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arXiv:cond-mat/0703189 · Physical Review E

Fluctuation theorems: Work is not an observable

Peter Talkner A, Eric Lutz, Peter Hänggi A

Published May 16, 2007• 754 citations
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Abstract

The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system's Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.

DOI
10.1103/physreve.75.050102

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The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system's Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.
We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4].We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4]. An interesting fact is that while the quantum version of the Jarzynski equality remains satisfied even in the presence of quantum correlations, the individual thermodynamical work moments in the expansion of the free energy are, in fact, sensitive to the genuine quantum correlations [5]. Whether this is a fortuitous coincidence remains to be seen, but it certainly goes towards explaining why the laws of thermodynamics happen to be so robust as to be independent of the underlying micro-physics.We intend to elucidate the subtle connection between Jarzynski’s relation and the “quantum arrow of time”. Finally we comment on the fact that our scheme for measuring the quantum work characteristic function belongs to the computational class known as DCQ1 [6], namely all computations that can be performed with only one pure qubit (and N maximally mixed ones).
The characteristic function for the joint measurement of the changes of two or more commuting observables upon an external forcing of a quantum system is derived. In particular, the statistics of the internal energy, the exchanged heat and the work of a quantum system that weakly couples to its environment are determined in terms of the energy changes of the system and the environment due to the action of a classical, external force on the system. If the system and the environment initially are in a canonical equilibrium, the work performed on the system is shown to satisfy the Tasaki–Crooks theorem and the Jarzynski equality.
Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states, explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the corresponding canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time-reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.
The characteristic function for the joint measurement of the changes of two commuting observables upon an external forcing of a quantum system is derived. In particular, the statistics of the internal energy, the exchanged heat and the work of a quantum system that weakly couples to its environment is determined in terms of the energy changes of the system and the environment due to the action of a classical, external force on the system. If the system and environment initially are in a canonical equilibrium, the work performed on the system is shown to satisfy the Tasaki-Crooks theorem and the Jarzynski equality. PACS numbers: 05.30.-d,05.70.Ln,05.40-a
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.
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.