A Quantum Fluctuation Theorem

Jorge Kurchan (P.M.M.H. Ecole Sup\'erieure de Physique et Chimie Industrielles)

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A Quantum Fluctuation Theorem

Jorge Kurchan (P.M.M.H. Ecole Supérieure de Physique et Chimie Industrielles)

Statistical Mechanics · Submitted 24 Jul 2000 · 17 references

Pith's verdict: The isolated-system theorem is a clean, correct piece of algebra; the thermostated half is a formal identity plus two physics assumptions that are asserted, not proved, and the abstract doesn't say so.

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arXiv IDcond-mat/0007360 (version 2)

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CitationJorge Kurchan (P.M.M.H. Ecole Sup\'erieure de Physique et Chimie Industrielles). "A Quantum Fluctuation Theorem." arXiv:cond-mat/0007360 (2000). https://pith.science/paper/IVQE6B5P

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Worth your time, with one caveat about where the weight sits. The core result — P(e)/P(-e) = e^{βe} for a periodically driven quantum system over an integer number of cycles — is right, and the derivation is short enough to check in an afternoon. The move is: write P(e) as a Fourier transform of Q(λ) = Tr[U† e^{λH} U e^{-(λ+β)H}], use U† = U* (which follows from real matrix elements plus symmetric cycles) to get the KMS-like symmetry Q(λ) = Q(-λ-β), bound Q on the strip -β < Re λ < 0 with a Cauchy-Schwarz estimate, and shift the contour. No fitting, no free parameters, no circular step. The Jarzynski corollary (Eq. 15) and the second-law inequality (Eq. 16) drop out immediately. That is a genuine quantum statement, and it is not in the classical literature he cites, nor in Yukawa [11], which is a different relation. Credit where due: this is the cleanest version of the argument I have seen for the quantum case, and the analyticity bound in Eqs. (12)-(13) is the part people usually hand-wave and he actually does it. The real-Hamiltonian and symmetric-cycle restriction is a real narrowing of scope — magnetic fields and spin-orbit coupling are out — but he states it in Eq. (1) and the surrounding text, so it is disclosed, not smuggled. My only complaint there is that the abstract's 'model-independent' oversells it slightly; a phrase like 'for time-reversal-symmetric drives' would be more honest. The soft spot is the thermostated section. The formal identity with D = H_bath + O is fine given the product initial state, and the trace conditions (24)-(26) are handled properly. But the step from that identity to a statement about total energy transferred to the bath needs conditions i) and ii) on p. 6 — finite ⟨O⟩ and bath energy growing linearly in cycle number — and those are asserted, not derived. Footnote [16] concedes that energy localization in kicked systems can break ii). So the thermostated claim is conditional on unproven physics, and the abstract presents both halves as one relation. That is the thing a referee should press on. Who should read it: anyone working on quantum thermodynamics, quantum work relations, or shot-noise/mesoscopic detection. The isolated result is citable as-is; the bath result is a target for follow-up. Send it out.

Referee report

Major objections (2)

[Abstract (s3); p. 6, conditions i)-ii); footnote [16]] The abstract presents P(e)/P(-e)=e^{βe} as a single relation covering both the isolated and thermostated cases, but the thermostated derivation on p. 6 requires conditions i) (finite ⟨O⟩) and ii) (linear-in-cycles bath energy growth), which are asserted rather than derived, and footnote [16] concedes that energy localization in isolated kicked systems can violate ii). The formal identity with D = H_bath + O (Eq. 23) is correct given the product initial state, but the step to 'the total amount of energy transferred to the bath' is conditional. Please qualify the abstract and state conditions i)-ii) as explicit assumptions with a discussion of when they hold and fail.
[Abstract (s3); Eq. (1) and the paragraph before Eq. (2)] The abstract calls the relation 'model-independent,' but the derivation of Eq. (3) requires the Hamiltonian to have real matrix elements at every instant and the cycles to be symmetric, H(t)=H(-t) (Eq. 1). This excludes magnetic fields and spin-orbit coupling, which are physically important. The authors state the restriction in the text, but the abstract does not reflect it. Please qualify 'model-independent' and add a brief remark on whether the theorem extends to complex Hamiltonians with an antiunitary time-reversal symmetry.

Minor objections (11)

[Eq. (1)] The first equality H(q,p,t)=H(q,p,t) is trivial as printed; presumably the intended statement is H(q,p,t)=H(q,p,t+t_o)=H(q,p,-t). Please correct.
[Eq. (2)] The sign convention U_r ≡ e^{i/ℏ τ H_r} differs from the standard U = e^{-iHt/ℏ}; please check and state the convention explicitly.
[Eq. (13)] The bound shows analyticity only on the open strip -β < Reλ < 0, while the text claims the closed strip -β ≤ Reλ ≤ 0; please comment on the boundary values for infinite-dimensional systems.
[Eq. (16)] The two sides of the inequality appear identical as printed; the second expression should be the annealed average -(1/β)ln⟨e^{-βe}⟩ and the last should be ⟨e⟩. Please correct the typo.
[Eq. (17)] The phrase 'annealed average ≤ quenched average' is Jensen's inequality in this context; the terminology is nonstandard here and should be clarified.
[Eq. (9)] Please specify the contour used in the integral representation of the delta function.
[Eq. (11)] Please show the intermediate steps in the chain Q_D(-λ*-β)* = Q_D(-λ-β), in particular the use of transposition and Eq. (3).
[Eq. (15)] Please define the overline precisely (average over quantum amplitudes and initial conditions).
[Footnote [16]] The energy-localization failure mode is important enough to be discussed in the main text, not only in a footnote.
[Reference [11]] Reference [11] is listed as 'to appear'; since Eq. (15) is derived here, the citation should be framed as related work rather than as the source of the result.
[General] The paper does not discuss the classical limit or the relation to the classical fluctuation theorem in detail; a brief remark would help the reader place the result.

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The paper

The paper claims

A driven quantum system's chance of gaining energy $e$ exceeds its chance of losing $e$ by exactly $e^{βe}$ — with no free parameters.

The reading

Kurchan asks whether the fluctuation theorems of classical non-equilibrium statistical mechanics survive in quantum mechanics, for a system pushed hard and periodically by an external force rather than nudged weakly around equilibrium. His answer is a single ratio: after an integer number of drive cycles, the probability $P(e)$ of an energy change $e$ and the probability $P(-e)$ of the reverse change are related by $P(e)/P(-e)=e^{βe}$, where $β$ is the inverse temperature of the initial preparation and nothing else appears — no details of the force, the potential, or the coupling. The same statement is claimed twice over: for an isolated system, where $e$ is the system's own energy increase, and for a system tied to a bath of harmonic oscillators, where $e$ is the energy handed to the bath, i.e. its entropy increase in units of $k_B$. If it holds, it converts a symmetry of the equations of motion into a checkable, model-independent prediction about the statistics of energy transfer in driven quantum devices, and it delivers as corollaries a quantum Jarzynski relation $⟨e^{-βe}⟩=1$ and a derivation of the second law as the statement that an annealed average cannot exceed a quenched one.

What carries the argument

The load-bearing object is the KMS-like identity $Q_D(λ)=Q_D(-λ-β)$, obtained by transposition, cyclic permutation of the trace, and the time-reversal property $U^{†}=U^{*}$ of the cycle evolution operator. It is not the ratio itself but the analyticity strip that makes the identity bite: the bound $|Q(λ)|^2 ≤ 2(Tr e^{-2{Re(λ)+β}D})(Tr e^{2Re(λ)D})$ keeps $Q$ finite between $Re(λ)=-β$ and $Re(λ)=0$, so the contour in the Laplace representation of $P(e)$ can be deformed onto the line where the identity converts $P(e)$ into $P(-e)e^{βe}$. In the thermostated case the same bound splits into a bath trace, which is just a partition function, and a system trace, which requires $∑_α p_α^{2μ}$ to be finite for $0<μ<1$ — a mild condition on the tail of the initial distribution.

Extended reading notes

Core claim

The paper establishes a quantum fluctuation theorem for a system governed by a time-dependent Hamiltonian whose explicit time dependence is periodic and symmetric under time reversal, $H(t)=H(-t)$, and whose matrix elements are real at every instant. Under those conditions the evolution operator over a whole number of cycles satisfies $U^{†}=U^{*}$. From that single symmetry Kurchan derives a KMS-like identity for any real symmetric operator $D$, $Q_D(λ)=Q_D(-λ-β)$ with $Q_D(λ)=Tr[U^{†}e^{λD}Ue^{-(λ+β)D}]$, and then shows that $Q$ is analytic in the strip $-β<Re(λ)<0$ by bounding it with traces of $e^{±2Re(λ)D}$. Since the distribution $P(e)$ is a Laplace transform of $Q$, the analyticity lets him slide the integration contour from $Re(λ)=0$ to $Re(λ)=-β$, and the ratio $P(e)/P(-e)=e^{βe}$ falls out. The thermostated version replaces $D$ by bath energy plus an arbitrary real symmetric system observable $O$, with the initial state a product of canonically distributed bath and system wavefunctions; the same identity then governs the joint energy change, and for long times, provided $O$ stays bounded and the bath energy grows linearly in the number of cycles, it becomes a statement about the entropy produced in the bath.

Load-bearing premise

The whole derivation rests on the evolution operator over a cycle obeying $U^{†}=U^{*}$, which requires the Hamiltonian to have real matrix elements at every instant and the drive to be symmetric in time, $H(t)=H(-t)$; systems with magnetic fields or spin-orbit coupling fall outside that condition, and the paper states the restriction without exploring how far it reaches.

Editorial extensions

If this is right

A quantum Jarzynski work relation follows directly: averaging $e^{-βe}$ over the distribution gives one, $⟨e^{-βe}⟩=1$, for any number of drive cycles and any strength of the drive.
The second law appears as a corollary of that identity, in the form $⟨e⟩ ≥ 0$, i.e. the average energy transferred to the bath cannot be negative over a cycle.
For a thermostated system at long times, the theorem becomes a statement about the entropy produced in the bath per cycle, so a measured ratio of forward and backward energy-transfer probabilities tests a property of the asymptotic periodic quantum state.
Because no model parameters enter the ratio, the relation is a parameter-free benchmark for experiments on driven quantum devices, such as the shot-noise measurements that motivated the work.

Reading between the lines

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

The theorem as written covers only Hamiltonians with real matrix elements, which excludes magnetic fields and spin-orbit coupling; extending the argument to the antiunitary time-reversal operator of spinful systems would be the natural next step, and the ratio's fate there is a live question the paper leaves open.
The derivation needs no assumption that the drive is slow, so the relation is a rare exact statement in a strongly driven regime where linear response is unavailable; that makes it a candidate calibration tool for numerics on periodically kicked quantum systems, where $P(e)$ can be computed exactly for small systems.
The classical limit of the same construction should recover the known work and entropy fluctuation relations, which suggests the quantum ratio could be used to identify where a driven system stops behaving classically rather than as an independent postulate.
The paper's own footnote flags that the long-time thermostated statement requires the absence of energy localization, which can occur in isolated kicked systems; that is a real boundary on the extension, and quantifying when a driven system localizes in energy would tell experimenters which drive parameters keep the theorem usable.

Original abstract

We consider a quantum system strongly driven by forces that are periodic in time. The theorem concerns the probability $P(e)$ of observing a given energy change $e$ after a number of cycles. If the system is thermostated by a (quantum) thermal bath, $e$ is the total amount of energy transferred to the bath, while for an isolated system $e$ is the increase in energy of the system itself. Then, we show that $P(e)/P(-e)=e^{\beta e}$, a parameter-free, model-independent relation.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We consider a quantum system strongly driven by forces that are periodic in time. The theorem concerns the probability $P(e)$ of observing a given energy change $e$ after a number of cycles. If the system is thermostated by a (quantum) thermal bath, $e$ is the total amount of energy transferred to the bath, while for an isolated system $e$ is the increase in energy of the system itself. Then, we show that $P(e)/P(-e)=e^{\beta e}$, a parameter-free, model-independent relation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider a quantum system strongly driven by forces that are periodic in time.	background 0.80	0.34	0.02	asserted 0.89
s1	The theorem concerns the probability $P(e)$ of observing a given energy change $e$ after a number of cycles.	contribution 0.71	0.31	0.04	asserted 0.98
s2	If the system is thermostated by a (quantum) thermal bath, $e$ is the total amount of energy transferred to the bath, while for an isolated system $e$ is the increase in energy of the system itself.	interpretation 0.41	0.15	0.03	asserted 1.00
s3	Then, we show that $P(e)/P(-e)=e^{\beta e}$, a parameter-free, model-independent relation.	result 0.99	0.94	0.94	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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full text
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References

Works cited (17)

[11]

S. Yukawa, cond-mat/0007456, to appear in Phys. Soc. Jpn. 69 (2000). I

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[16]

For this to happen there should be no ‘localisation’ in energy, as c an happen in an isolated quantum ‘kicked’ system[13]

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J.Kurchan, J.Phys.A 31, 3719 (1998)

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[10]

G.E.Crooks, cond-mat/9901352, to appear in Phys.Rev.E; also “ Path Ensemble Averages in Systems Driven Far From Equilibrium”, unpublished preprint

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U. Gavish, Y. Levinson and Y. Imry, Phys. Rev. B 62 R10639 (2000)

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[13]

For low dimensional ‘kicked’ systems the literature is vast. The r eader may ﬁnd useful articles and references in the volume in honour of B. Chirikov, Physica D131 ( 1999)

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[14]

R. Kubo, M. Toda and N. Hashitume; Statistical Physics II. Nonequilibrium Statistical Me- chanics, Springer-Verlag, 1992 S. R. De Groot and P. Mazur; Non-equilibrium thermodynamics, Dover Pub., New York, 1984. G. Parisi, Statistical Field Theory, Frontiers in Physics, Lecture Notes Series, Addison-Wesley (1988)

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G. W. Ford, M. Kac and M. Mazur, J. Math. Phys. 6, 504 (1965) G. W. Ford and M. Kac, J. Stat. Phys 46, 803 (1987). Two reviews are: G. Zhou, Z. Su, B Hao, Y. Lu, Phys. Rep. 118, 1 (1985) H. Grabert, P. Schramm and G-L Ingold, Phys. Rep. 168, 115 (1988) see also: A. Caldeira and A. Legget, Phys Rev. A31, 1059 (1985). B. L. Hu, J. P. Paz and Y. Zhang, Phys....

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[17]

Note that whenever we invoke the limit of large time-diﬀerences, we understand it as taken after the limit of large heat bath size M . 8

View on Pith

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How later papers used it

253 later papers cite it

Evidence: substantial. 252 of the citing papers share no author with it, and 32 of those carry a confident label from Pith's citation classifier.

Pith read 33 of the 423 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 19 papers with this label

Used the definition due to this paper and another work, which has a transparent physical meaning as the energy change in the system.

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics (2015) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

253 arXiv papers cite this in 423 sentences

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Papers citing it · 243 sentences without a confident label

Contests it 1 paper

This result demands us to change the derivation of the quantum Jarzynski equality from the previous ones [ref] based on the conventional approach, which employed the internal unitary time evolution and regarded the measured value of the energy loss of the internal system as the “work”, whereas the energy gain of the external system is the actual work that we can use.

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Specifically, we present generalised versions of the Tasaki-Crooks relation (TCR) [ref] and the quantum Jarzynski equality (QJE) [ref] suitable to describe fluctuations in systems with an arbitrary, and possibly variable, number of charges.

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The above relation is a quantum version of the Jarzynski equation [ref] (see also [ref] ) adapted to our initial state in the form of a grand-canonical equilibrium state.

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We note that the same argument is also valid for quantum systems that obey unitary dynamics [ref] , by replacing the Shannon entropy by the von Neumann one.

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Based on the two-point measurement scheme [ref] , the probability distribution function of a quantum work at time $t$ is written as P(W, t) = ∑_nkδ(W-δE_kn(t))P_kn(t)

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Under these assumptions, we can then take advantage of a formal mathematical apparatus recently developed to study the thermodynamics of quantum systems driven out of equilibrium by unitary evolution [ref] .

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Therefore we will use the two energy measurement approach as an operational definition of work [ref] and heat [ref] .

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Since we apply a time reversal rule which exactly reverses the quantum state dynamics, and the arrow of time is computed without any projective end point measurement, the resulting FT with absolute irreversibility is different from its previous appearences [ref] , and from other quantum generalizations of FTs in general [ref] .

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Hence, we obtain the same form as the previous derivations [ref] of the Jarzynski equality under the cyclic process.

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Our calculation is structurally similar to the others in the literature [ref] .

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However, it can be shown that if one imposes that the statistics of work is consistent with the standard fluctuation theorems, the distribution of work should be defined by the two-point measurement protocol [ref] .

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We now review the quantum Jarzynski equality (QJE) [ref] in order to establish the necessary background for the rest of the paper.

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In this case the work $W$ is defined as a difference between the final and initial energies of the quantum system [ref] .

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In the theoretical investigations of the transient quantum fluctuation theorems [ref] , these measurements are projective [ref] .

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [11]

S. Yukawa, cond-mat/0007456, to appear in Phys. Soc. Jpn. 69 (2000). I

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For this to happen there should be no ‘localisation’ in energy, as c an happen in an isolated quantum ‘kicked’ system[13]

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U. Gavish, Y. Levinson and Y. Imry, Phys. Rev. B 62 R10639 (2000)

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For low dimensional ‘kicked’ systems the literature is vast. The r eader may ﬁnd useful articles and references in the volume in honour of B. Chirikov, Physica D131 ( 1999)

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Note that whenever we invoke the limit of large time-diﬀerences, we understand it as taken after the limit of large heat bath size M . 8

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