Bulg. J. Phys. vol.45 no.2 (2018), pp. 221-246

Quantum Fluctuation Theorems and Work–Energy Relationships with Due Regard for Convergence, Dissipation and Irreversibility

C.M. Van Vliet
Department of Physics, University of Miami, Coral Gables, Florida 33124-0530, USA
Abstract. In this article, firstly the fluctuation theorems (FT) for expended work in a driven nonequilibrium system, isolated or thermostatted, as formulated originally by Crooks and Tasaki, together with the ensuing Jarzynski work–energy (W–E) relationships, will be discussed and reobtained. Secondly, the fluctuation theorems for entropy flow due to Evans, Cohen and Morriss with extensions by many researchers, a.o. Evans and Searles, Gallavotti and Cohen, Kurchan, Lebowitz and Spohn, and Harris and Schütz will be reconsidered. Our treatment will be fully quantum-statistical, being an extension of our previous research reported in Phys. Rev. E, 2012. While a true explosion of papers took place after the initial articles at the turn of the century, virtually all of these suffered from one or more of the following deficiencies: (i). The arguments are based on classical trajectories in phase space; this is true for Christopher Jarzynski's original work, as well as for Crooks' paper; better fares Tasaki's quantum paper in the arXiv. (ii). Many quantum treatments involve the 'pure' von Neumann equation or 'non-reduced' Heisenberg operators. This is regrettable particularly for an otherwise beautiful derivation in the complex plane by Talkner and Hänggi; correlation functions for non-reduced Heisenberg operators do not converge. As we pointed out in many papers and in our recent book: Kubo Linear Response Theory (LRT) is a hollow shell until proper randomization (Kubo: 'stochasticization') is introduced and carried out. Hence, the interactions λσ with the reservoir or internal causes must explicitly be considered. Taking the trace over these, the resulting semigroup has complete positivity, exhibits non-unitarity for the time evolution, dissipation and irreversibility, the general result being the Lindblad quantum master equation (QME). In the physical literature a more explicit result is obtained after application of the 'weak coupling–long time' limit, developed long ago by Leon Van Hove. In our cited paper these results have been extended to non-stationary processes, the result being concordant with work by Gaspard. (iii). Whereas a few dozen papers use a stochastic approach with some Master Equation as leitmotiv, this author found most treatments wanting and not in accord with the general tenets spelled out by Lindblad and others, e.g. Breuer and Petruccione. In particular, a stochastic treatment with 'jump-induced' random trajectories as by Harris and Schütz is begging the question. While a number of their relationships will still be employed, our Markov probability P(σf ,tfσ0 ,t0) shall only denote the two state-points, with no reference whatsoever to stochastic trajectories, these being meaningless in a quantum description. A straightforward reasoning gives the desired results.

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