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REVIEW 3 major objections 7 minor 93 references

Fluctuation Theorems for Heat exchanges between passive and active baths

T0 review · 3 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A heat-exchange fluctuation theorem whose slope is set by inverse-temperature differences still holds when one bath is active, but the recovered temperature depends on how heat is defined.

desk verdict A clean numerical demonstration that heat-fluctuation-theorem temperatures for an active bath are definition-dependent; the qualitative result holds up, but the quantitative values need uncertainty and convergence analysis before they should be trusted as benchmarks. read the letter →

arxiv 2412.07706 v1 pith:HDK2PNZ3 submitted 2024-12-10 cond-mat.stat-mech

Massimiliano Semeraro , Antonio Suma , Giuseppe Negro This is my paper · ORCID
classification cond-mat.stat-mech PACS 05.40.-a05.70.Ln
keywords fluctuationtheoremheatexchangeactiveOrnstein-Uhlenbeckparticledouble-wellpotentialkinetictemperatureeffectivelargedeviationprincipleout-of-equilibrium
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 whether the standard heat-exchange fluctuation theorem, $I(-q)-I(q)=(1/T_1-1/T_2)q$, which relates the slope of the large-deviation rate function to the bath temperatures, survives when one of the two baths is an active, out-of-equilibrium bath. The setup is a single particle in a double-well potential, with a passive bath in one well and either another passive bath or an active bath (a passive bath plus an Ornstein-Uhlenbeck coloured noise) in the other; the heat per unit time $q$ is measured two ways: as the kinetic energy the particle carries when it jumps between wells, and as the work it does on the passive bath. The paper finds the theorem holds in every case, for both definitions. In the active case the temperature extracted from the fluctuation-theorem slope is definition-dependent: for jump-energy heat it matches the kinetic temperature of the active bath, while for work heat it is larger, lying between the kinetic and effective temperatures. If correct, this provides a workable way to assign a temperature to an active bath and shows that out of equilibrium the temperature registered by a fluctuation theorem depends on the observable used to define heat.

What carries the argument

The argument is carried by the large-deviation rate function $I(q)$ for the heat per unit time, through the identity $I(-q)-I(q)=(1/T_r-1/T_l)q$, evaluated operationally as the slope of $\ln[p(q)/p(-q)]/\tau$ after the sub-exponential contributions decay. The second element is the double-well geometry that spatially separates the two baths, with the particle mediating heat through jumps. The third is the choice of two heat observables: $q_E$, the sum of kinetic energies carried by the particle at each well crossing, and $q_W$, the Stratonovich work done by the particle against the passive bath. For the active case, the fourth element is the Ornstein-Uhlenbeck coloured noise added to the white-noise bath in the left well (an Active Ornstein-Uhlenbeck particle), with Péclet number controlling activity. Together these pieces turn a fluctuation theorem into a thermometer whose reading depends on the time scale of energy release captured by the chosen heat definition.

What would settle it

Compute the fitted slope of Eq. (21) at sampling times extending well beyond $\tau=3\times10^4$ for $\mathrm{Pe}=50$, or obtain the rate functions by an independent large-deviation estimator; if the slope changes with $\tau$ once the sub-exponential shift is removed, then $T_{FT}$ is a finite-time fit parameter rather than a thermodynamic temperature.

Watch

Extended reading notes

Core claim

The paper's central claim is that the finite-time form of the heat fluctuation theorem, $\ln[p(q)/p(-q)]/\tau \simeq (1/T_r-1/T_l)q$, holds for heat exchanged between two spatially separated baths even when one bath is active, and that the fitted temperature $T_{FT}$ is physically meaningful but depends on the heat observable. For two equilibrium baths, both heat definitions give slopes equal to the true bath temperatures, and at long times the two rate functions $I(q_E)$ and $I(q_W)$ coincide. With the active bath (an Ornstein-Uhlenbeck noise added in the left well, with $T_1=T_2=0.2$), the theorem still holds, but at Péclet number 50 the temperature extracted from the jump-kinetic-energy heat is $T_{FT}^{q_E}\simeq 0.3$, equal to the kinetic temperature of the active bath, while the temperature extracted from the work heat is $T_{FT}^{q_W}\simeq 0.42$, between the kinetic and effective temperatures. The explanation offered is the active noise's run-up effect: the particle is pushed toward the barrier, arrives at the right well with extra velocity, and dissipates the excess not instantaneously at the crossing but during its descent, so instantaneous jump heat registers the kinetic temperature while work heat includes the delayed dissipation.

Load-bearing premise

The temperature extraction assumes that the longest simulation times already sit in the asymptotic regime where the heat distributions have their final exponential decay shape, so the fitted slope equals the true infinite-time rate-function difference; this convergence is checked only by visual overlap of the curves, not by an independent test.

Editorial extensions

If this is right

  • The fluctuation theorem works for spatially separated equilibrium baths, extending the earlier simultaneous-bath results to the double-well geometry.
  • For the active bath, the fluctuation-theorem temperature extracted from jump-energy heat $q_E$ equals the kinetic temperature, so this heat definition gives an instantaneous, velocity-based temperature.
  • For work heat $q_W$, the fluctuation-theorem temperature is systematically larger than the kinetic temperature and smaller than the effective one, so different heat observables probe different relaxation timescales of the active bath.
  • At all Péclet numbers studied, the ordering $T_{kin}^{dw}\simeq T_{FT}^{q_E}<T_{FT}^{q_W}<T_{eff}^{dw}$ holds, suggesting that the active bath exhibits a hierarchy of fluctuation-theorem temperatures rather than a single one.
  • The equality of rate functions $I(q_E)=I(q_W)$ in the passive case and their inequality in the active case provides a quantitative signature of active nonequilibrium behavior in heat statistics.

Reading between the lines

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

  • An implication the paper leaves implicit: the gap between $T_{FT}^{q_E}$ and $T_{FT}^{q_W}$ is a ready-made experimental probe of active memory; it should grow with the persistence time $\tau_p$ and collapse to zero as $\tau_p\to 0$, where the active bath becomes passive.
  • A testable extension would apply the same slope analysis to the joint heat that includes the active work, $q_L^W - w_a$; whether Eq. (21) remains linear there would decide if the fluctuation-theorem temperature is a property of the passive component alone or of the whole active bath.
  • If the linear-slope behavior persists for $q_E$ at higher Péclet numbers and in other potentials, fluctuation-theorem thermometry could be used in experiments on Janus-particle suspensions or optically trapped tracers to read the kinetic temperature of an active environment through heat statistics alone.
  • Read as a statement about temperature, the results suggest that an active bath does not have a unique fluctuation-theorem temperature; the temperature is a function of the observable and the timescale over which energy is exchanged.
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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 / 7 minor

Summary. The paper numerically studies heat exchange between two baths that are spatially separated by a double-well potential, with a Brownian particle hopping between the wells. Two heat definitions are used: the kinetic energy carried during jumps (q_E) and the work performed by the particle on the passive bath (q_W). The authors test the fluctuation theorem (Eq. 21), which relates the large-deviation slope to the inverse temperatures of the two baths. In the passive-passive case both heat definitions are reported to satisfy the theorem with the fitted temperatures matching the bath temperatures. In the active case, where the left bath contains an additional Ornstein-Uhlenbeck active noise, the theorem is reported to hold for both heat definitions; the extracted temperature T_FT equals the kinetic temperature for q_E and lies between the kinetic and effective temperatures for q_W. The results are interpreted through the jump phenomenology of the particle, in particular the run-up effect of the active force before a jump.

Significance. If the results are robust, the paper makes a useful contribution to the debate on out-of-equilibrium temperatures for active baths: it provides a concrete setup in which a fluctuation-theorem-based temperature is measurable and differs according to the observable, showing that different heat observables probe different time scales of energy exchange. The manuscript is clearly written and the numerical protocol is well specified, including the parameter choices, the restart protocol for noises, and the comparison with analytical harmonic-limit expressions for kinetic and effective temperatures. The comparison of T_FT with independently defined kinetic and effective temperatures is a genuine external benchmark. The main weakness is that the active-bath claim rests entirely on finite-time convergence of fitted large-deviation curves, and the manuscript does not provide quantitative convergence diagnostics or uncertainties for the extracted temperatures, which is load-bearing for its central message.

major comments (3)
  1. [§3.2, Figs. 7(a)-7(c)] The central active-bath claim for q_W rests on the assertion that -ln(p(q_R^W))/tau has converged to the rate function I(q) at the largest sampling times. The evidence in Fig. 7(a) and its inset is not sufficient: the main panel shows curves that are still separating as tau grows, and the inset is described only as showing overlap without a quantitative threshold. Fig. 7(c) shows p(q_R^W) shifted relative to p(q_R^E) at tau = 3e4; the authors interpret this as a physical effect, but a finite-time contribution to the rate function would produce the same observation. Since T_FT^{q_R^W} about 0.42 is obtained by fitting Eq. (21) at tau = 3e4 with T_r = T_1 fixed, this value could be a fit parameter rather than a thermodynamic temperature. Please provide quantitative convergence diagnostics (e.g., collapse of the curves at successive tau, local estimates of I(q) and I(-q)-I(q) with error bars), report the fitted q-range, and propagate the resulting uncertainty to T_FT.
  2. [§3.2, Table 1] The quantitative claims that T_FT^{q_R^E} coincides with T_kin^dw and that T_FT^{q_R^W} lies strictly between the kinetic and effective temperatures are made without any reported uncertainties. Table 1 lists point values only, and the text describes agreement at the level of two significant digits. For a numerical paper whose main result is a set of temperature comparisons, bootstrap or trajectory-to-trajectory standard errors on T_FT, T_kin, and T_eff are needed; without them the 'exact coincidence' claim is not assessable.
  3. [§2.3, Eq. (21)] The extraction of T_FT assumes that the finite-time ratio (1/tau) ln[p(q)/p(-q)] has reached the asymptotic form I(-q)-I(q). For q_E the inset of Fig. 6(b) gives only visual evidence of overlap at tau > 1e4, with no estimate of the sub-exponential contribution (c(q)-c(-q))/tau. Since the paper offers no analytical derivation of the active-case FT, the numerical convergence is the only support for the claim. Please add a quantitative convergence test for q_E as well, and state the uncertainty of the fit slope for each entry in Table 1.
minor comments (7)
  1. [§2.2, Eq. (14)] In the definition of Delta U(x(tau)) the text writes U(x(tau)) - U(x(tau)); this should be U(x(tau)) - U(x(0)).
  2. [§2.1] The word 'tempexrature' appears in the description of the right-well bath; it should be 'temperature'.
  3. [§2.3] The word 'superscrpits' should be 'superscripts'.
  4. [§2.2, Eq. (20)] The phrase 'come to depend depend' contains a duplicated word; it should read 'come to depend on'.
  5. [§3.2, parenthetical after Fig. 6(a)] The parenthetical remark that p(q_R^W) is 'just symmetrical' is inconsistent with Fig. 7(c), which shows an asymmetric p(q_R^W); please correct the remark.
  6. [Fig. 7(a) and accompanying text] The expression '-ln(q_R^W)/tau' appears to be missing the distribution symbol; it should read '-ln(p(q_R^W))/tau'.
  7. [References] Reference [57] has a malformed URL ('https://bookstore.ams.orgsza/view?...'); please update it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the active-bath T_FT is explicitly a fit parameter, and the central comparison to T_kin and T_eff is benchmarked by separately defined temperatures.

full rationale

The paper's derivation chain is not circular. In the passive-bath case, Eq. 21 is tested against known bath temperatures T1 and T2, providing an external check of the numerical heat-exchange fluctuation theorem. In the active-bath case, Section 2.3 states plainly that "the estimates for Tr and Tl, in the following denoted as T_FT, are obtained by ... performing at each of such times a linear fit of the resulting curves," and Section 3.2 reports the fitted values T_FT ~ 0.3 and T_FT ~ 0.42. The subsequent claim that T_FT^E matches T_dw_kin and T_FT^W lies between T_dw_kin and T_dw_eff is an external comparison, because T_kin and T_eff are computed independently from equipartition (Eq. A.7) and the fluctuation-dissipation relation (Eq. A.5), not from the fluctuation-theorem slope. The assertion that an FT-like relation holds for the active bath rests on the empirical linearity of ln(p(q)/p(-q))/tau and the large-time overlap of the rate-function curves (Figs. 6b and 7a), which are not forced by the fit. Self-citations such as [31] and [70] supply background model context and are not load-bearing. The paper acknowledges in Section 2.3 that "we have no a priori indications" for the active-bath temperatures and in Section 4 that "analytical approaches could ... provide further validation," meaning the absence of an analytic derivation of the active-case FT is a validity and finite-time-convergence caveat, not a circular step.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on fitting parameters from the fluctuation theorem slope, on the assumed validity of the large deviation principle, and on modeling choices about noise restart and stochastic conventions. No new physical entities are introduced.

free parameters (4)
  • T_FT^{q_E} (left active bath temperature from jump-kinetic heat) = 0.30, 0.35, 0.39, 0.43, 0.63 for Pe=50,65,75,100,150 (Table 1)
    Extracted by linear fits of Eq. 21 slopes with Tr=T1=0.2 fixed; the central claim compares these to Tkin and Teff.
  • T_FT^{q_W} (left active bath temperature from work-based heat) = 0.42, 0.45, 0.47, 0.60, 1.59 for Pe=50,65,75,100,150 (Table 1)
    Same fitting procedure; differs from q_E result, driving the paper's main conclusion.
  • Right-well reference temperature Tr = 0.2 (fixed equal to T1)
    Assumed equal to the passive bath temperature when fitting; all slope deviation is assigned to the left bath, a choice that could bias T_FT if the right bath is perturbed.
  • Model parameters (a, b, gamma, T1, T2, Fa, dt, tau, tau_eq, Np) = a=1, b=2, gamma=10, T1=T2=0.2, dt=0.01, tau=3e4, tau_eq=1e4, Np=1e6
    Chosen by hand as simulation inputs; the quantitative T_FT values depend on them, but they are not fit to the fluctuation theorem.
assumptions (5)
  • domain assumption Large deviation principle for q_R^E and q_R^W
    Used to interpret -ln p(q)/tau as the rate function I(q) and to justify Eq. 21; convergence is only checked numerically (Sec. 2.3, Figs. 4b, 6b, 7a).
  • domain assumption Fluctuation theorem Eq. 1 of Bodineau-Derrida applies to spatially separated baths
    Assumed as the starting point for extracting temperatures; for the active case its validity is exactly what is being tested, so using it to define T_FT is an assumption.
  • domain assumption Stratonovich convention for heat integrals and Itô convention for numerical integration
    Heat definitions in Eqs. 13, 15, 16 use Stratonovich, while velocity-Verlet/Itô numerics give Eq. 22; convention choice affects stationary position distribution and heat statistics.
  • ad hoc to paper Noise restart from stationary distributions at each well entry
    Section 2.1 states noise processes restart at each hop; this is a modeling choice that defines how jumps are counted and may not reflect physical baths.
  • domain assumption Kramers residence time formula for passive well and numerical residence times for active well
    Used to ensure thermalization tau_r > tau_p > tau_I; the active formula is explicitly stated not to hold, so residence times are numerical estimates (Sec. 2.3).

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Pith. "Pith review of Fluctuation Theorems for Heat exchanges between passive and active baths." pith.science (2026). https://pith.science/paper/HDK2PNZ3

@misc{pith2026241207706,
  author       = {Pith},
  title        = {Pith review of: Fluctuation Theorems for Heat exchanges between passive and active baths},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDK2PNZ3}},
  note         = {Machine review of arXiv:2412.07706}
}
read the original abstract

In addition to providing general constraints on probability distributions, fluctuation theorems allow to infer essential information on the role played by temperature in heat exchange phenomena. In this numerical study, we measure the temperature of an out of equilibrium active bath using a fluctuation theorem that relates the fluctuations of the heat exchanged between two baths to their temperatures. Our setup consists of a single particle moving between two wells of a quartic potential accommodating two different baths. The heat exchanged between the two baths is monitored according to two definitions: as the kinetic energy carried by the particle whenever it jumps from one well to the other and as the work performed by the particle on one of the two baths when immersed in it. First, we consider two equilibrium baths at two different temperatures and verify that a fluctuation theorem featuring the baths temperatures holds for both heat definitions. Then, we introduce an additional Gaussian coloured noise in one of the baths, so as to make it effectively an active (out-of-equilibrium) bath. We find that a fluctuation theorem is still satisfied with both heat definitions. Interestingly, in this case the temperature obtained through the fluctuation theorem for the active bath corresponds to the kinetic temperature when considering the first heat definition, while it is larger with the second one. We interpret these results by looking at the particle jump phenomenology.

Figures

Figures reproduced from arXiv: 2412.07706 by the authors.

Figure 1
Figure 1. Schematic depiction of our idealized setup. The black line denotes the quartic double well potential Equation 2 with minima and local maxima at ±xm and xu, respectively, and depth ∆U, the red and blue areas and labels below and above xu denote the action of baths with different features in the two wells and the gray circle denotes the Brownian particle while jumping from the left to the right well, as suggested by t… view at source ↗
Figure 2
Figure 2. (a): typical trajectory of a Brownian particle from case a) at sampling time τ = 5 · 103 . The black dashed lines denote the location of the left and right potential minima at ±xm = ± p b/a = ± √ 2. (b): time evolution of QR E and QR W corresponding to the trajectory in panel (a). Parameters are a = 1.0, b = 2.0, γ = 10, T1 = 0.2 and T2 = 0.3. and Q L W ≡ − Z τ 0 (−γx˙(s) + p 2γT2 ξ2(s))(1 − θ(x(t))) ◦ dx(s) = − X N… view at source ↗
Figure 3
Figure 3. (a): stationary position distributions for case a) with T1 = 0.2 and T2 = 1.4 at sampling time τ = 3 · 104 . The black solid and green dashed lines are the stationary solutions Equation 22 and Equation 23, respectively, while the blue histogram is the position distribution numerically sampled, as denoted by the legend. (b): stationary position distributions for cases a) and b) and Equation 22, as denoted by the lege… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (a): distribution p(q R E ) for case a) at sampling time τ = 3 · 104 for T2 = 0.22, 0.3 and 0.4, as denoted by the legend. (b): curves − ln(p(q R E )/Aτ )/τ for T2 = 0.3 at different sampling times, as denoted by the legend. Aτ denotes the maximum of the distribution a…
Figure 5
Figure 5. Figure 5: (a) and (b): comparison between the distributions p(q R E ) and p(q R W ) at sampling times τ = 103 and τ = 3 · 104 , respectively, with same parameters as in [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: (a): distribution p(q R E ) for case b) at sampling time τ = 3·104 for P e = 5, 50 and 75, as denoted by the legend. (b): curves − ln(p(q R E )/Aτ )/τ for P e = 50 at different sampling times, as denoted by the legend. As in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: (a): curves − ln(p(q R W ))/τ for P e = 50 at different sampling times, as denoted by the legend. Aτ denotes the maximum of the distribution at each sampling time. The inset shows instead the trend of the same curves at the largest sampling times considered. (b): ratio…

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.