse lengths $\lambda_i(s)$ obtained from a quartic equation. Because an exact inverse Laplace transform is impractical, the paper proposes the interpolations $\lambda_i(s)\approx\sqrt{s/T+\lambda_i^2}$ and $B_i(s)=A_i/s+1/(4T)$, which are exact in the short- and long-time limits and reproduce the numerically computed eigenvalues for the three parameter sets tested. From these follow the mode relaxation times $\tau_i=(\lambda_i^2 T)^{-1}$, the Gaussian decaying part of the transient distribution, the relaxation fronts moving at $v_i^*=2T\lambda_i$, a closed-form mean-square displacement, and the entropy-production rate expressed through the mode weights and inverse lengths.

Load-bearing premise

The time-dependent predictions rest on the approximation $\lambda_i(s)\approx\sqrt{s/T+\lambda_i^2}$ and on a hand-set initial amplitude of $1/4$ for each mode, which the paper checks numerically for only three parameter sets; if these fail elsewhere, the relaxation times, front speeds, and MSD formula are not guaranteed.

Editorial extensions

If this is right

  • The steady state of a thermally coupled run-and-tumble particle in a linear trap is exactly a two-exponential mixture, so no single effective temperature reproduces the density; a second, shorter correlation length always contributes near the origin.
  • Since $\tau_i=(\lambda_i^2 T)^{-1}$, the mode with the larger $\lambda_i$ relaxes faster; the minority short-range mode therefore decays quickly while the dominant tail mode controls the late-time approach to steady state.
  • Relaxation is center-outward: close to the origin the local relaxation time is constant, while at large distances it grows linearly with $|x|$ as a front propagates at speed $v_i^*=2T\lambda_i$; the same pattern, including a non-monotonic dip, is found for a purely Brownian particle in the same potential.
  • The mean-square displacement interpolates between the short-time thermal law $\langle x^2\rangle\approx 2T t$ and a steady-state value through a closed approximate formula that matches Langevin simulations for the parameter sets tested.
  • The steady-state entropy production rate is fixed by the two-mode splitting, $\dot S = \frac{v_0^2}{\mu k_B T}+\mu b^2\left(\frac{\phi_1}{k_B T_1}+\frac{\phi_2}{k_B T_2}-\frac{1}{k_B T}\right)$, so the density shape alone determines the dissipation.

Reading between the lines

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

  • An inference beyond the paper: if the interpolation ansatz $\lambda_i(s)\approx\sqrt{s/T+\lambda_i^2}$ holds beyond the three tested parameter sets, the same two-mode structure should appear for any piecewise-linear confining potential, and for smooth potentials one would expect a discrete or continuous spectrum of relaxation times with $\tau\sim(\lambda^2 T)^{-1}$; this is a testable extension th
  • An inference beyond the paper: the non-monotonic local relaxation time, with its dip at intermediate distances, arises from the crossing of an overshooting core and an undershooting tail; this signature should be observable in experiments that release a confined active colloid nearly from a point and track the density toward steady state.
  • An inference beyond the paper: the entropy-production formula gives a stochastic-thermodynamic reading in which the two modes act like two reservoirs at effective temperatures $k_B T_i=b/\lambda_i$; this suggests that steady-state dissipation could be inferred from density measurements alone, a connection the paper states but does not develop into a measurement protocol.
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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

3 major / 3 minor

Summary. The paper studies a one-dimensional run-and-tumble particle in a piecewise linear potential U(x)=b|x|, with both telegraphic active noise and thermal white noise. It obtains an exact steady-state position distribution as a sum of two Laplace distributions, whose inverse length scales are the two negative eigenvalues of a 3x3 matrix (Sec. 2). It then analyzes relaxation from a delta-function initial condition. The Laplace-space solution reduces to a quartic eigenvalue problem (Sec. 3); the authors approximate the two relevant eigenvalues by Eq. (24) and the mode amplitudes by Eq. (25), and use these to derive an approximate time-dependent distribution, MSD, local relaxation times, and a propagating relaxation-front velocity (Sec. 4). The paper also derives the steady-state entropy production rate in terms of the mode weights (Sec. 5).

Significance. The steady-state derivation is an elegant, exact result that is verified against Langevin simulations and has no free parameters; this part is a genuine contribution. The time-dependent analysis, if valid, gives a simple two-mode relaxation picture with testable predictions (front speed, MSD, local relaxation time). However, the time-dependent conclusions rest on two uncontrolled approximations, and one of them (Eq. (24)) demonstrably fails in a limit that is part of the model, so the unqualified relaxation-time claim in the abstract is not supported. The paper is worth revising, not rejecting: the exact steady-state core and the simulation methodology are sound.

major comments (3)
  1. [Sec. 4, Eq. (24)] The interpolation ansatz Eq. (24) is not merely unverified away from the three parameter sets: it is provably incorrect in the thermal (weak-activity) limit that is part of the model. For r→0 and v0→0, the exact Laplace-space root of the quartic (18) tends to the Brownian half-line root λ_Br(s)=(b+√(b²+4sT))/(2T), which the authors themselves write before Eq. (22). That root has a branch point at s=-b²/(4T), giving a relaxation time 4T/b². Substituting λ_i=b/T into Eq. (24) gives √(s/T+b²/T²), with branch point at s=-b²/T and hence τ_i=T/b², a factor of 4 smaller. Because τ_i=(λ_i²T)^{-1} (Eq. (28)), the front speed v_i^*=2√(T/τ_i), and the MSD (34) all inherit this error, the paper's unqualified statement in the abstract that the mode relaxation time is (λ_i²T)^{-1} is not established in the weakly active regime. The numerical checks in Figs. 3 and 4 use Tac comparable to T and thus cannot detect the failure; the authors should either restrict the claim to the regime where the approximation is controlled or provide a better approximation with a rigorous error estimate.
  2. [Sec. 4, Eq. (25)] The approximation B_i(s)≈A_i/s+α_i/T with α_i=1/4 is introduced without derivation; the text immediately acknowledges (Figs. 3(c-d)) that it deviates at intermediate s and that the resulting P_app is not normalized at intermediate times. Since the mode amplitudes control the relative weights of the two Laplace components and the short-time Gaussian splitting, the approximate time-dependent distribution (26) and the MSD (34) inherit an uncontrolled error at the very times where the two modes exchange probability. The paper should quantify the error, or better, determine α_i from the exact expression (20) at a matching point (e.g., by matching the small-s expansion), rather than setting it by hand.
  3. [Sec. 4.3] The relaxation-front velocity v_i^*=2√(T/τ_i) is derived by saddle-point evaluation of the approximate integrals in Eq. (36), so it is a property of the interpolation ansatz, not a directly computed consequence of the exact quartic solution (18). The local relaxation time t(x) in Fig. 5 is defined by an arbitrary threshold ε=0.1 and tested for one parameter set; the non-monotonic dip is also present in the Brownian case of Fig. 5(b), so this observation does not by itself discriminate between the model and ordinary Brownian motion. I would like to see at least one additional parameter set for t(x) and a statement of how t(x) depends on ε.
minor comments (3)
  1. [Sec. 4.1, Eq. (34)] The last term inside the second bracket appears to be t/τ1 e^{-t/τ2}; it should presumably be t/τ2 e^{-t/τ2}.
  2. [Throughout] There are several typos: 'Lapalce' (Sec. 3), 'normalization' (Sec. 2), 'Botzmann-like' (Sec. 6), 'confiding potential' (Sec. 5), and '10 6' should be '10^6' in figure captions.
  3. [Sec. 2, after Eq. (8)] The statement that all eigenvalues are real 'as absence of any boundaries forbids oscillatory solutions' is not self-evident; for a non-Hermitian matrix M3, realness of eigenvalues should be justified (or it can be checked from the discriminant of the cubic).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: steady-state and Laplace-space results are derived from the model; time-domain approximations are explicitly labeled as approximations and tested against simulation.

full rationale

Walking the claimed derivation chain: the steady-state distribution is obtained by solving the coupled Fokker-Planck equations exactly, with decay rates lambda_1, lambda_2 as roots of the cubic (8) and coefficients A_1, A_2 fixed by symmetry and normalization (12)-(13); no parameter is fitted to the data being predicted. The time-dependent Laplace-space solution (19)-(21) follows from the exact quartic (18), so the central two-mode structure is an exact consequence of the model. The closed-form time-domain results then use two approximations that the paper explicitly labels as such: Eq. (24) for lambda_i(s) and Eq. (25) for B_i(s), with alpha_i = 1/4 set 'based on the numerical results' and checked against Langevin dynamics in Figs. 3-5. These are presented as approximations, not as first-principles predictions, and they are benchmarked against independent simulation data; therefore they do not reduce by construction to their inputs. The self-citations present (refs. [42], [58], [70]) are background or future-work mentions and are not load-bearing; no uniqueness theorem or prior result is invoked to force the model choice. The skeptic's point that Eq. (24) may give the wrong branch point in the Brownian limit is a correctness/accuracy concern about an unchecked approximation, not circularity, because the paper never claims Eq. (24) is exact. No circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The two 'modes' are mathematical components of the exact and approximate distributions, with amplitudes and decay rates derived from the model parameters.

free parameters (1)
  • alpha_i (mode amplitude relaxation coefficient) = 1/4
    In Eq. (25), B_i(s) approximately A_i/s + alpha_i/T. The paper sets alpha1=alpha2=1/4 using numerical results in Figs. 3(c-d) and the normalization condition alpha1+alpha2=1/2. This is a hand-chosen coefficient in the approximate time-dependent solution.
assumptions (5)
  • domain assumption The active and thermal noises in Eq. (1) are independent, Markovian, and fully described by the Fokker-Planck equations (3).
    Standard RTP model with white thermal noise, cited to refs [43,59]. Underlies all derivations.
  • domain assumption For a piecewise linear potential, the steady-state solution vector must be a combination of decaying exponentials; the positive eigenvalue of M3 is discarded because it would diverge at large x.
    Section 2, after Eq. (9). Relies on absence of boundaries and normalizability.
  • ad hoc to paper The interpolation lambda_i(s) approximately sqrt(s/T+lambda_i^2) captures the exact eigenvalues over the whole Laplace domain.
    Eq. (24). Only correct in the limits s->0 and s->infinity; verified numerically for three parameter sets in Fig. 3(a-b).
  • ad hoc to paper The coefficients B_i(s) are approximated as A_i/s + alpha_i/T with alpha_i=1/4.
    Eq. (25). Chosen to match asymptotic limits and normalization; deviates at intermediate s as admitted in Section 4.
  • standard math Saddle-point approximation is valid for the integrals in Eq. (29) at large times.
    Used to derive the relaxation front and Gaussian tail in Section 4.3.

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Pith. "Pith review of Steady state and relaxation dynamics of run and tumble particles in contact with a heat bath." pith.science (2026). https://pith.science/paper/6A7OKHPA

@misc{pith2026250602645,
  author       = {Pith},
  title        = {Pith review of: Steady state and relaxation dynamics of run and tumble particles in contact with a heat bath},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6A7OKHPA}},
  note         = {Machine review of arXiv:2506.02645}
}
abstract

We study the relaxation dynamics of a run and tumble particle in a one-dimensional piecewise linear potential $U(x)=b|x|$, from delta-function initial conditions at $x=0$ to steady state. In addition to experiencing active telegraphic noise, the particle is in contact with a heat bath at temperature $T$ that applies white thermal noise. We find that the position distribution of the RTP is described by a sum of two distributions ("modes"), each of which of the form $P(x,t\to\infty)\sim e^{-\lambda_i|x|}$ ($i=1,2$) at steady state. The two modes are dynamically coupled: At very short times ($t\to 0$), each mode stores half of the probability, and exhibits thermal diffusive spreading with a Gaussian profile. With progressing time and evolution toward steady state, the partition of probability between the modes becomes increasingly uneven and, depending on the model parameters, the mode with the smaller value of $\lambda_i$ may carry an overwhelming majority of the probability. Moreover, we identify that the characteristic relaxation time of each mode is $\tau_i=(\lambda_i^2T)^{-1}$, which implies that the minority mode also relaxes much faster than the dominant one. A more detailed analysis reveals that $\tau_i$ is characteristic of the mode relaxation only close to the origin at the core of the distribution, while further away it increases linearly with $|x|$ as if a relaxation front is propagating at constant speed $v_i^*=2\sqrt{T/\tau_i}$ in the system. The rate of non-equilibrium entropy production can be related to the two-mode splitting of the probability distribution and be expressed in terms of their correlation-lengths $\lambda_i$ and their contributions to the steady state distribution.

Figures

Figures reproduced from arXiv: 2506.02645 by the authors.

Figure 1
Figure 1. (a) The SSD, P(x), of the RTP obtained from Langevin dynamics simulations (red circles) and the analytical result in Eq. (11) (black dashed line). All the model parameters are set to unity: r = 1, v0 = 1, b = 1, and T = 1, for which we have (see details in the text): λ1 ≈ 0.69, λ2 ≈ 2.48, A1/λ1 ≈ 0.32, and A2/λ2 ≈ 0.07. (b) Enlargement of the central region in (a). The green dashed lines depicts the large |x| asympt… view at source ↗
Figure 2
Figure 2. (a) The eigenvalues and (b) the normalizations coefficients defining of the SSD, as a function of the tumbling rate r. The model parameter are set to: b = 1, T = 1 and Tac = v 2 0/2r = 1/2. A1/λ1 ≈ 0.32 and A2/λ2 ≈ 0.07, i.e., A1 ≈ 0.22 and A2 ≈ 0.17. For comparison, the inverse length scales corresponding to Brownian and RTP without thermal contact dynamics would be, respectively, λBr = b/T = 1, and λRTP = b/T∗ = 2… view at source ↗
Figure 3
Figure 3. (a-b) The dependence of the eigenvalues λ1 and λ2, and (c-d) the coefficients B1 and B2, on the Laplace variable s. Symbols indicate the numerically determined values and the dashed lines represent the approximation of Eq. (24). Model parameter values are: r = 1, T = 1, b = 1, v0 = 1 (green, squares); r = 4, T = 2, b = 1, v0 = 3 (yellow, circles); r = 3, T = 4, b = 1, v0 = 2 (red, triangles). the temperatures is muc… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: MSD, ⟨x 2 (t)⟩, of the RTP evaluated from Langevin dynamics simulations (red circles) vs. the predictions of Eq. (34) (black dashed line). Model parameter values are (a) r = 1, T = 1, b = 1, and v0 = 1; (b) r = 4, T = 2, b = 1, and v0 = 3; and (c) r = 3, T = 4, b = 1, …
Figure 5
Figure 5. Figure 5: (a) Local relaxation time, t(x), for the RTP obtained by Langevin dynamics simulations with parameters b = 1, r = 1, v0 = 1 and T = 1. (b) The same for a Brownian particle with b = 1 and T = 1. The RTP results are obtained from Langevin dynamics simulations, while the …
Figure 6
Figure 6. Figure 6: Probability distribution,P(x, t), of the RTP at times (a) t = 0.5, (b) t = 1.5, (c) t = 2.5, and (d) t = 4.5. Red lines - Langevin dynamics simulations; Black dashed line - the SSD. Langevin dynamics simulations were conduced with the Euler integration method, with tim…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.