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arXiv:0705.1252 · Journal of Physics A Mathematical and Theoretical

The Tasaki–Crooks quantum fluctuation theorem

Peter Talkner A, Peter Hänggi A

Published Jun 12, 2007• 143 citations
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Abstract

Starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system in Talkner et al (2007 Phys. Rev. E 75 050102 ( Preprint cond-mat/0703213)) the quantum version of the Crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse Fourier transformation.

DOI
10.1088/1751-8113/40/26/f08

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Starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system in Talkner et al (2007 Phys. Rev. E 75 050102 ( Preprint cond-mat/0703213)) the quantum version of the Crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse Fourier transformation.
Crooks fluctuation theorem in PT{\mathscr{PT}}PT-symmetric quantum mechanics
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 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.