Path integral approach to the calculation of the characteristic function of work

H. T. Quan; Rui Pan; Tian Qiu; Zhaoyu Fei

REVIEW 3 major objections 5 minor 63 references

Path integral approach to the calculation of the characteristic function of work

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Path integrals reproduce the two-point-measurement work statistics exactly for the driven oscillator and the expanding piston.

desk verdict A solid derivation paper: the harmonic oscillator work statistics reproduce Ref. [44] and the expanding-piston characteristic function is new, with the main soft spot being unproved interchange of infinite sums and Fourier inversion that Appendix B only partially mitigates. read the letter →

arxiv 1908.09731 v2 pith:P6F4HMWR submitted 2019-08-26 cond-mat.stat-mech quant-ph

Tian Qiu , Zhaoyu Fei , Rui Pan , H. T. Quan This is my paper · ORCID

classification cond-mat.stat-mechquant-ph

keywords workstatisticspathintegralcharacteristicfunctiontwo-pointmeasurementquantumthermodynamicstime-dependentharmonicoscillatorexpandingpistonquantum-classicalcorrespondence

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 show that work statistics in quantum thermodynamics can be computed directly from path integrals, rather than by solving wave equations for the system's states. It derives the characteristic function of work under the two-point measurement scheme for two solvable models: a harmonic oscillator with a time-dependent frequency and a free particle in a rigid box with one uniformly moving wall. For the oscillator, the path-integral result is closed-form and identical to the previously known wave-mechanics result; for the piston, it is an infinite sum over collision classes of classical paths, and its Fourier transform gives the work distribution and transition probabilities, which an appendix proves agree with the wave-mechanics answer. The same path-integral route also gives the classical characteristic function of the driven oscillator analytically, and numerical evaluation of the piston sums shows convergence to the known classical distribution as $\hbar\to 0$.

What carries the argument

The load-bearing object is the path-integral representation of the characteristic function, Eq. (11): $\chi_W(\nu)=\int e^{\frac{i}{\hbar}(S_1[x]-S_2[y])}\rho(x_i,y_i)\delta(x_f-y_f)$, with forward and backward actions $S_1$, $S_2$ that each contain a segment of duration $\hbar\nu$ at fixed work parameter. This split turns the problem into two propagators: for the oscillator the semiclassical propagator is exact because only one classical path contributes, while for the piston the exact propagator is a sum over infinitely many classical paths classified into four collision classes with sign factors from half-wave loss. Gaussian integrations over the intermediate and boundary positions reduce the oscillator case to Eq. (33); the piston case introduces the imaginary error function through the collision-class sums and produces the four-fold sum Eq. (47). The same forward/backward split is then reduced to the classical work functional in the $\hbar\to0$ limit.

What would settle it

Take the right-hand side of Eq. (47) with upper summation limits $N$ on all four indices, Fourier-transform the truncated sum numerically, and compare the resulting weights with the explicit wave-mechanics transition probabilities in Appendix B for modest $\hbar$; if the truncated sums do not converge as $N$ grows, or converge to different weights, the claimed equality fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the characteristic function of work for the two-point measurement scheme admits an exact path-integral evaluation for both prototype systems. For the time-dependent harmonic oscillator, the expression reduces to Eq. (33), which is exactly Eq. (17) of Ref. [44]. For the free particle in an expanding piston, the characteristic function is the four-fold infinite sum in Eq. (47), built from products of imaginary error functions and the four collision classes of classical paths; Fourier inversion gives the delta-weighted work distribution Eq. (50), and the individual coefficients are the transition probabilities Eq. (51). Appendix B proves that these transition probabilities coincide with those obtained from the time-dependent wave-equation solution. The paper also obtains the classical characteristic function of the driven oscillator directly by path integration over classical trajectories, Eq. (60).

Load-bearing premise

The piston result depends on being allowed to take an infinite four-fold sum of terms that grow exponentially with the summation indices, move it inside a Fourier integral, and reorder it term by term; if those sums need regularization or do not converge, Eqs. (50) and (51) are not established.

Editorial extensions

If this is right

For the driven harmonic oscillator, work statistics can be obtained by evaluating classical-path actions and Gaussian integrals, without expanding in instantaneous eigenstates.
For the expanding piston, Eqs. (50) and (51) give explicit transition probabilities between instantaneous energy levels of the moving box, an object otherwise available only through numerical wave-packet propagation.
The classical work distribution of the driven oscillator follows from the same path-integral framework directly, rather than by taking the $\hbar\to0$ limit of the quantum characteristic function.
The numerical accumulation of Eq. (50) approaches the known classical expanding-piston distribution as $\hbar\to0$, supporting a trajectory-level quantum-to-classical correspondence.
The paper states that the forward/backward action split is in principle applicable to open quantum systems and quantum fields as well.

Reading between the lines

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

One can read Eq. (47) as defining the work distribution by analytic continuation of the sums; a natural step not taken in the paper would be to study the convergence radius or to regularize the sums and check that the result is independent of the regularization.
The same collision-class enumeration would apply to a box whose wall motion is not uniform, or to higher-dimensional pistons, as long as the classical paths remain piecewise linear; whether the imaginary-error-function integrals would still close is a testable question.
The paper's equivalence between path-integral and wave-mechanics results suggests that the trajectory work defined by the forward/backward action difference can be assigned a well-defined probability at finite $\hbar$ in these models; proving that for general potentials would require an additional argument beyond what the paper supplies.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The useful thing here is the expanding-piston calculation. The harmonic oscillator section re-derives a known result, so the novelty is the method rather than the formula, but the piston characteristic function in Eq. (47) and the transition probabilities in Eq. (51) are new, and Appendix B independently checks them against the Schrödinger-based results of Doescher–Rice and Quan–Jarzynski. That cross-check is the strongest part of the paper: no free parameters, no fitting, and the final probabilities match an independent formalism. The classical harmonic oscillator limit also agrees with Ref. [44], and the numerical quantum-to-classical plot for the piston is a reasonable sanity check.

The soft spots are real but not fatal. The step from Eq. (47) to the delta-comb Eq. (50) involves Fourier inverting a four-fold infinite sum whose terms are only conditionally convergent; the authors never justify the interchange. Appendix B proves the final probabilities are correct, which actually covers much of the concern, because the TPM work distribution is a delta-comb by definition once the transition probabilities are known. What remains unproven is the path-integral evaluation itself, which also commutes sums with integrals to reach Eq. (47). Since the final result is externally verified, I read this as a rigor gap rather than a correctness failure, but the authors should add a convergence argument or at least a numerical partial-sum test against the TDSE results. Appendix A is also an algebra dump — many steps are just 'after some simplification' — though the final expression checks out against a published result. The classical-limit discussion in Sec. IV is more asserted than proved; for the piston they only have numerics, which is honest but weaker than the quantum part.

The citation pattern is fine. The paper leans on Ref. [52] for the path-integral representation and on the authors' own earlier work, but the relation to Ref. [44] and the independent check in Appendix B are handled fairly.

Who is this for? People working on quantum work statistics and path-integral methods in stochastic thermodynamics. It is a competent, useful toolbox paper that extends an existing formalism to a nontrivial model. It deserves serious peer review; I would send it out and ask the authors to tighten the convergence issue and expand Appendix A. I'd probably cite the piston result if I worked in this area.

Referee Report

3 major / 5 minor

Summary. The manuscript derives the characteristic function of work for two isolated quantum systems—a harmonic oscillator with a time-dependent frequency and a free particle in a uniformly expanding piston—using Feynman path integrals, and compares the results with Schrödinger-based calculations. It also obtains classical work statistics by taking the classical limit of the path-integral expression and illustrates the quantum-to-classical crossover numerically for the piston. The main claimed achievements are exact analytical work distributions for these two models and a demonstration that the path-integral approach reproduces the two-point-measurement work statistics.

Significance. If the derivations are valid, the paper offers a useful demonstration that Feynman's path integral can serve as a unified tool for quantum and classical work statistics. The harmonic-oscillator characteristic function exactly matches the known result of Deffner and Lutz (Ref. [44]), and the piston transition probabilities are cross-checked against time-dependent Schrödinger solutions in Appendix B. The manuscript contains no fitted parameters and the central quantum results are checked against independent references, which is a genuine strength. The remaining risk is concentrated in the piston derivation, where infinite sums and Fourier transforms are interchanged without a rigorous justification; because of this, the quantitative status of Eq. (50) and the classical-limit demonstration is not yet fully established.

major comments (3)

[Section III.B, Eqs. (47)-(50)] The step from the characteristic function to the work distribution is a term-by-term Fourier inversion of an infinite four-fold sum. The Erfi-containing coefficients do not decay absolutely (the relevant combinations oscillate and decay only like n^{-1} e^{-i n^2/2} in the large-argument regime), so the distributional interchange of the sum and the Fourier integral is not automatic. Since Eq. (50) is the central piston work distribution and is used in Fig. 1, the authors need to justify the interchange by proving convergence in a suitable test-function space or by introducing a regulator and taking the limit, or they should verify Eq. (50) numerically against a direct Schrödinger evaluation of P(W) for finite truncations. Appendix B checks Eq. (51), but it does not certify the interchange that produces Eqs. (47) and (50).
[Section III.B, Eqs. (37)-(39) and (39)-(47)] The derivation also integrates the infinite propagator sums term by term over the intermediate position x_b and over the four coordinate variables. No convergence theorem is cited for these operations, and the sums are only conditionally convergent. If any of these interchanges fails, Eq. (47) is not an established characteristic function, even though Eq. (51) is later confirmed by the Schrödinger calculation. Please provide a justification of the interchanges or a numerical check of Eq. (47), for instance by comparing partial sums with the exact TPM characteristic function obtained from the Schrödinger transition probabilities.
[Section IV, Eq. (54) and Fig. 1] The classical limit is introduced through a stationary-phase argument, but for the expanding piston the authors state that the usual classical work functional is not applicable, and the quantum-to-classical transition is concluded from a single numerical figure with no quantitative convergence test. This is load-bearing for the paper's claim that the path-integral approach works in both quantum and classical thermodynamics; either provide a quantitative analysis of the ℏ→0 limit or explicitly restrict the conclusion to the harmonic-oscillator case.

minor comments (5)

[Eqs. (29)-(30)] The harmonic-oscillator eigenstate sums start at n=1 while the spectrum E_n^0 = ℏω0(n+1/2) begins at n=0; the normalization of the density matrix appears inconsistent as written. Please correct the index convention or the prefactor.
[Appendix A] The passages 'After some simplification' and 'After further simplifications' hide the most involved algebra, from Eqs. (A1) through (A8). Since Eq. (33) is a central exact result, please include a more detailed derivation or provide a symbolic-checkable supplementary file.
[Section III.B, Eq. (39)] The factor (-1)^{5/4} requires a branch specification, since different choices change the overall phase of the propagator.
[Fig. 1] The caption states 'Color online', but the text does not identify which curve corresponds to which value of ℏ; please add a legend or explicit description. Also define the quantum accumulated work distribution more precisely, including the range of W′ and the meaning of W_min.
[Throughout] There are minor typos, including 'protocal' for 'protocol' and 'forth coming' for 'forthcoming'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: path-integral derivation is self-contained; the sole self-citation [52] is an exact representation of the TPM characteristic function and is not used to force the target results.

full rationale

The paper's central results are not obtained by fitting a parameter or by defining the output into the input. The characteristic function starts from the standard two-point-measurement expression Eq. (9). The path-integral expression Eq. (11) is attributed to Ref. [52], which shares author H. T. Quan, but the paper immediately states the explicit actions S1 and S2 in Eq. (10), and the rewrite is an exact identity, not an ansatz fitted to the answers. This self-citation supplies a mathematical representation, not a uniqueness argument and not a fitted quantity, so under the review rules it is not load-bearing circularity. For the harmonic oscillator, Eq. (33) is explicitly compared with Eq. (17) of the independent Ref. [44], and the classical analogue Eq. (60) with Eq. (25) of the same Ref. [44]; these are external benchmarks, not renamings of the paper's inputs. For the expanding piston, Eq. (47) is followed by Eq. (50), and Eq. (51) is independently rederived in Appendix B from the time-dependent Schrödinger equation using the exact solutions of Refs. [46,56]. Although Ref. [46] also has an author overlap, Appendix B contains the derivation and the exact solutions originate with the external Ref. [56], so this is a cross-check rather than a circular reduction. The potential lack of justification for interchanging infinite sums and Fourier transforms between Eqs. (47) and (50) is a mathematical-rigor and convergence concern, not a circularity: it does not make the claimed result identical to an input by construction. No fitted parameters, no input-dependent predictions, no uniqueness theorems imported from the authors, and no renamed known results masquerading as derivations were found.

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

No free parameters are fitted. The only inputs are the initial temperature, the protocol, and the physical constants, all specified by the model. The relevant axioms are listed above.

assumptions (6)

domain assumption Semiclassical propagators are exact for the two models.
Used to evaluate all path integrals; true for harmonic oscillator and for the free particle in a box, but imported from Refs. [53,54].
domain assumption The action sum in Eq. (35) with four collision classes gives the exact propagator for the expanding piston.
Imported from Ref. [54]; supports the derivation of Eqs. (37)-(43).
domain assumption The initial density matrix is thermal and the work is defined by two projective measurements.
The two-point measurement scheme is the framework of Sec. II and is not derived.
ad hoc to paper The Fourier transform and the infinite sums in Eq. (47) may be interchanged.
No convergence or regularization argument is given; this is the weakest load-bearing assumption.
domain assumption In the classical limit the two quantum paths collapse to the classical trajectory satisfying Newton's equation.
Sec. IV Eqs. (52)-(56); given as a proof sketch but not fully demonstrated.
standard math Gaussian and Fresnel integrations are valid for complex parameters.
Used in Eq. (26) and Appendix A; assumes the usual analytic continuation and regularization of oscillatory integrals.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Path integral approach to the calculation of the characteristic function of work." pith.science (2026). https://pith.science/paper/P6F4HMWR

@misc{pith2026190809731,
  author       = {Pith},
  title        = {Pith review of: Path integral approach to the calculation of the characteristic function of work},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6F4HMWR}},
  note         = {Machine review of arXiv:1908.09731}
}

read the original abstract

Work statistics characterizes important features of a non-equilibrium thermodynamic process. But the calculation of the work statistics in an arbitrary non-equilibrium process is usually a cumbersome task. In this work, we study the work statistics in quantum systems by employing Feynman's path-integral approach. We derive the analytical work distributions of two prototype quantum systems. The results are proved to be equivalent to the results obtained based on Schr\"{o}dinger's formalism. We also calculate the work distributions in their classical counterparts by employing the path-integral approach. Our study demonstrates the effectiveness of the path-integral approach to the calculation of work statistics in both quantum and classical thermodynamics, and brings important insights to the understanding of the trajectory work in quantum systems.

Figures

Figures reproduced from arXiv: 1908.09731 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 52 canonical work pages

[44]

D´ ora, A

B. D´ ora, A. B´ acsi, and G. Zar´ and, Phys. Rev. B 86, 161109 (2012)

work page 2012
[52]

Liu, Phys

F. Liu, Phys. Rev. E 86, 010103 (2012)

work page 2012
[1]

Sekimoto, Stochastic Energetics, Lecture Notes in Physics, V ol

K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics, V ol. 799 (Springer-V erlag, Berlin, 2010)

work page 2010
[2]

(25) in Ref

This result is identical to Eq. (25) in Ref. [44]. We would like to emphasize that we obtain the classical char- acteristic function of work (Eq. (60)) by doing path integra l in the classical trajectory space. But in Ref. [44], this resul t was obtained by taking the classical limit ( ℏ → 0) of the quantum characteristic function of work (33). We believe...

work page
[3]

Jarzynski, Annu

C. Jarzynski, Annu. Rev. Phys. Cond. Matt. 2, 329 (2011)

work page 2011
[4]

trajectory work

and the Crooks ﬂuctuation theorem [7]. By analogy with the partition function, which characterize s completely the properties of a thermal equilibrium state, t he work statistics is an essential function which encodes impo r- tant information about the non-equilibrium thermodynamic process [42, 43]. Nevertheless, the calculation of the work statistics in ...

work page
[5]

Seifert, Rep

U. Seifert, Rep. Prog. Phys. 75, 126001 (2012)

2012
[6]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)

1997

Show all 63 references

[7]

Klages, W

Nonequilibrium Statistical Physics of Small Systems: Fluc tua- tion Relations and Beyond, edited by R. Klages, W. Just, and C. Jarzynski (Wiley-VCH, New Y ork, 2013)

2013
[8]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. E 56, 5018 (1997)

1997
[9]

G. E. Crooks, Phys. Rev. E 60, 2721 (1999)

1999
[10]

G. E. Crooks, Phys. Rev. E 61, 2361 (2000)

2000
[11]

Hummer and A

G. Hummer and A. Szabo, Proc. Natl. Acad. Sci. U.S.A. 98, 3658 (2001)

2001
[12]

Liphardt, S

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr, and C. B usta- mante, Science 296, 1832 (2002)

2002
[13]

Collin, F

D. Collin, F. Ritort, C. Jarzynski, S. Smith, I. Tinoco, and C. Bustamante, Nature (London) 437, 231 (2005)

2005
[14]

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J . Evans, Phys. Rev. Lett. 89, 050601 (2002)

2002
[15]

Ciliberto, R

S. Ciliberto, R. Gomez-Solano, and A. Petrosyan, Annu. Rev. Cond. Matt. 4, 235 (2013)

2013
[16]

J. P . Pekola and I. M. Khaymovich, Annu. Rev. Cond. Matt. 10, 193 (2019)

2019
[17]

Junier, A

I. Junier, A. Mossa, M. Manosas, and F. Ritort, Phys. Rev . Lett. 102, 070602 (2009)

2009
[18]

Blickle, T

V . Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechi nger, Phys. Rev. Lett. 96, 070603 (2006)

2006
[19]

S. An, J. N. Zhang, M. Um, D. Lv, Y . Lu, J. Zhang, Z. Q. Yin, H. T. Quan, K. Kim, Nat. Phys. 11, 193 (2015)

2015
[20]

T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. T. Quan, and T. Li, Phys. Rev. Lett. 120, 080602 (2018)

2018
[21]

Talkner, E

P . Talkner, E. Lutz, and P . H¨ anggi, Phys. Rev. E 75, 050102 (2007)

2007
[22]

Tasaki, arXiv:cond-mat/0009244 [cond-mat.stat-m ech]

H. Tasaki, arXiv:cond-mat/0009244 [cond-mat.stat-m ech]

arXiv
[23]

Kurchan, arXiv:cond-mat/0007360 [cond-mat.stat- mech]

J. Kurchan, arXiv:cond-mat/0007360 [cond-mat.stat- mech]

arXiv
[24]

Talkner and P

P . Talkner and P . H¨ anggi, Phys. Rev. E93, 022131 (2016)

2016
[25]

H. K. Yadalam and U. Harbola, Phys. Rev. A 99, 063802 (2019)

2019
[26]

Sampaio, S

R. Sampaio, S. Suomela, and T. Ala-Nissila, Phys. Rev. E 94, 062122 (2016)

2016
[27]

Brandner and U

K. Brandner and U. Seifert, Phys. Rev. E 93, 062134 (2016)

2016
[28]

Perarnau-Llobet, E

M. Perarnau-Llobet, E. B¨ aumer, K. V . Hovhannisyan, M. Hu- ber, and A. Acin, Phys. Rev. Lett. 118, 070601 (2017)

2017
[29]

K. Funo, M. Ueda, T. Sagawa, arXiv:1803.04778v2 [cond- mat.stat-mech]

arXiv
[30]

Kwon and M

H. Kwon and M. S. Kim, arXiv:1810.03150v1 [quant-ph]

arXiv
[31]

Liu, arXiv:1710.02311v2 [cond-mat.stat-mech]

F. Liu, arXiv:1710.02311v2 [cond-mat.stat-mech]

arXiv
[32]

Suomela, J

S. Suomela, J. Salmilehto, I. G. Savenko, T. Ala-Nissil a, and M. M¨ ott¨ onen, Phys. Rev. E91, 022126 (2015)

2015
[33]

Engel, Europhys

A. Engel, Europhys. Lett. 79, 10003 (2007)

2007
[34]

Subas ¸ı and B

Y . Subas ¸ı and B. L. Hu, Phys. Rev. E85, 011112 (2012)

2012
[35]

F. W. J. Hekking and J. P . Pekola, Phys. Rev. Lett. 111, 093602 (2013)

2013
[36]

Solinas and S

P . Solinas and S. Gasparinetti, Phys. Rev. E 92, 042150 (2015)

2015
[37]

B¨ aumer, M

E. B¨ aumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sampaio, arXiv:1805.10096v2 [quant-ph]

arXiv
[38]

Sampaio, S

R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, and T . G. Philbin, Phys. Rev. A 97, 012131 (2018)

2018
[39]

Guarnieri, N

G. Guarnieri, N. H. Y . Ng, K. Modi, J. Eisert, M. Paternos tro, and J. Goold, Phys. Rev. E 99, 050101 (2019)

2019
[40]

Strasberg, arXiv:1810.00698v4 [quant-ph]

P . Strasberg, arXiv:1810.00698v4 [quant-ph]

arXiv
[41]

B. P . V enkatesh, G. Watanabe, and P . Talkner, New J. Phys. 17, 075018 (2015)

2015
[42]

A. E. Allahverdyan, Phys. Rev. E 90, 032137 (2014)

2014
[43]

H. J. D. Miller and J. Anders, New J. Phys. 19, 062001 (2017)

2017
[45]

Goold, F

J. Goold, F. Plastina, A. Gambassi, and A. Silva, arXiv:1804.02805 (2018)

2018 arXiv
[46]

Deffner and E

S. Deffner and E. Lutz, Phys. Rev. E 77, 021128 (2008)

2008
[47]

Deffner, O

S. Deffner, O. Abah, and E. Lutz, Chem. Phys. 375, 200 (2010)

2010
[48]

H. T. Quan and C. Jarzynski, Phys. Rev. E 85, 031102 (2011)

2011
[49]

Z. Gong, S. Deffner, and H. T. Quan, Phys. Rev. E 90, 062121 (2014)

2014
[50]

Talkner, P

P . Talkner, P . S. Burada, and P . H¨ anggi, Phys. Rev. E78, 011115 (2008)

2008
[51]

Fei and H

Z. Fei and H. T. Quan, to be published
[53]

Z. Fei, H. T. Quan, and F. Liu, Phys. Rev. E 98, 012132 (2018)

2018
[54]

Funo and H

K. Funo and H. T. Quan, Phys. Rev. Lett. 121, 040602 (2018)

2018
[55]

Husimi, Prog

K. Husimi, Prog. Theor. Phys. 9, 381 (1953)

1953
[56]

M. G. E. Da Luz and B. K. Cheng, J. Phys. A: Math. Gen. 25, L1043 (1992)

1992
[57]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, Cambridge, Massachusetts, 2007)

2007
[58]

S. W. Doescher and M. H. Rice, Am. J. Phys. 37, 1246 (1969)

1969
[59]

Z. Gong, Y . Lan, H. T. Quan, Phys. Rev. Lett. 117, 180603 (2016)

2016
[60]

R. C. Lua and A. Y . Grosberg, J. Phys. Chem. B. 109, 6805 (2005)

2005
[61]

Bartolotta and S

A. Bartolotta and S. Deffner, Phys. Rev. X 8, 011033 (2018)

2018
[62]

Ortega, E

A. Ortega, E. McKay, ´A. M. Alhambra, and E. Mart´ ın- Mart´ ınez, Phys. Rev. Lett.122, 240604 (2019)

2019
[63]

J. J. Dong and Y . F. Yang, Phys. Rev. B 100, 035124 (2019)

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [44]

D´ ora, A

B. D´ ora, A. B´ acsi, and G. Zar´ and, Phys. Rev. B 86, 161109 (2012)

work page 2012

[2] [52]

Liu, Phys

F. Liu, Phys. Rev. E 86, 010103 (2012)

work page 2012

[3] [1]

Sekimoto, Stochastic Energetics, Lecture Notes in Physics, V ol

K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics, V ol. 799 (Springer-V erlag, Berlin, 2010)

work page 2010

[4] [2]

(25) in Ref

This result is identical to Eq. (25) in Ref. [44]. We would like to emphasize that we obtain the classical char- acteristic function of work (Eq. (60)) by doing path integra l in the classical trajectory space. But in Ref. [44], this resul t was obtained by taking the classical limit ( ℏ → 0) of the quantum characteristic function of work (33). We believe...

work page

[5] [3]

Jarzynski, Annu

C. Jarzynski, Annu. Rev. Phys. Cond. Matt. 2, 329 (2011)

work page 2011

[6] [4]

trajectory work

and the Crooks ﬂuctuation theorem [7]. By analogy with the partition function, which characterize s completely the properties of a thermal equilibrium state, t he work statistics is an essential function which encodes impo r- tant information about the non-equilibrium thermodynamic process [42, 43]. Nevertheless, the calculation of the work statistics in ...

work page

[7] [5]

Seifert, Rep

U. Seifert, Rep. Prog. Phys. 75, 126001 (2012)

2012

[8] [6]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)

1997