Maximum entropy production rate in quantum thermodynamics

Abstract

In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrödinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsager's reciprocity and Callen's fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.