Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1506.01185 · Physical Review E

Principle of minimal work fluctuations

Gaoyang Xiao, Jiangbin Gong

Published Aug 19, 2015• 10 citations
arXivHTMLPDF

Abstract

Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$, a change in the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ may impact how rapidly the statistical average of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ converges towards the theoretical value ${e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$, where $W$ is the work, $\ensuremath{\beta}$ is the inverse temperature, and $\mathrm{\ensuremath{\Delta}}F$ is the free energy difference between two equilibrium states. Motivated by our previous study aiming at the suppression of work fluctuations, here we obtain a principle of minimal work fluctuations. In brief, adiabatic processes as treated in quantum and classical adiabatic theorems yield the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$. In the quantum domain, if a system initially prepared at thermal equilibrium is subjected to a work protocol but isolated from a bath during the time evolution, then a quantum adiabatic process without energy level crossing (or an assisted adiabatic process reaching the same final states as in a conventional adiabatic process) yields the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where $W$ is the quantum work defined by two energy measurements at the beginning and at the end of the process. In the classical domain where the classical work protocol is realizable by an adiabatic process, then the classical adiabatic process also yields the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$. Numerical experiments based on a Landau-Zener process confirm our theory in the quantum domain, and our theory in the classical domain explains our previous numerical findings regarding the suppression of classical work fluctuations [G. Y. Xiao and J. B. Gong, Phys. Rev. E 90, 052132 (2014)].

DOI
10.1103/physreve.92.022130

Related papers

Quantum work fluctuations in connection with the Jarzynski equality

Juan D. Jaramillo, Jiawen Deng, Jiangbin Gong

A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\ensuremath{\Delta}}F$ of a system at inverse temperature $\ensuremath{\beta}$ from an ensemble average of nonequilibrium exponential work, i.e., $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\ensuremath{\Delta}}F$ is determined by the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, denoted $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$. We discover in this work that $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.

arXiv:1701.07603•Physical review. E•Oct 9, 2017•13 citations

Experimental study on the principle of minimal work fluctuations

Wei Cheng, Wenquan Liu, Yang Wu +5 more

The central quantity in the celebrated quantum Jarzynski equality is ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where $W$ is work and $\ensuremath{\beta}$ is the inverse temperature. The impact of quantum randomness on the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and hence on the predictive power of the Jarzynski estimator is an important problem. Working on a single nitrogen-vacancy center in diamond and riding on an implementation of the two-point measurement of nonequilibrium work with single-shot readouts, we have conducted a direct experimental investigation of the relationship between the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and adiabaticity of nonequilibrium work protocols. It is observed that adiabatic processes minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, thus verifying an early theoretical concept, the so-called principle of minimal work fluctuations. Furthermore, it is experimentally demonstrated that shortcuts-to-adiabaticity control can be exploited to minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ in fast work protocols. Our work should stimulate further experimental studies of quantum effects on the bias and error in the estimates of free-energy differences based on the Jarzynski equality.

arXiv:2310.14461•Physical Review A•Oct 23, 2023•5 citations

Nonequilibrium quantum fluctuations of work

ARMEN E. ALLAHVERDYAN

The concept of work is basic for statistical thermodynamics. To gain a fuller understanding of work and its (quantum) features, it needs to be represented as an average of a fluctuating quantity. Here I focus on the work done between two moments of time for a thermally isolated quantum system driven by a time-dependent Hamiltonian. I formulate two natural conditions needed for the fluctuating work to be physically meaningful for a system that starts its evolution from a nonequilibrium state. The existing definitions do not satisfy these conditions due to issues that are traced back to noncommutativity. I propose a definition of fluctuating work that is free of previous drawbacks and that applies for a wide class of nonequilibrium initial states. It allows the deduction of a generalized work-fluctuation theorem that applies for an arbitrary (out-of-equilibrium) initial state.

arXiv:1404.4190•Physical Review E•Sep 26, 2014•158 citations

Quantum Rényi-Jarzynski Equality

Benjamin Bobell, Mert Okyay, Rahul Nandkishore

The Jarzynski equality provides a strict link between nonequilibrium work and equilibrium free energy changes. Its typical quantum formulations, however, rely on measurement protocols that destroy coherence. In this Letter, we use the resource-theoretic approach to derive a non-destructive quantum Jarzynski equality conditioned on the outcomes of an arbitrary bath observable. This yields the Rényi-Jarzynski equality, which quantifies a finite bath's drift from equilibrium under a non-adiabatic drive via the Rényi $k$-divergence. We further demonstrate that the Rényi-Jarzynski equality provides a tunable cost function for quantum optimal control problems where minimizing bath drift is desired, such as state preparation and gate design, enabling the minimization of cross-talk in finite quantum systems. Our toy model exhibits a transition between competing minima for some critical value of $k$, illustrating how the Rényi order tunes sensitivity to different regions of a bath distribution. Strikingly, when drive parameters vary across bath energy levels, minimizing bath drift requires generating system-bath entanglement.

cond-mat.stat-mechCross-listedarXiv:2608.19320•Aug 21, 2026•0 citations

Operational Work Fluctuation Theorem for Open Quantum Systems

Konstantin Beyer, Walter T. Strunz

The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.

arXiv:2408.13417•Physical Review Letters•Apr 9, 2025•2 citations

From Quantum Dynamics to the Second Law of Thermodynamics

Hal Tasaki

In quantum systems which satisfy the hypothesis of equal weights for eigenstates [4], the maximum work principle (for extremely slow and relatively fast operation) is derived by using quantum dynamics alone. This may be a crucial step in establishing a firm connection between macroscopic thermodynamics and microscopic quantum dynamics. For special models introduced in [4,5], the derivation of the maximum work principle can be executed without introducing any unproved assumptions.

arXiv:cond-mat/0005128•May 8, 2000•0 citations
Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality \ensuremath⟨e\ensuremath−\ensuremathβW\ensuremath⟩=e\ensuremath−\ensuremathβ\ensuremathΔF\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}\ensuremath⟨e\ensuremath−\ensuremathβW\ensuremath⟩=e\ensuremath−\ensuremathβ\ensuremathΔF, a change in the fluctuations of e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW may impact how rapidly the statistical average of e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW converges towards the theoretical value e\ensuremath−\ensuremathβ\ensuremathΔF{e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}e\ensuremath−\ensuremathβ\ensuremathΔF, where WWW is the work, \ensuremathβ\ensuremath{\beta}\ensuremathβ is the inverse temperature, and \ensuremathΔF\mathrm{\ensuremath{\Delta}}F\ensuremathΔF is the free energy difference between two equilibrium states. Motivated by our previous study aiming at the suppression of work fluctuations, here we obtain a principle of minimal work fluctuations. In brief, adiabatic processes as treated in quantum and classical adiabatic theorems yield the minimal fluctuations in e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW. In the quantum domain, if a system initially prepared at thermal equilibrium is subjected to a work protocol but isolated from a bath during the time evolution, then a quantum adiabatic process without energy level crossing (or an assisted adiabatic process reaching the same final states as in a conventional adiabatic process) yields the minimal fluctuations in e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW, where WWW is the quantum work defined by two energy measurements at the beginning and at the end of the process. In the classical domain where the classical work protocol is realizable by an adiabatic process, then the classical adiabatic process also yields the minimal fluctuations in e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW. Numerical experiments based on a Landau-Zener process confirm our theory in the quantum domain, and our theory in the classical domain explains our previous numerical findings regarding the suppression of classical work fluctuations [G. Y. Xiao and J. B. Gong, Phys. Rev. E 90, 052132 (2014)].
A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\ensuremath{\Delta}}F$ of a system at inverse temperature $\ensuremath{\beta}$ from an ensemble average of nonequilibrium exponential work, i.e., $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\ensuremath{\Delta}}F$ is determined by the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, denoted $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$. We discover in this work that $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.
The central quantity in the celebrated quantum Jarzynski equality is ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where WWW is work and $\ensuremath{\beta}$ is the inverse temperature. The impact of quantum randomness on the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and hence on the predictive power of the Jarzynski estimator is an important problem. Working on a single nitrogen-vacancy center in diamond and riding on an implementation of the two-point measurement of nonequilibrium work with single-shot readouts, we have conducted a direct experimental investigation of the relationship between the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and adiabaticity of nonequilibrium work protocols. It is observed that adiabatic processes minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, thus verifying an early theoretical concept, the so-called principle of minimal work fluctuations. Furthermore, it is experimentally demonstrated that shortcuts-to-adiabaticity control can be exploited to minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ in fast work protocols. Our work should stimulate further experimental studies of quantum effects on the bias and error in the estimates of free-energy differences based on the Jarzynski equality.
The concept of work is basic for statistical thermodynamics. To gain a fuller understanding of work and its (quantum) features, it needs to be represented as an average of a fluctuating quantity. Here I focus on the work done between two moments of time for a thermally isolated quantum system driven by a time-dependent Hamiltonian. I formulate two natural conditions needed for the fluctuating work to be physically meaningful for a system that starts its evolution from a nonequilibrium state. The existing definitions do not satisfy these conditions due to issues that are traced back to noncommutativity. I propose a definition of fluctuating work that is free of previous drawbacks and that applies for a wide class of nonequilibrium initial states. It allows the deduction of a generalized work-fluctuation theorem that applies for an arbitrary (out-of-equilibrium) initial state.
The Jarzynski equality provides a strict link between nonequilibrium work and equilibrium free energy changes. Its typical quantum formulations, however, rely on measurement protocols that destroy coherence. In this Letter, we use the resource-theoretic approach to derive a non-destructive quantum Jarzynski equality conditioned on the outcomes of an arbitrary bath observable. This yields the Rényi-Jarzynski equality, which quantifies a finite bath's drift from equilibrium under a non-adiabatic drive via the Rényi kkk-divergence. We further demonstrate that the Rényi-Jarzynski equality provides a tunable cost function for quantum optimal control problems where minimizing bath drift is desired, such as state preparation and gate design, enabling the minimization of cross-talk in finite quantum systems. Our toy model exhibits a transition between competing minima for some critical value of kkk, illustrating how the Rényi order tunes sensitivity to different regions of a bath distribution. Strikingly, when drive parameters vary across bath energy levels, minimizing bath drift requires generating system-bath entanglement.
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.