Entropy-production fluctuation theorem for a generalized Langevin particle in crossed electric and magnetic fields

I. P\'erez Castillo; J. I. Jim\'enez-Aquino; L. C. Gonz\'alez-Morales

REVIEW 3 major objections 5 minor 41 references

Entropy-production fluctuation theorem for a generalized Langevin particle in crossed electric and magnetic fields

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For a charged Brownian particle in a harmonic trap driven by time-dependent electric fields and a constant magnetic field, the paper proves that the total entropy production obeys the detailed fluctuation theorem even when the surrounding b

desk verdict Careful exact DFT for the non-Markovian GLE; the result is right under the authors' position-marginal entropy definition, which itself needs a defense. read the letter →

arxiv 2512.16026 v2 pith:G5TGBAX7 submitted 2025-12-17 cond-mat.stat-mech

L. C. González-Morales , I. Pérez Castillo , J. I. Jiménez-Aquino This is my paper · ORCID

classification cond-mat.stat-mech PACS 05.10.Gg05.40.Jc

keywords entropyproductiondetailedfluctuationtheoremgeneralizedLangevinequationnon-MarkoviandynamicsmagneticfieldharmonictrapGaussianprocessstochasticthermodynamics

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 establishes that the detailed fluctuation theorem — P(Δs)/P(−Δs) = e^{Δs} — holds exactly for the total entropy production of a charged Brownian particle in a harmonic trap, driven by a time-dependent electric field, with a constant magnetic field and a non-Markovian (memory) bath. The memory kernel and the magnetic field complicate the dynamics, but the linearity of the generalized Langevin equation plus Gaussian noise keeps the phase-space distribution Gaussian. The authors prove the key relation σ² = 2⟨Δs⟩ by showing the fluctuation distribution stays canonical at all times, and this immediately gives the detailed fluctuation theorem. This matters because it extends exact entropy-production statistics to structured, viscoelastic-like baths, where standard Markovian proofs do not apply. The result holds for two protocols: a prescribed time-dependent force and an equivalently dragged trap center.

What carries the argument

The proof hinges on the exact Laplace-transform solution of the linear generalized Langevin equation in crossed fields, expressed through response functions H0, H1, and H2. The covariance matrix of the phase-space fluctuations is computed and, after lengthy algebra, shown to reduce to the canonical form in Eq. (B14). This canonical-preservation identity is what forces the variance–mean relation σ²_s = 2⟨Δs_tot⟩, which, together with Gaussianity, yields the detailed fluctuation theorem. The same machinery yields a work-variance identity that provides the matching mean.

What would settle it

Choose a specific memory kernel, such as an exponential γ(t) = (γ0/τ)e^{−t/τ}, compute the exact covariance matrix F, G, H, I from Eqs. (B2)–(B5) at a finite time, and check whether H = I = 0 and F = k_BT/ω², G = k_BT hold. Any deviation would break σ² = 2⟨Δs⟩ and, with it, the detailed fluctuation theorem. Numerically simulating the GLE for a non-harmonic trap and testing the ratio P(Δs)/P(−Δs) would also settle it.

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Extended reading notes

Core claim

The central discovery is that for equilibrium initial conditions, the phase-space probability density of the fluctuations (X, Y, Vx, Vy) is exactly the canonical Gibbs distribution at every time t, with position covariance k_BT/k and velocity covariance k_BT, independent of the memory kernel and the magnetic field. Consequently, the trajectory-dependent total entropy production is a Gaussian random variable whose variance equals twice its mean, σ²_s = 2⟨Δs_tot⟩. For a Gaussian variable this relation is equivalent to the detailed fluctuation theorem P(Δs)/P(−Δs) = e^{Δs}. The paper shows this explicitly for two protocols: direct time-dependent forcing and dragging the trap center.

Load-bearing premise

The load-bearing premise is that the fluctuation phase-space distribution remains exactly canonical at all times — that the cross-correlations H and I vanish and the variances stay fixed at their equilibrium values, which is asserted after lengthy algebra rather than displayed in full.

Editorial extensions

If this is right

The detailed fluctuation theorem and the integral fluctuation theorem ⟨e^{−Δs}⟩ = 1 hold for a non-Markovian harmonic charged particle in crossed electric and magnetic fields.
The result holds for arbitrary memory kernels consistent with the fluctuation–dissipation theorem, so it covers viscoelastic and other structured baths.
The two protocols—force driving and trap dragging—are proven thermodynamically equivalent at the level of entropy-production statistics.
The Gaussian character of entropy production is exact for this linear system, enabling testable predictions for the full distribution, not just the mean.

Reading between the lines

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

If the canonical-preservation identity extends to any linear system whose response functions satisfy the same algebraic relations, the detailed fluctuation theorem would hold for a wider class of non-Markovian Gaussian processes; this is worth testing.
The definition of system entropy here uses only the position marginal; a phase-space definition that includes the velocity marginal would likely give the same variance–mean relation because the velocity marginal is also canonical, but this is not shown in the paper.
Anharmonic confining potentials would break the exact Gaussian-canonical structure; the detailed fluctuation theorem may then fail or require corrections, suggesting a testable boundary for the theorem's validity.
The magnetic field enters the proof only through the response-function algebra; the apparent irrelevance of B to the final distribution hints that the result may extend to non-uniform or time-dependent fields if time-reversal symmetry is respected.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First: this is a careful, exact calculation that does what it advertises. For a non-Markovian GLE with memory, harmonic trap, constant magnetic field, and time-dependent electric driving, the authors prove a detailed fluctuation theorem for the total entropy production under two protocols. The mechanism is the usual one for linear systems: show Δs is Gaussian, compute the variance, and verify σ² = 2⟨Δs⟩. The genuinely new piece is the explicit phase-space density calculation in Appendix B, which shows the one-time distribution remains exactly the canonical Gibbs form despite the memory kernel and magnetic field. That is not trivial, and the paper gets credit for carrying it through. No free parameters, no fitted data; the work-variance relations are consistent with Jarzynski.

Where it gets softer. The biggest issue is the definition of total entropy production. The paper uses the position marginal P(r,t) in the system entropy and a first law with potential energy only. That is not the standard underdamped Seifert entropy, which uses the full phase-space distribution and includes kinetic energy. The authors have the phase-space distribution available—they just computed it—so this is a modeling choice, not an accident. But it is a substantial one: the object they prove the DFT for may not be the physically observed entropy production of the charged particle. This should be flagged and defended.

Second, the key algebraic cancellation in Appendix B (H=0, I=0, F=kBT/k, G=kBT) is reported as 'after a long algebra,' with intermediate steps suppressed. A referee cannot easily verify it. That is a presentation gap, and since the entire result rests on it, the authors should be asked to show more.

Third, the two-time stationarity assertion leading to Eq. (49) and (C1)—that equilibrium initial conditions make correlation functions depend only on time differences—is stated without proof. I believe it is true for a GLE when the initial state is the full equilibrium, but the paper does not establish it, and it is load-bearing for Eq. (57). The text's casual wording invites exactly the objection the stress-test raises. It should be either proven or referenced.

The self-citations to [31] and [34] are appropriate; the method is a direct extension, and the paper says so. No fitting, no circularity. The novelty is real but incremental: this is a useful benchmark for non-Markovian charged-particle setups, not a conceptual shift. It belongs in a specialized journal; the referees should ask for the extra derivations and a clearer discussion of the entropy definition.

Referee Report

3 major / 5 minor

Summary. The paper studies a charged Brownian particle in a harmonic trap, driven by a time-dependent electric field in a constant magnetic field, with non-Markovian dynamics given by a generalized Langevin equation with memory and Gaussian noise. Using the exact linear solution, the authors obtain a Gaussian phase-space probability density and, from it, compute the trajectory-dependent total entropy production. For two driving protocols—direct forcing and dragged trap center—they claim to prove a detailed fluctuation theorem P(Δs_tot)/P(−Δs_tot)=e^{Δs_tot}, relying on Gaussianity of Δs_tot and a variance–mean identity σ²_s=2⟨Δs_tot⟩. The proof is built on two technical pillars: a canonical (equilibrium) phase-space density for the fluctuations at every time, and a two-time stationarity property of the reduced fluctuation dynamics.

Significance. If the central result were correct, it would be a useful exact extension of detailed fluctuation theorems to non-Markovian charged-particle dynamics in crossed fields, going beyond the Markovian analyses of Refs. [31] and [34]. The paper is self-contained in its setup, provides explicit Laplace-domain response functions, and derives work-variance relations in closed form, with no free parameters. However, the two load-bearing technical steps—the stationary canonical PSPD of Appendix B and the two-time stationarity assumed in Eq. (49)/Appendix C—are not established for the stated model and are generally false under the manuscript's own treatment of initial conditions and noise. The result may be salvageable under a different, properly defined equilibrium-bath initialization, but as written the proof does not support the claim.

major comments (3)

[§III.A.1, Eq. (49); Appendix C, Eq. (C1)] The variance–mean identity (58) depends on Eq. (57), which is derived from the two-time correlation identity ⟨R(τ)R(t)⟩=⟨R₀R(τ−t)⟩ asserted in Eq. (49) and used again in Eq. (C1). This is a strict time-translation-invariance property of the reduced fluctuation process. The text states it 'for equilibrium initial conditions' after Eq. (48), but Appendix B only establishes the one-time canonical marginal (B14). For the finite-horizon GLE with the memory integral starting at 0 and with noise treated as independent of the initial fluctuations (e.g., ⟨X₀f(t)⟩=0), the reduced (X,Y,Vx,Vy) process is not time-homogeneous: there is an initial-slip transient. The one-time canonical property constrains only equal-time covariances and does not determine two-time correlations. Without a proof—or an explicit change of initialization to a stationary process—Eq. (C1) is an additional unproved assumption
[Appendix B, Eq. (B14)] The derivation of the canonical PSPD P(R,S) drops all initial-condition–noise cross-correlations (⟨X₀f(t)⟩=⟨Y₀f(t)⟩=⟨Vx₀f(t)⟩=⟨Vy₀f(t)⟩=0) and then asserts 'after a long algebra' that F=k_BT/ω², G=k_BT, H=I=0. This is not a consequence of the stated assumptions for a general memory kernel. For a colored-noise GLE with independent initial conditions, the equal-time variance has an initial-slip correction. For example, in the one-dimensional version with a smooth memory kernel, a small-t expansion of ⟨X²(t)⟩ gives an O(t⁴) contribution proportional to γ(0), while the deterministic initial-condition terms cancel at that order, so F(t)≠k_BT/ω² unless the initial conditions are correlated with the future noise. Thus the central premise (B14) is not merely undisplayed algebra; it is generally false under the paper's treatment of the noise. The 'equilibrium initial conditions' need to specify t
[§III, Eqs. (36)–(40)] The paper defines the system entropy as −ln P(r,t), using the marginal density of position only, and the first-law balance Q=W−ΔU with U taken as the potential energy only. In the underdamped stochastic thermodynamics of Seifert (as used in Refs. [27,30,31]), the system entropy is −ln P(x,v,t) and the internal energy includes kinetic energy. The manuscript does not justify why the position-marginal object is 'the' total entropy production. Even if the Gaussian variance–mean calculation were correct, the theorem would apply to an entropy-like functional constructed from the position marginal, not to the standard underdamped total entropy production. The terminology and scope should be clarified, or the definition should be matched to the standard one.

minor comments (5)

[§III.A.2, Eq. (59)] The detailed fluctuation theorem is written as P(Δs_tot)/P(−Δs_tot)=e^{Δs_tot} with the same symbol on both sides; using, e.g., Σ on the left would avoid notational confusion.
[Appendix headings] The appendix headings use Spanish ('Apéndice A', 'Apéndice B', 'Apéndice C'); these should be 'Appendix' in an English-language journal.
[References [7], [9], [27]] Several references contain incomplete or malformed bibliographic data, e.g., Ref. [27] is cited as 'Phy. Rev. Lett.5, 040602' (should be PRL 95, 040602), and Refs. [7] and [9] have broken DOI strings. Please check all references.
[§III.A.2, Eq. (56)] The notation ⟨W r̃(τ)⟩ appears before the covariance is defined; introduce the cross-covariance explicitly to avoid ambiguity.
[§II, after Eq. (31)] The factorization P(R,S)=P_z(Z,V_z)P(R̃,S̃) is stated but the z-component details are deferred; a sentence noting that the z-axis decouples because the magnetic force is perpendicular would help readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step found: the DFT proof is self-contained from GLE, Gaussianity and Appendix B algebra; cited self-works are background only.

full rationale

The derivation chain is not circular. The paper starts from the linear GLE (1) with Gaussian colored noise and equilibrium canonical/Maxwellian initial conditions, solves the dynamics explicitly, and in Appendix B obtains the phase-space density in canonical form via the algebraic identities H=0, I=0 (Eqs. B9–B13). The total entropy production is then defined through the standard Seifert decomposition, and Δs_tot is shown to be a Gaussian linear functional of the process. The decisive step σ_s^2 = 2⟨Δs_tot⟩ follows from direct calculation of the work variance and the correlation identity in Appendix C, not from assuming the fluctuation theorem. No fitted parameter is renamed as a prediction, and no equation is constructed so that the target DFT holds by definition. The self-citations [31] and [34] provide background and the Markovian analogue or the long-time stationarity statement; they are not used to import the transient DFT. A technical concern does exist but is a correctness issue, not circularity: the two-time stationarity assertion in Eq. (49) and Appendix C Eq. (C1) is stated for equilibrium initial conditions but is not derived in the displayed algebra, so it may fail for general memory kernels. That would undermine the proof but does not make the proof equivalent to its inputs. Accordingly, the only reason the score is not 0 is the presence of minor, non-load-bearing self-citations.

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

The central claim rests on standard generalized-Langevin heat-bath assumptions (FDT, Gaussian noise, equilibrium initial state) plus one ad hoc modeling choice: the definition of system entropy as a position-marginal quantity with a potential-energy-only first law. No free parameters are fitted to data, and no new physical entities are introduced.

assumptions (5)

domain assumption Kubo's second fluctuation-dissipation theorem, Eq. (2): ⟨f_i(t)f_j(t')⟩=k_BT δ_ij γ(t−t').
This equips the generalized Langevin equation with a thermal heat bath; it is needed for the PSPD to stay canonical and for all later variance cancellations.
domain assumption The initial state is the equilibrium canonical distribution in position and Maxwellian in velocity.
Transient fluctuation theorems require an equilibrium starting state; here it also guarantees that the fluctuation process (X,V) starts in the stationary state.
domain assumption The noise f(t) is Gaussian and the dynamics is linear, so the phase-space distribution remains Gaussian for all times.
This is used to conclude that the total entropy production is a Gaussian random variable, which is the hinge of the proof.
ad hoc to paper Total entropy production is defined via the position-marginal density P(r̃,t) and a first-law heat balance Q=W−ΔU with U taken as potential energy only.
For an underdamped generalized Langevin dynamics the standard phase-space entropy includes velocity degrees of freedom and the energy balance includes kinetic energy; the paper does not justify this reduced definition.
standard math The inverse Laplace transforms H0, H1, H2 and their time derivatives satisfy the initial-value identities of Appendix A.
Needed to write the explicit solutions (6)–(8) and (17)–(19) and to evaluate boundary terms.

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Pith. "Pith review of Entropy-production fluctuation theorem for a generalized Langevin particle in crossed electric and magnetic fields." pith.science (2026). https://pith.science/paper/G5TGBAX7

@misc{pith2026251216026,
  author       = {Pith},
  title        = {Pith review of: Entropy-production fluctuation theorem for a generalized Langevin particle in crossed electric and magnetic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5TGBAX7}},
  note         = {Machine review of arXiv:2512.16026}
}

read the original abstract

We study fluctuations of entropy production for a charged Brownian particle confined in a harmonic trap and driven out of equilibrium by crossed electric and magnetic fields. The magnetic field is constant and perpendicular to the plane of motion, while the electric field is time dependent and provides the driving. The non-Markovian dynamics is modeled by a generalized Langevin equation with memory and Gaussian noise. This setting represents a charged Brownian degree of freedom in a structured bath, where delayed friction modifies relaxation while the magnetic field couples the transverse coordinates. Using the exact solution of this linear dynamics, we obtain the time-dependent Gaussian phase-space probability density and from it compute the trajectory-dependent total entropy production. For two solvable driving protocols -- direct forcing by a prescribed time-dependent electric force and dragging of the harmonic trap center -- we prove analytically that the total entropy production obeys a detailed fluctuation theorem.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 10 canonical work pages

[31]

Binder, L

F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the quantum regime, Fun- damental Theories of Physics195, 1 (2018)

2018
[34]

Sekimoto,Stochastic EnergeticsVol

K. Sekimoto,Stochastic EnergeticsVol. 799 (Springer, Heidelberg, Germany, 2010)

2010
[1]

According to Eq

Work statistics In this case, the time-dependent protocol is ˜λ(t) = eFe(t) = (Fxe(t), Fye(t)), and the effective potential is gi- ven byV( ˜r, ˜λ(t)) = (k/2)| ˜r|2 − eFe(t)· ˜r. According to Eq. (35), the total work done on the system over a finite timeτreads W=− Z τ 0 ˙eFe(t)· ˜r(t)dt=− Z τ 0 ˙Fxe(t)x(t)dt − Z τ 0 ˙Fye(t)y(t)dt ,(43) and the change in t...
[2]

(40), we use Eqs

Entropy production statistics To evaluate the total entropy production∆s tot in Eq. (40), we use Eqs. (41), (42), and (44). This yields ∆stot = 1 T W+ 1 2 k|⟨˜r(τ)⟩| 2 −k ˜r(τ)· ⟨˜r(τ)⟩ + ˜r(τ)· eFe(τ) ! ,(53) where we have setkB = 1, so thatTdenotesk BTand ∆stot is dimensionless. From Eqs. (43) and (53),Wand ∆stot are linear functionals of the Gaussian p...
[3]

In this case, the effective potential becomesU(˜r, ˜λ(t)) = (k/2)|˜r− ˜λ(t)|2, where ˜λ(t) = eFe(t) k

Work statistics We next consider a second solvable model, in which the center of the harmonic trap is dragged with an arbitrary velocity during a finite time interval. In this case, the effective potential becomesU(˜r, ˜λ(t)) = (k/2)|˜r− ˜λ(t)|2, where ˜λ(t) = eFe(t) k . The total work done on the system over a finite timeτthen reads ˆW= Z τ 0 ∂U( ˜r, ˜λ(...
[4]

(68) Again, ˆWand∆ˆs tot are linear functionals of the Gaus- sian process ˜r(t), and thereforeP(∆ˆstot)has the same Gaussian form as in Eq

Entropy production statistics For the dragged-trap protocol, the total entropy pro- duction along a trajectory over timeτcan be written as ∆ˆstot = 1 T ˆW+ 1 2 k|⟨˜r(τ)⟩| 2 −k ˜r(τ)· ⟨˜r(τ)⟩ + ˜r(τ)· eFe(τ)− 1 2k |eFe(τ)| 2 ! , (67) and its mean value is ⟨∆ˆstot⟩= 1 T ⟨ ˆW⟩ −1 2 k|⟨˜r(τ)⟩| 2 +⟨ ˜r(τ)⟩ ·eFe(τ)− 1 2k |eFe(τ)| 2 ! . (68) Again, ˆWand∆ˆs tot ...
[5]

By definingF≡σ xx(t),G≡σ vxvx (t),H≡σ xvx (t), andI≡σ xvy (t), we thus have according to Eqs. (26), (27), (29), and (30) that F=⟨[x− ⟨x⟩] 2⟩=⟨X 2(t)⟩ =⟨X 2 0 ⟩[χ0(t) + Ω2ω2χ2(t)]2 +⟨Y 2 0 ⟩Ω2ω4H 2 1 (t) +⟨V 2 x0⟩[H0(t)−Ω 2H2(t)]2 +⟨V 2 y0⟩Ω2H2 1(t) + 2 β Z t 0 H0(t′)dt′ Z t′ 0 H0(t′′)γ(t′ −t ′′)dt′′ + Ω 2 Z t 0 H1(t′)dt′ Z t′ 0 H1(t′′)γ(t′ −t ′′)dt′′ + Ω ...
[6]

G. N. Bochkov and Yu. E. Kuzovlev, PhysicaA106, 443 (1981);106, 480 (1981)

1981

Show all 41 references

[7]

D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett.71, 2401 (1993)

1993
[8]

Gallavotti and E

G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys.80, 931 (1995)

1995
[9]

Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation ap- proach

C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation ap- proach. Phys. Rev. E56, 5018 (1997)

1997
[10]

G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy dif- ferences, Phys. Rev. E60, 2721 (1999)

1999
[11]

Kurchan, J

J. Kurchan, J. Phys. A: Math. Gen.31, 3719 (1999)

1999
[12]

van Zon, S

R. van Zon, S. Ciliberto, and E. G . D. Cohen, Power and Heat Fluctuation Theorem for Electric Circuit, Phys. Rev. Lett.92, 130601 (2004). DOI: https://doi.org/10.1103/PhysRevLett.92.130601

2004 doi
[13]

Jae Dong Noh and Jong-Min Park, Fluctuation Rela- tion for Heat, Phys. Rev. Lett.108, 240603 (2012). DOI:https://doi.org/10.1103/PhysRevLett.108.240603

2012 doi
[14]

J. I. Jiménez-Aquino and R. M. Velasco, Po- wer fluctuation theorem for a Brownian harmo- nic oscillator, Phys. Rev. E,87, 022112 (2013). DOI://doi.org/10.1103/PhysRevE.87022112

2013 doi
[15]

G. N. Bochkov and Yu. E. Kuzovlev, Physics-Uspekhi 56, 590 (2013)

2013
[16]

https://doi:10.3390/e20090635

Ricardo Rao and Massimilano Esposito, Detailed Fluc- tuation Theorems: A Unifying Perspective, Entropy20, 635 (2018). https://doi:10.3390/e20090635

2018 doi
[17]

Lemos, A

C.G.O. Lemos, A. L. Ferreira, W. Figueiredo and M. Santos, Testing ground for fluctuation theorems: The one-dimensional Ising model, Phys. Rev. E. 97, 042121 (2018)

2018
[18]

https://doi.org/10.3390/e25010165

Jung Jun Park and Hyunchul Nha, Fluctuation Theorem for Information Thermodynamics of Quan- tum Correlated Systems, Entropy25, 165 (2023). https://doi.org/10.3390/e25010165

2023 doi
[19]

https://doi.org/10.3390/e2606439

Massimiliano Semeraro, Antonio Suma, and Giuseppe Negro, Fluctuation Theorems for Heat Exchanges bet- ween Passive and Active Baths, Entropy26, 439 (2024). https://doi.org/10.3390/e2606439

2024 doi
[20]

Alessandro Coretti, Lamberto randoni, and Sara Bo- nella, Phys. Rev. E102, 030101(R) (2020). DOI: 10.1103/PhysRevE.102.030101

2020 doi
[21]

Peter Sidajaya, Jovan Hsuen Khai Low, Clive Cenxin Aw, and Valerio Scarani, Emergence of fluctuation re- lations in UNO, Phys. Rev. Res.7, 023054 (2025). DOI:10.1103/PhysRevResearch.7.023054

2025 doi
[22]

P. Hack, C. Lindig-Leon, S. Gottwald, and D. A. Braun, Thermodynamic fluctuation theorems govern hu- man sensorimotor learning, Sci. Rep.13, 869 (2023). doi.org/10.1038/s41598-023-27736-8

2023 doi
[23]

A topological fluctuation theorem

Benoit Mahault, Evelyn Tang, and Ramin Golestanian. A topological fluctuation theorem. Nat. Commun.13, 3036 (2022). doi.org/10.1038/s41467-022-30644-6

2022 doi
[24]

Adv.7, 1807 (2021)

Qian Zeng and Jin Wang, New fluctuation theorems on Maxwell’s demon, Sci. Adv.7, 1807 (2021)

2021
[25]

Hayashi, Application of the fluctuation theorem to motor proteins: from F1-ATPase to axonal cargo trans- port by kinesin and dynein, Biophys

K. Hayashi, Application of the fluctuation theorem to motor proteins: from F1-ATPase to axonal cargo trans- port by kinesin and dynein, Biophys. Rev.10, 1311 (2018)

2018
[26]

Physica A511302-315 (2018)

Subhasish Chaki, Rajarshi Chakrabarti, Entropy produc- tion and work fluctuation relations for a single particle in active bath. Physica A511302-315 (2018)

2018
[27]

Physica A525223- 233 (2019)

Koushik Goswami, Work fluctuation relations for a drag- ged Brownian particle in active bath. Physica A525223- 233 (2019)

2019
[28]

Narinder, P

N. Narinder, P. Shuvojit, and C. Bechinger, Work fluc- tuation relation of an active Brownian particle in a vis- coelastic fluid. Phys. Rev. E104, 034605, (2021)

2021
[29]

Argun, A

A. Argun, A. R. Moradi, E. Pince, G.B. Bagci, G. Volpe, Experimental evidence of the failure of Jarzynski equality in active baths. arXiv:1601.01123 https://doi.org/10.48550/arXiv.1601.01123 13

doi arXiv
[30]

K. Funo, M. Ueda, and T. Sagawa, Quantum fluctua- tion theorems, Thermodynamics in the Quantum Re- gime: Fundamental Aspects and New Directions,249 (2018)

2018
[32]

Seifert, Entropy Production along a Stochastic Tra- jectory and an Integral Fluctuation Theorem, Phy

U. Seifert, Entropy Production along a Stochastic Tra- jectory and an Integral Fluctuation Theorem, Phy. Rev. Lett.5, 040602, (2005)

2005
[33]

Marcos L. W. Basso, Jonas Maziero, and Lucas C. Ceĺe- ri, Quantum Detailed Fluctuation Theorem in Curved Spacetimes: The Observer Dependent Nature of Entropy Production, Phys. Rev. Lett. 134, 050406 (2025)

2025
[35]

A. Saha, S. Lahiri, and A. M. Jayannavar, Phys. Rev. E 80, 011117 (2009)

2009
[36]

J. I. Jiménez-Aquino, Entropy production theorem for a charged particle in an electromagnetic field, Phys. Rev. E82, 051118 (2010)

2010
[37]

Mai and A

T. Mai and A. Dhar, Nonequilibrium work fluctuations for oscillators in non-Markovian baths. Phys. Rev. E75, 061101 (2007)

2007
[38]

Ohkuma and T

T. Ohkuma and T. Ohta, Fluctuation theorems for non-linear generalized Langevin systems. J. Stat. Mech. P10010 (2007)

2007
[39]

J. I. Jiménez-Aquino, Non-Markovian work fluctuation theorem in crossed electric and magnetic fields. Phys. Rev. E92, 022149 (2015)

2015
[40]

Bappa Ghosh, Srabanti Chaudhury, Fluctuation theo- remsfortotalentropyproductioningeneralizedLangevin systems, Physica A466, 133 (2017)

2017
[41]

R. Kubo. The fluctuation-dissipation theorem. Rep. Prog. Phys.29, 1 (1966)

1966

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [31]

Binder, L

F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the quantum regime, Fun- damental Theories of Physics195, 1 (2018)

2018

[2] [34]

Sekimoto,Stochastic EnergeticsVol

K. Sekimoto,Stochastic EnergeticsVol. 799 (Springer, Heidelberg, Germany, 2010)

2010

[3] [1]

According to Eq

Work statistics In this case, the time-dependent protocol is ˜λ(t) = eFe(t) = (Fxe(t), Fye(t)), and the effective potential is gi- ven byV( ˜r, ˜λ(t)) = (k/2)| ˜r|2 − eFe(t)· ˜r. According to Eq. (35), the total work done on the system over a finite timeτreads W=− Z τ 0 ˙eFe(t)· ˜r(t)dt=− Z τ 0 ˙Fxe(t)x(t)dt − Z τ 0 ˙Fye(t)y(t)dt ,(43) and the change in t...

[4] [2]

(40), we use Eqs

Entropy production statistics To evaluate the total entropy production∆s tot in Eq. (40), we use Eqs. (41), (42), and (44). This yields ∆stot = 1 T W+ 1 2 k|⟨˜r(τ)⟩| 2 −k ˜r(τ)· ⟨˜r(τ)⟩ + ˜r(τ)· eFe(τ) ! ,(53) where we have setkB = 1, so thatTdenotesk BTand ∆stot is dimensionless. From Eqs. (43) and (53),Wand ∆stot are linear functionals of the Gaussian p...

[5] [3]

In this case, the effective potential becomesU(˜r, ˜λ(t)) = (k/2)|˜r− ˜λ(t)|2, where ˜λ(t) = eFe(t) k

Work statistics We next consider a second solvable model, in which the center of the harmonic trap is dragged with an arbitrary velocity during a finite time interval. In this case, the effective potential becomesU(˜r, ˜λ(t)) = (k/2)|˜r− ˜λ(t)|2, where ˜λ(t) = eFe(t) k . The total work done on the system over a finite timeτthen reads ˆW= Z τ 0 ∂U( ˜r, ˜λ(...

[6] [4]

(68) Again, ˆWand∆ˆs tot are linear functionals of the Gaus- sian process ˜r(t), and thereforeP(∆ˆstot)has the same Gaussian form as in Eq

Entropy production statistics For the dragged-trap protocol, the total entropy pro- duction along a trajectory over timeτcan be written as ∆ˆstot = 1 T ˆW+ 1 2 k|⟨˜r(τ)⟩| 2 −k ˜r(τ)· ⟨˜r(τ)⟩ + ˜r(τ)· eFe(τ)− 1 2k |eFe(τ)| 2 ! , (67) and its mean value is ⟨∆ˆstot⟩= 1 T ⟨ ˆW⟩ −1 2 k|⟨˜r(τ)⟩| 2 +⟨ ˜r(τ)⟩ ·eFe(τ)− 1 2k |eFe(τ)| 2 ! . (68) Again, ˆWand∆ˆs tot ...

[7] [5]

By definingF≡σ xx(t),G≡σ vxvx (t),H≡σ xvx (t), andI≡σ xvy (t), we thus have according to Eqs. (26), (27), (29), and (30) that F=⟨[x− ⟨x⟩] 2⟩=⟨X 2(t)⟩ =⟨X 2 0 ⟩[χ0(t) + Ω2ω2χ2(t)]2 +⟨Y 2 0 ⟩Ω2ω4H 2 1 (t) +⟨V 2 x0⟩[H0(t)−Ω 2H2(t)]2 +⟨V 2 y0⟩Ω2H2 1(t) + 2 β Z t 0 H0(t′)dt′ Z t′ 0 H0(t′′)γ(t′ −t ′′)dt′′ + Ω 2 Z t 0 H1(t′)dt′ Z t′ 0 H1(t′′)γ(t′ −t ′′)dt′′ + Ω ...

[8] [6]

G. N. Bochkov and Yu. E. Kuzovlev, PhysicaA106, 443 (1981);106, 480 (1981)

1981