Time-reversal symmetry in nonstationary Markov processes with application to some fluctuation theorems

Abstract

Nonequilibrium processes require that the density operator of an interacting system with Hamiltonian H(t) = H(0)(t)+λV converges and produces entropy. Employing projection operators in the state space, the density operator is developed to all orders of perturbation and then resummed. In contrast to earlier treatments by Van Hove [Physica 21, 517 (1955)] and others [U. Fano, Rev. Mod. Phys. 29, 74 (1959); U. Fano, in Lectures on the Many-Body Problem, Vol 2, edited by E. R. Caniello (Academic Press, New York, 1964); R. Zwanzig, in Lectures in Theoretical Physics, Vol. III, edited by W. E. Britten, B. W. Downs, and J. Downs (Wiley Interscience, New York, 1961), pp. 116-141; K. M. Van Vliet, J. Math. Phys. 19, 1345 (1978); K. M. Van Vliet, Can. J. Phys. 56, 1206 (1978)], closed expressions are obtained. From these we establish the time-reversal symmetry property P(γ,t|γ',t') = Pγ',t'|γ,t), where the tilde refers to the time-reversed protocol; also a nonstationary Markovian master equation is derived. Time-reversal symmetry is then applied to thermostatted systems yielding the Crooks-Tasaki fluctuation theorem (FT) and the quantum Jarzynski work-energy theorem, as well as the general entropy FT. The quantum mechanical concepts of work and entropy are discussed in detail. Finally, we present a nonequilibrium extension of Mazo's lemma of linear response theory, obtaining some applications via this alternate route.