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Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents

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Pith's one-line read This paper develops a martingale-based coarse-graining method to bound the entropy-production rate of a stationary Markov chain from the first-passage statistics of a fluctuating current, yielding a refined dissipation inequality and…

desk verdict Genuinely new refined dissipation bounds and a necessary speed-symmetry condition for optimal currents, with the main finite-state results looking solid; the paper overclaims infinite-state generality in Sec. 7.1 but is well worth refereeing. read the letter →

arxiv 2507.03752 v6 pith:6YOJFU2Q submitted 2025-07-04 cond-mat.stat-mech

Adarsh Raghu , Izaak Neri This is my paper · ORCID
classification cond-mat.stat-mech MSC 60J2760F1060G4082C31 PACS 05.70.Ln05.40.-a02.50.Ga
keywords stochasticthermodynamicsfirst-passagetimesfluctuatingcurrentsentropyproductionboundseffectiveaffinitymartingalemethodslargedeviationsspeed-accuracytrade-off
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

The paper aims to establish a tighter thermodynamic bound for first-passage problems of fluctuating currents: the dissipation rate $\dot{s}$ is bounded below by $(\ell_+/\langle T\rangle)(|\ln p_-|/\ell_- + I_-(1/j))$, where $p_-$ is the splitting probability, $\langle T\rangle$ the mean first-passage time, and $I_-$ the large-deviation rate function of the first-passage time at the negative threshold. This refines the known speed-accuracy-dissipation trade-off, which involves only $p_-$ and $\langle T\rangle$, by additionally capturing the shape of the first-passage time distribution. A direct consequence is that optimal currents, for which dissipation is fully accounted for by the effective affinity, must satisfy a speed symmetry: the average speed toward the positive threshold equals the average speed toward the negative threshold. The derivation works for both continuous- and discrete-time Markov chains, thereby extending the notion of effective affinity to discrete time. This matters because the relevant quantities are experimentally accessible in systems such as molecular motors, where dwell times and backward-step probabilities can be measured.

What carries the argument

The central object is the average entropy production evaluated at the stopping time, $\langle S(T)\rangle$, expressed as a Kullback-Leibler divergence between the forward and time-reversed distributions over stopped trajectories $X_0^T$. Coarse-graining this divergence with the observable $D=\operatorname{sign}(J(T))$ yields the standard bound; coarse-graining with the pair $(D,T)$ yields the refined bound. Martingale theory carries the time-reversal step: the tilted martingale $M(t)=\phi_a(X(t))e^{-aJ(t)-\lambda_J(a)t}$, together with Doob's optional stopping theorem, connects first-passage quantities in the time-reversed chain to those in the forward chain, producing the identities $|\ln p^\dagger_+|/\ell_+ = |\ln p_-|/\ell_-$ and $I_- = I^\dagger_+$. The dual process, the Doob transform of the tilted chain at the effective affinity $a^*$, provides the conjugate process in which the generalized symmetry holds.

What would settle it

Simulate a finite-state nonequilibrium Markov chain with a known non-optimal current, evaluate $\dot{s}$, $p_-$, $\langle T\rangle$, and the rate function $I_-$ at $\tau=1/j$ for large thresholds, and check the inequality $\dot{s} \ge (\ell_+/\langle T\rangle)(|\ln p_-|/\ell_- + I_-(1/j))$; a violation at large $\ell_{\min}$ would falsify the refined bound. Alternatively, for a current predicted to be optimal, measure the two mean first-passage speeds and check whether they are equal.

Watch

Extended reading notes

Core claim

The central claim is the refined asymptotic dissipation bound (Eq. 17): for stationary Markov chains on a finite state space, with a fluctuating current $J(t)$ of positive average rate $j$, the entropy production rate satisfies $$\dot{s} \ge (\ell_+/\langle T\rangle)\left(\frac{|\ln p_-|}{\ell_-} + I_-(1/j)\right)(1+o_{\ell_{\min}}(1)),$$ which is equivalent to $\dot{s}\ge I_J(-j)$, the large-deviation rate function of the current evaluated against its typical direction. The earlier bound $\dot{s}\ge (\ell_+/\ell_-)|\ln p_-|/\langle T\rangle$ corresponds to dropping the positive term $I_-(1/j)$. The new term encodes the fluctuations of the first-passage time at the negative threshold, and its inclusion forces optimal currents (those with $\dot{s}=j a^*$) to satisfy the speed symmetry $\lim_{\ell_+\to\infty}\langle T\rangle_+/\ell_+ = \lim_{\ell_-\to\infty}\langle T\rangle_-/\ell_-$. The paper also establishes the generalized first-passage symmetry $I_-(\tau)=\hat{I}^\dagger_+(\tau)$ for generic currents, with respect to the time-reversal of the dual process defined by the Doob transform at the effective affinity.

Load-bearing premise

The derivation assumes the state space of the Markov chain is finite, so the eigenvector $\phi_a$ of the tilted martingale is bounded and optional stopping applies; the key time-reversal identity (58) is proved only under that finite-cardinality assumption, as the authors note in Section 7.1.

Editorial extensions

If this is right

  • The refined inequality $\dot{s} \ge (\ell_+/\langle T\rangle)(|\ln p_-|/\ell_- + I_-(1/j))$ strictly improves the earlier first-passage trade-off relation whenever $I_-(1/j)>0$, giving a tighter lower bound on dissipation from the same kind of measurements.
  • Optimal currents must obey the speed symmetry $\lim_{\ell_+\to\infty}\langle T\rangle_+/\ell_+ = \lim_{\ell_-\to\infty}\langle T\rangle_-/\ell_-$; this is a necessary condition for optimality, but not sufficient.
  • The effective affinity $a^*$, the exponential decay constant of the splitting probability, is well defined for discrete-time Markov chains as well, so dissipation bounds and inference schemes based on it apply beyond continuous time.
  • Every fluctuating current satisfies the generalized symmetry $I_-(\tau) = \hat{I}^\dagger_+(\tau)$, meaning the negative-threshold first-passage statistics equal the positive-threshold statistics in the time-reversed dual process.
  • For observables that are not fluctuating currents (e.g., in systems with magnetic fields), the inequalities (112) and (113) still hold, showing that the trade-off is fundamentally between dissipation and accuracy in the time-reversed dynamics.

Reading between the lines

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

  • The speed symmetry is directly testable in single-molecule experiments: if mean dwell times for forward and backward steps of a motor are found equal but the current is not optimal, the motor's positional current lies in the set $J_v\setminus J_{\rm opt}$, which would constrain thermodynamically consistent coarse-grained models.
  • The same coarse-graining-of-Kullback-Leibler-divergence-at-stopping-times scheme could generate a hierarchy of bounds by conditioning on richer functionals of $X_0^T$, such as the full empirical distribution of states, potentially approaching $\dot{s}$ from below with more detailed observations.
  • The equivalence $\dot{s}\ge I_J(-j)$ suggests a link between the refined first-passage bound and fixed-time thermodynamic uncertainty relations for current fluctuations; one could test whether the bound remains tight for currents that saturate the Gallavotti-Cohen symmetry.
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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 / 4 minor

Summary. The paper studies first-passage times T = inf{t : J(t) ∉ (−ℓ−, ℓ+)} for fluctuating currents J in stationary Markov chains with finite state space, in both discrete and continuous time. Its central result is the refined asymptotic dissipation bound s˙ ≥ (ℓ+/⟨T⟩)(|ln p−|/ℓ− + I−(1/j))(1 + o(1)), stated as Eq. (17), together with the equivalent fixed-time form s˙ ≥ I_J(−j), Eq. (18). The derivation combines a coarse-graining of the Kullback-Leibler representation of ⟨S(T)⟩, an asymptotic Wald equality, and martingale/large-deviation relations for the tilted process. The paper also derives the speed symmetry (19)/(80) for optimal currents, extends the effective affinity concept to discrete-time Markov chains, and obtains a generalized first-passage symmetry I−(τ) = Ά+(τ) via the time reversal of the dual Doob-transformed process.

Significance. If the results hold, the paper makes a genuine advance in stochastic thermodynamics: Eq. (17) refines the known first-passage trade-off relation by adding the large-deviation term I−(1/j), providing a strictly stronger bound that is verified in the numerical examples. The derivation of the time-reversal identity (58), previously conjectured, is a valuable technical contribution, and the coarse-graining-at-stopping-times method is conceptually clean. The extension of the effective affinity to discrete time is also useful, especially because the standard parabolic bound does not hold there. The finite-state central theorem is internally consistent and is built on published lemmas rather than on fitting to data; the numerical illustrations support the claims for the toy models studied. The main reservations concern the scope of the claimed generality and a few technical steps in the proof of the refined bound, all of which appear repairable.

major comments (2)
  1. [§7.1 and Appendix B.1] The statement in §7.1 that Eqs. (2) and (17) 'apply in general' to Markov jump processes on infinite-cardinality state spaces and to overdamped Langevin processes is not supported by the proof as written. Appendix B.1 uses finiteness of X to bound the Perron eigenvector φ_a and to justify Doob's optional stopping theorem, and the time-reversal identity (58) is load-bearing for passing from the time-reversed bound (52) to the forward bound (2), and similarly from (64)/(69) to (17). Since §7.1 itself concedes that (58) was derived under finite cardinality and defers the infinite-state analysis to future work, the claim of general applicability should be either removed or explicitly labeled as a conjecture.
  2. [§4.2, Eqs. (64)–(67)] The derivation of (64) relies on the replacement p_T(ℓ+τ|+) ≈ δ(τ − 1/j), which is used to evaluate the Kullback-Leibler term as ℓ+ I†+(1/j). This is a saddle-point/Laplace approximation that is not justified in the text: a large deviation principle gives exponential decay of p_T but does not by itself imply convergence of the integral of p_T against I†+ without additional assumptions on the rate functions (e.g., regularity, uniqueness of the minimizer, and control of subexponential prefactors). Since this step is necessary to obtain the refined bound (17), the authors should supply a justification or state the required assumptions explicitly.
minor comments (4)
  1. [§4.1, Eq. (62)] The second term on the right-hand side of Eq. (62) duplicates the first term, p+ ln(p+/p†+); presumably it should be p− ln(p−/p†−).
  2. [§4.2, Eq. (64)] As printed, Eq. (64) does not reduce to (17) after substitution of (58). The preceding estimates (65)–(67) suggest that the intended intermediate bound is s˙ ≥ [|ln p†+| + ℓ+ I†+(1/j)]/⟨T⟩, with the prefactor ℓ+ attached only to the rate-function term, not to |ln p†+|/ℓ−.
  3. [§6.1, text near Fig. 6] The text describing the right-hand panels of Fig. 6 writes the plotted quantity as (ŝFPR − ŝiFPR)/s˙, whereas the figure caption and the definition of ŝiFPR in Eq. (99) indicate the plotted quantity is (ŝiFPR − ŝFPR)/s˙; the sign convention should be made consistent.
  4. [§7.2, Eq. (115)] The discrete-time thermodynamic uncertainty relation (115) is stated with a brief citation to Ref. [54], but the identity (117) relating the variance of J to the conditional Fano factor of T is only sketched; a few lines of derivation would make the discrete-time extension easier to verify.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: the refined bound (17) follows from KL coarse-graining plus martingale large-deviation lemmas rederived in the appendices; self-citations are contextual, and the finite-state caveat of Sec. 7.1 is a generality gap rather than a circular step.

full rationale

The central derivation is not circular. The refined bound (17) is assembled from (i) the representation of ⟨S(T)⟩ as a Kullback-Leibler divergence on stopped trajectories (Sec. 3.1.1), (ii) Jensen coarse-graining over D=(sign J(T)) and (D,T) giving Eqs. (41) and (62), (iii) the Wald-type asymptotic ⟨S(T)⟩=sdot⟨T⟩(1+o) derived in Appendix C, (iv) the large-threshold scalings of p− and p†_+ derived in Appendix B.1, and (v) the martingale-derived relations (58) and (69) connecting forward and time-reversed first-passage quantities. The load-bearing martingale identities, including the inverse relations (70) and (138), are proved in Appendix B.2 rather than merely imported by citation; Refs. [25,31] are used as prior context and as sources of standard large-deviation facts, but the paper's own appendices carry the argument. No parameter is fitted to a subset of data, and no first-passage observable is defined in terms of the dissipation bound it is used to prove; the speed-symmetry corollary (19)/(80) follows from (17) by forcing I_-(1/j)=0 under the optimality condition sdot=ja*. The one flagged weakness is an acknowledged scope gap rather than circularity: Eq. (58) is proved under finite |X| because Appendix B.1 bounds the Perron eigenvector φ_a, while Sec. 7.1 states that 'some of the results — such as Eqs. (58) — were derived under the assumption of finite cardinality. Therefore, caution should be exercised when extending these results to cases with infinite cardinality' and nevertheless asserts extension to infinite-cardinality chains and overdamped Langevin systems. That unsupported generality claim is a correctness risk, not a circular step. The apparent typo in Eq. (64) and the unstated saddle-point/Legendre steps are presentational and repairable without changing the finite-state logical chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The bounds are derived for stationary finite Markov chains with antisymmetric currents and positive mean current, assuming local detailed balance and large-threshold asymptotics. No free parameters are fitted; the example parameters in Sec. 6 are illustrative. The refined bound uses previously published large-deviation identities from Refs. [30,31] as lemmas. No new physical entities are introduced.

assumptions (6)
  • domain assumption X is a finite, ergodic, time-reversible Markov chain in the stationary regime
    Sec. 2.1; required for optional stopping and for exponential decay of splitting probabilities.
  • domain assumption The fluctuating current satisfies c_xy = -c_yx and has nonzero mean, taken positive (j > 0)
    Sec. 2.2; the antisymmetry is used for λ_J(a) = λ†_J(-a) (Eq. 39), and j > 0 is assumed without loss of generality.
  • domain assumption Local detailed balance, so entropy production is given by Eq. (33) and exp(-S) is a Radon-Nikodym derivative
    Sec. 2.3; physical assumption of a thermal environment and even parity under time reversal.
  • standard math First-passage times satisfy large deviation principles (Eqs. 8, 9) and the martingale inverse relations (Eqs. 70, 76)
    Imported from Refs. [30, 31]; these published results are used as lemmas in deriving the refined bound.
  • standard math Doob's optional stopping theorem applies to the martingale M(t) at T, and the eigenvector φ_a is bounded
    Appendix B; boundedness relies on finiteness of the state space.
  • ad hoc to paper The saddle-point approximation p_T(ℓ+τ|+) ≈ δ(τ - ⟨T|+⟩/ℓ+) holds in the large threshold limit
    Sec. 4.2; a heuristic step used to pass from Eq. (67) to Eq. (64)/(17), requiring a unique minimum of the rate function I_+.

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Pith. "Pith review of Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents." pith.science (2026). https://pith.science/paper/6YOJFU2Q

@misc{pith2026250703752,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YOJFU2Q}},
  note         = {Machine review of arXiv:2507.03752}
}
read the original abstract

We develop a method for deriving thermodynamic bounds for first-passage problems of currents with two boundaries in Markov chains. Using this method, we derive a thermodynamic bound on the rate of dissipation in terms of the splitting probability and the first-passage time statistics of a fluctuating current, which is a refinement of a previously derived inequality. We also show that the concept of effective affinity, originally developed for continuous-time Markov chains, naturally extends to discrete-time Markov chains. Furthermore, we analyse symmetries in first-passage problems of fluctuating currents with two boundaries. We show that optimal currents -- those for which the effective affinity fully accounts for the dissipation -- satisfy a symmetry property: the current's average speed to reach the positive threshold equals the current's speed to reach the negative threshold. The developed approach uses a coarse-graining procedure for the average entropy production at random times and uses martingale methods to perform time-reversal of first-passage quantities.

Figures

Figures reproduced from arXiv: 2507.03752 by the authors.

Figure 1
Figure 1. Graphical illustration of the first passage problem Eq. (1). Multiple [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Illustrated application of the first-passage problem (1) for the case [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. A graphical illustration of the refined bound Eq. (18) for the rate [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Classification of currents according to symmetries and optimality. [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 4
Figure 4. Figure 4: 27 [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: Diagram illustrating the effects of the Doob transform (89) and the [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: Top Panel: Graphical illustration of the random walker model on a [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: (a) Graphical illustration of four state model being studied: the [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]
Figure 8
Figure 8. Figure 8: (a) Plot of ˆsFPR/s˙ (red dotted line) and ˆsiFPR/s˙ (blue dashed line) for the four state model illustrated in [PITH_FULL_IMAGE:figures/full_fig_p048_8.png]

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