Maximal-entropy-production-rate nonlinear quantum dynamics compatible with second law, reciprocity, fluctuation-dissipation, and time-energy uncertainty relations

In view of the recent quest for well-behaved nonlinear extensions of the traditional Schroedinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, in this paper, together with a review of the general features of the nonlinear quantum (thermo)dynamics I proposed in a series of papers [see references in G.P. Beretta, Found.Phys. 17, 365 (1987)], I show its exact equivalence with the maximal-entropy-production variational-principle formulation recently derived in S. Gheorghiu-Svirschevski, Phys.Rev. A 63, 022105 (2001). In addition, based on the formalism of general interest I developed for the analysis of composite systems, I show how the variational derivation can be extended to the case of a composite system to obtain the general form of my equation of motion, that turns out to be consistent with the demanding requirements of strong separability. Moreover, I propose a new intriguing fundamental ansatz: that the time evolution along the direction of steepest entropy ascent unfolds at the fastest rate compatible with the time-energy Heisenberg uncertainty relation. This ansatz provides a possible well-behaved general closure of the nonlinear dynamics, compatible with the nontrivial requirements of strong separability, and with no need of new physical constants. In any case, the time-energy uncertainty relation provides lower bounds to the internal-relaxation-time functionals and, therefore, upper bounds to the rate of entropy production.Comment: RevTeX; 19 pages; submitted to Phys.Rev.A on Feb.9, 2001; revised version submitted on Sept.14, 2001 with slightly modified derivation in Section III, improved discussion on strong separability in Sections X and IX, added Eqs. 64b, 64c and 11

Steepest Entropy Ascent Model for Far-Non-Equilibrium Thermodynamics. Unified Implementation of the Maximum Entropy Production Principle

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of non-equilibrium dynamics into a unified formulation, which extends to such frameworks the concept of Steepest Entropy Ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. The present formulation constitutes a generalization also for the quantum thermodynamics framework. In the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near equilibrium limit, the metric tensor is related to the Onsager's generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far non-equilibrium domain, most of the existing theories of non-equilibrium thermodynamics can be cast in such a way that the state exhibits a spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models is intrinsically and strongly consistent with the second law of thermodynamics. Non-negativity of the entropy production is a readily proved general feature of SEA dynamics. In several of the different approaches to non-equilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called Maximum Entropy Production Principle. It is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far non-equilibrium states.Comment: 15 pages, 4 figures, to appear in Physical Review

Rigorous and General Definition of Thermodynamic Entropy

The physical foundations of a variety of emerging technologies --- ranging from the applications of quantum entanglement in quantum information to the applications of nonequilibrium bulk and interface phenomena in microfluidics, biology, materials science, energy engineering, etc. --- require understanding thermodynamic entropy beyond the equilibrium realm of its traditional definition. This paper presents a rigorous logical scheme that provides a generalized definition of entropy free of the usual unnecessary assumptions which constrain the theory to the equilibrium domain. The scheme is based on carefully worded operative definitions for all the fundamental concepts employed, including those of system, property, state, isolated system, environment, process, separable system, system uncorrelated from its environment, and parameters of a system. The treatment considers also systems with movable internal walls and/or semipermeable walls, with chemical reactions and/or external force fields, and with small numbers of particles. The definition of reversible process is revised by introducing the new concept of scenario. The definition of entropy involves neither the concept of heat nor that of quasistatic process; it applies to both equilibrium and nonequilibrium states. The role of correlations on the domain of definition and on the additivity of energy and entropy is discussed: it is proved that energy is defined and additive for all separable systems, while entropy is defined and additive only for separable systems uncorrelated from their environment; decorrelation entropy is defined. The definitions of energy and entropy are extended rigorously to open systems. Finally, to complete the discussion, the existence of the fundamental relation for stable equilibrium states is proved, in our context, for both closed and open systems.Comment: 19 pages, RevTex

Recent Progress in the Definition of Thermodynamic Entropy

The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carath\'eodory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: three Postulates, which are considered as having a quite general validity, and four Assumptions, which identify the domains of validity of the definitions of energy (Assumption 1) and entropy (Assumptions 2, 3, 4). The domain of validity of the present definition of entropy is not restricted to stable equilibrium states. For collections of simple systems, it coincides with that of the proof of existence and uniqueness of an entropy function which characterizes the relation of adiabatic accessibility proposed by Lieb and Yngvason. However, our treatment does not require the formation of scaled copies so that it applies not only to collections of simple systems, but also to systems contained in electric or magnetic fields and to small and few-particle systems.Comment: 23 pages, 5 figure