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Self-propulsion symmetries determine entropy production of active particles with hidden states

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Pith's one-line read The partial entropy production rate of an overdamped active particle with hidden self-propulsion is determined by the parity and time-reversal symmetries of the hidden propulsion, and first becomes nonzero only at sixth or eighth order in…

desk verdict A useful symmetry principle for partial entropy production in hidden-state active particles, with solid derivations but asymptotic prefactors that still deserve an independent check. read the letter →

arxiv 2507.22199 v1 pith:HUZMUUZX submitted 2025-07-29 cond-mat.stat-mech

Jacob Knight , Farid Kaveh , Gunnar Pruessner This is my paper · ORCID
classification cond-mat.stat-mech
keywords partialentropyproductionrateactiveparticleshiddenself-propulsionrun-and-tumbleprocessdiffusionwithstochasticresettingtime-reversalsymmetryparityOnsager-Machluppathintegral
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 much time-irreversibility remains visible in the position of an active particle when its self-propulsion is hidden. For a particle obeying $\dot{x} = \nu w(t) + \xi(t)$ with stationary hidden $w$, it derives a perturbative expansion of the partial entropy production rate of the position process alone. The central result is that only the parity ($P$) and time-reversal ($T$) symmetries of $w$ matter: if $w$ is $PT$-symmetric the partial EPR vanishes exactly; if $w$ is $T$-symmetric but parity-broken the first nonzero term is sixth order in $\nu$; if $w$ is $P$-symmetric but time-reversal-broken it is eighth order. This explains why free active Ornstein-Uhlenbeck particles have zero partial EPR and yields explicit leading coefficients for an asymmetric run-and-tumble process and for diffusion with stochastic resetting. The practical stakes: observed spatial trajectories can look nearly reversible even when the full system dissipates strongly, so position-only estimates must reach into high-order correlations.

What carries the argument

The load-bearing object is the cumulant expansion of the partial EPR, Eq. (10), together with its symmetrised leading-order form Eq. (13). The expansion writes $\dot{S}_x$ as a sum over $n$ of integrated position-velocity correlators times differences of $n$-point cumulants of $w$, evaluated under the modified path measure $P^*[\{w(t)\}] \propto \exp\!\left(-\frac{\nu^2}{4D}\int_0^T w(t)^2\, dt\right) P[\{w(t)\}]$. The extra exponential factor is $PT$-invariant, so $P^*$ has exactly the same $P$ and $T$ symmetries as $w$ itself. The mechanism: $P$-symmetry makes all odd $w$-cumulants vanish, $T$-symmetry makes the even-order forward/reverse differences vanish, and the second cumulant cancels because stationarity makes it a function only of $|t_1-t_2|$. The first non-vanishing order $n^*$ therefore encodes which symmetry is broken, and Eq. (13) turns that order into an explicit squared moment difference, which is what the two worked examples integrate.

What would settle it

Simulate the drift-free asymmetric telegraph model and estimate $\dot{S}_x$ from the Kullback-Leibler divergence between forward and time-reversed position path distributions at several small $\nu$; if the rate scales like $(\nu^2/D)^2$, or the coefficient of $(\nu^2/D)^3$ disagrees with Eq. (15), the symmetry-selection argument fails. Alternatively, search for a stationary hidden process that is not $PT$-symmetric but still yields zero partial EPR; finding one would disprove the claimed converse.

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Extended reading notes

Core claim

The paper's central claim is that for the overdamped active particle $\dot{x} = \nu w(t) + \xi(t)$, the partial entropy production rate $\dot{S}_x$ of the position process is fixed, to leading order, by whether the hidden stationary process $w(t)$ is symmetric under sign reversal ($P$), under time reversal ($T$), or both. Writing the marginal position path probability as an Onsager-Machlup integral and expanding the logarithm of the forward/reverse path-probability ratio in cumulants of $w$ under a modified measure, the paper obtains an expansion in which odd-order terms are killed by $P$-symmetry and even-order terms by $T$-symmetry, while the $n=2$ term always vanishes in steady state. Consequently $\dot{S}_x = 0$ when $w$ is $PT$-symmetric, and the leading nonzero contribution is of order $(\nu^2/D)^3$ when only $T$ holds (asymmetric telegraph with zero mean drift, Eq. (15)) and of order $(\nu^2/D)^4$ when only $P$ holds (diffusion with stochastic resetting, Eq. (16)). The paper also argues the converse: a vanishing partial EPR forces $PT$-symmetry of the hidden propulsion, so the position process is time-reversible exactly when the hidden self-propulsion is invariant under simultaneous parity and time reversal.

Load-bearing premise

The derivation assumes that the Onsager-Machlup path integral for the marginal non-Markovian position process is well defined and that the series can be truncated at the first nonzero order, with symmetry-forbidden lower-order terms vanishing exactly even though the starred measure already differs from the bare one at order $\nu^2/D$; no rigorous error bound is given.

Editorial extensions

If this is right

  • For any hidden self-propulsion with $PT$ symmetry, the position trajectory alone shows zero entropy production even when the fully observed $(x,w)$ system dissipates at rate $\nu^2\langle w^2\rangle/D$.
  • For drift-free hidden processes, the partial EPR is suppressed to at least order $(\nu^2/D)^3$, so ordinary two-point velocity statistics cannot detect it; fourth- or sixth-order correlations are required.
  • Free active Ornstein-Uhlenbeck particles have vanishing partial EPR because their hidden process is Gaussian and only the second cumulant exists, recovering a known result without special-case path integration.
  • In an external harmonic potential the partial EPR remains zero, while anharmonic potentials generally make it nonzero at higher orders.
  • The explicit leading coefficients in Eqs. (15) and (16) give quantitative predictions that can be tested directly against simulations of asymmetric run-and-tumble particles and resetting propulsion.

Reading between the lines

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

  • Beyond the paper, the converse $PT$ criterion suggests a practical symmetry witness: if long position traces show no time asymmetry, one may certify that the hidden propulsion is invariant under simultaneous sign flip and time reversal, independent of its microscopic details.
  • Beyond the paper, the $n=2$ cancellation is generic, so every stationary Gaussian hidden propulsion in free space will have zero partial EPR; non-Gaussianity appears to be the minimal requirement for time-irreversibility to show up in the position alone.
  • Beyond the paper, the ordering of leading orders proposes a hierarchy of thermodynamic visibility: processes whose irreversibility is hardest to spot (stochastic resetting) pay the highest power of $\nu$, which may connect to known coarse-graining bounds on entropy production in partially observed systems.
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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

2 major / 5 minor

Summary. The manuscript introduces a perturbative framework for computing the partial entropy production rate (EPR) of the position process of an overdamped active particle, ẋ = νw(t) + ξ(t), where w(t) is a hidden stationary self-propulsion process. The central result is the cumulant expansion in Eq. (10), from which the authors deduce that the leading non-vanishing contribution to the partial EPR is controlled by the parity (P) and time-reversal (T) symmetries of w(t): if w(t) is PT-symmetric the partial EPR vanishes identically; otherwise the leading order n* is even for P-symmetric w(t) and odd for T-symmetric w(t). The authors apply the formalism to two concrete models, an asymmetric telegraph process with zero net drift (predicting a sixth-order contribution, Eq. (15)) and diffusion with stochastic resetting (predicting an eighth-order contribution, Eq. (16)), with explicit correlation-function calculations relegated to the supplement.

Significance. If the perturbative truncation is justified, this is a valuable and elegant contribution to the thermodynamics of partially observable systems. The symmetry criterion (P, T, PT) provides a simple organizing principle for when hidden degrees of freedom mask time-irreversibility, and the explicit leading-order predictions in Eqs. (15) and (16) are falsifiable by simulation or experiment. The supplement contains detailed, explicit correlation-function derivations for both applications, and the dimensional analysis of the results is consistent. The framework is likely to be useful for other hidden-state active-particle models, and the paper is clearly written with appropriate references to prior work.

major comments (2)
  1. [Supplement SIIA and Eq. (13)] The reduction of Eq. (10) to Eq. (13) rests on the assertion that the correlators X_n and the cumulant differences C_n vanish for all n < n* and that the leading contribution is exactly (ν^{2}/(2D))^{n*} W_{n*}^{2}, with remainder O(ν^{2n*+2}/D^{n*+1}). The supplement's argument that C_{n*} = W_{n*} + O(ν^{2}/D) and that X_{n*+1} or C_{n*+1} cannot contribute at lower order is heuristic; the text itself calls the truncation 'delicate'. No rigorous error bound or convergence control is provided for the cumulant series, despite the fact that Eqs. (15) and (16) make specific quantitative predictions with fixed coefficients. I recommend that the authors either supply a rigorous remainder estimate or verify the leading-order coefficients numerically by simulating the full Langevin dynamics at small ν^{2}/D.
  2. [Main text, after Eq. (7)] The converse statement that vanishing partial EPR forces PT-symmetry of w(t) is not established. Vanishing of the Kullback-Leibler divergence in Eq. (5) only implies equality of the forward and reversed path measures on the support of P[{x}], not that the ratio inside the logarithm in Eq. (7) equals unity for every path. The implicit characteristic-functional uniqueness argument would require the ratio to hold for the full space of periodic velocity paths, which is not shown. The 'if' direction (PT-symmetry implies zero partial EPR) is sound, but the 'only if' direction should either be proven rigorously or explicitly stated as a conjecture.
minor comments (5)
  1. [Paragraph after Eq. (5)] There is a typo: 'postion' should be 'position'.
  2. [Table I, row 2] In the entry for ˙Sx in the 'asymmetric telegraph, ⟨w⟩≠0' row, the expression '⟨w⟩^2 ν ν^2/D' contains a stray 'ν'; it should read '⟨w⟩^2 ν^2/D'.
  3. [Abstract] The phrase 'at least at sixth order in the self-propulsion velocity' is correct as a lower bound, but it could be misread as applying to the resetting example, whose leading order is eighth order. A brief clarification that the actual order depends on which symmetry is broken would help.
  4. [Supplement SIIIB] The resetting propagator in Eq. (S66) is attributed to Ref. [35], a PhD thesis. The original contribution by Evans and Majumdar (Ref. [34]) should also be cited for the propagator.
  5. [Eqs. (7)-(8)] The notation ⟨•⟩* is used in Eq. (7) before its definition in Eq. (8); consider referencing Eq. (8) directly in the text preceding Eq. (7) to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the partial EPR expansion is derived from the stated Langevin equation and the symmetry classification is read off the resulting cumulant series, not imposed as an input.

full rationale

The paper's central claim is not circular. Equation (10), and hence the leading-order Equation (13), follows by substituting the Onsager-Machlup conditional weight Eq. (3) and the independent hidden-state weight P[{w(t)}] into the definition of partial EPR Eq. (5); the velocity correlators and w-cumulants are then evaluated from the specified telegraph/resetting dynamics, not fitted. The P/T/PT symmetry classification is obtained by checking whether the cumulant difference C_n and moment difference W_n vanish for odd or even n, which is a property of the expansion rather than an input. The forward PT statement is argued directly from Eq. (7) and the converse from the non-negativity of (x−1)ln x; the same-group citation [21] is a pointer, not the load-bearing step. The two worked examples produce explicit leading-order coefficients, Eqs. (15) and (16), by direct evaluation of the relevant three- and four-time correlators. The acknowledged delicacy of truncating at n* in Suppl. SIIB is an absence of a rigorous error bound, i.e. a correctness risk, not a circular reduction. Accordingly no circular step is identified.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim is derived from the stated Langevin equation and standard stochastic thermodynamics, introducing no free parameters or invented entities. It relies on formal path-integral methods, a steady-state assumption, and several cited results for the full EPR and the resetting propagator. The most fragile premise is the unproven converse of the PT-symmetry criterion and the absence of a rigorous remainder bound for the perturbative expansion.

assumptions (8)
  • standard math The Onsager-Machlup path integral P[{x}|{w}] ∝ exp(-1/(4D)∫(ẋ-νw)^2 dt) is a valid representation of the conditional path probability.
    Invoked in Eq. (3) and Suppl. Eq. (S28); follows from Gaussian white noise statistics, but is a formal functional integral.
  • domain assumption Partial entropy production rate is defined as the rate of KL divergence between forward and time-reversed path measures (Eq. 5), and the full EPR satisfies Sx,w ≥ Sx.
    Standard stochastic thermodynamics definition, cited to [25-27]; underpins the whole calculation.
  • domain assumption The hidden process w(t) is statistically independent of x(t) and of the thermal noise ξ(t), and has a stationary distribution.
    Stated after Eq. (1); if w depended on x or ξ the factorization P[{x},{w}] = P[{x}|{w}]P[{w}] and the expansion would fail.
  • domain assumption A steady state exists on the periodic domain [0,L), and boundary and transient terms vanish in the T→∞ limit.
    Used to justify steady-state initialisation and to drop finite-horizon corrections; relies on the ring geometry.
  • standard math The cumulant expansion of the logarithm in Eq. (10) is a valid asymptotic series in ν, and the lowest non-vanishing order n* determines the leading EPR.
    Formal power series assumption; the paper does not prove convergence or bound the remainder. See Suppl. SII.
  • domain assumption The full EPR of the joint process for telegraphic w(t) is Sx,w = ν²⟨w²⟩/D, quoted from [26,32].
    Used in Table I for comparison; not re-derived in this paper.
  • domain assumption The Doi-Peliti propagator for diffusion with stochastic resetting, Eq. (S66), taken from [35], correctly describes the w-process.
    Basis for the four-point correlation function used in Eq. (16); a PhD thesis source of limited accessibility.
  • ad hoc to paper Zero partial EPR implies PT-symmetry of w(t), via uniqueness of the characteristic functional on periodic path velocities.
    Stated after Eq. (7) with a one-line proof sketch; not fully demonstrated for the restricted test-function space in a ring geometry.

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Pith. "Pith review of Self-propulsion symmetries determine entropy production of active particles with hidden states." pith.science (2026). https://pith.science/paper/HUZMUUZX

@misc{pith2026250722199,
  author       = {Pith},
  title        = {Pith review of: Self-propulsion symmetries determine entropy production of active particles with hidden states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUZMUUZX}},
  note         = {Machine review of arXiv:2507.22199}
}
read the original abstract

Entropy production distinguishes equilibrium from non-equilibrium. Calculating the entropy production rate (EPR) is challenging in systems where some degrees of freedom cannot be observed. Here we introduce a perturbative framework to calculate the ``partial EPR'' of a canonical hidden-state system, a generic self-propelled active particle with hidden self-propulsion. We find that the parity symmetry, P, and (time-)reversibility, T, of the hidden variable determine partial entropy production. Non-trivial entropy production appears at least at sixth order in the self-propulsion velocity. We apply our framework to two processes which break P- and T-symmetries respectively: an asymmetric telegraph process and diffusion with stochastic resetting.

Figures

Figures reproduced from arXiv: 2507.22199 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of three processes with (i) fully resolved [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

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