The irreversibility of quantum mechanical systems perturbed by random forces is investigated by means of phenomonological equations derived from the Schr\"odinger equation. A semiquantitative discussion depending on an assumption of time reversal symmetry of the ensemble of random functions leads to a description of the irreversibility of a system with a finite number of states. A more general quantitative method utilizing "smoothed" density matrices produces a macroscopic description of systems tending to equilibrium states.
The irreversibility of quantum mechanical systems perturbed by random forces is investigated by means of phenomonological equations derived from the Schr"odinger equation. A semiquantitative discussion depending on an assumption of time reversal symmetry of the ensemble of random functions leads to a description of the irreversibility of a system with a finite number of states. A more general quantitative method utilizing "smoothed" density matrices produces a macroscopic description of systems tending to equilibrium states.
The transition probability of an isolated system for a time-dependent unitary evolution is invariant under the reversal of protocols. In this paper, we generalize the expression of microscopic reversibility to externally perturbed large quantum open systems, which provides a model-independent equality between time forward and reversed joint transition probabilities. A time-dependent external perturbation acts on the subsystem during a transient duration, and subsequently the perturbation is switched off so that the total system would thermalize. We concern ourselves with the net transition probability for the subsystem from the initial to final states after a time evolution during which the energy is irreversibly exchanged between the subsystem and reservoir. The time-reversed probability is given by the reversal of the forcing protocol and the initial ensemble. Microscopic reversibility equates the time forward and reversed probabilities, and therefore appears as a thermodynamic symmetry for open quantum systems.
The Pauli master equation describes the statistical equilibration of a closed quantum system. Simplifying and generalizing an approach developed in two previous papers, we present a derivation of that equation using concepts developed in quantum chaos and random-matrix theory. We assume that the system consists of subsystems with strong internal mixing. We can then model the system as an ensemble of random matrices. Equilibration results from averaging over the ensemble. The direction of the arrow of time is determined by an (ever-so-small) coupling to the outside world. The master equation holds for sufficiently large times if the average level densities in all subsystems are sufficiently smooth. These conditions are quantified in the text, and leading-order correction terms are given.
We propose a new approach concerning the introduction of time-irreversibility in statistical mechanics. It is based on a transition function defined in terms of path integral and verifying a time-irreversible equation. We show first how dynamic processes may enter in the description of equilibrium states. In order to do that a characteristic time is associated with closed paths. For large isolated systems at equilibrium or for systems in contact with a thermostat our results are identical with those obtained with the Gibbs ensemble methods. For a model used in the microscopic approaches of the brownian motion no new basic assumption is required to predict a transition from a quantum state to a classical one exhibiting a time-irreversible behavior. This demonstration is sufficient to show that very well accepted approximations can lead to time-irreversible behaviors for a large class of systems. The difference between our work and the system+reservoir approaches is underlined. Here equilibrium states and irreversible processes are described on the same footing representing a progress in the question of time-irreversibility in statistical physics. The transition function is also used for describing a small system for which there is no thermodynamics. By adding to the transition function a second one characterizing the reverse motion we may describe time-reversible systems. In a simple case we replace two real valued transition functions by a complex function verifying a Schrodinger like equation. From this we see how to break the time-reversibility of this equation and how to investigate the connection quantum mechanics-thermodynamics from a very fundamental point of view.
This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.
Nonintegrable Poincaré systems with continuous spectrum (so-called Large Poincaré Systems, LPS) lead to the appearance of diffusive terms in the framework of dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory dynamics and of wave functions. These diffusive terms correspond to well-defined classes of dynamical processes (i.e., so-called “vacuum-vacuum” transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include already in the fundamental dynamical description the two aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulations of classical or quantum mechanics. Instead of being irreducible concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville-von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature we inherited from the 19th century, based on the one hand on dynamical time-reversible laws and on the other on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory avoiding the conventional dual structure based on Schrödinger’s equation on the one hand, and on the “collapse” of the wave function on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as the many-world theory, coarse graining or averaging over the environment). There is a striking parallelism between classical and quantum theory. For LPS we have, in general, both a “collapse” of trajectories and of wave functions for LPS. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we know that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of "laws of physics" dealing no more with certitudes but with probabilities. The formulation is appropriate to describe an open, evolving universe.