With the help of simple probabilistic models of Kac and McKean, we discuss the meaning of the generalized expression for entropy that was recently introduced by our group and compare it with Boltzmann's expression. We emphasize the fact that Boltzmann's formulation in terms of the single particle distribution function, f(1), requires very restricted assumptions about the preparation of the system (chaos) and the nature of the collision mechanism (Markov processes).Our generalized [unk]-theorem, however, refers to the complete system; in general, it does not lead to an [unk]-theorem for the single particle distribution function, f(1). It is valid whatever the preparation of the system. In McKean's model, situations exist where it gives the correct behavior while the Boltzmann's expression for entropy becomes meaningless. In addition, in Kac's model, we show that correlations reach equilibrium more rapidly than f(1) and that there is an asymptotic regime where both formulations give the same result.

Journal Article↗

PubMed1970

Dynamics of systems with large number of degrees of freedom and generalized transformation theory.

A generalized transformation theory which leads to a non-Hamiltonian description of dynamics is introduced. The transformation is such that all averages of observables remain invariant. However, the time evolution of the density matrix can no longer be expressed in terms of a commutator with the Hamiltonian. Therefore such transformations are not canonical in the usual sense. An explicit "two components" representation of the equations of motion is given which has the following properties: (a) each of the components satisfies a separate equation of motion, and (b) one component satisfies a kinetic equation of a generalized Boltzmann type.WE OBTAIN, THEREFORE, THE MOST REMARKABLE RESULT THAT THE RELATION BETWEEN DYNAMICS AND STATISTICAL MECHANICS (OR THERMODYNAMICS) TAKES A SPECIALLY TRANSPARENT AND SIMPLE FORM: thermodynamics appears in a precise sense as the random phase approximation of dynamics.Other problems such as the meaning of diagonalization of the Hamiltonian and definition of excitations will be treated in a forthcoming paper.

Journal Article↗

PubMed1970

Physical particle representation and generalized transformation theory.

The investigation of the recently described generalized transformation theory which leads to a non-Hamiltonian description of dynamics is pursued. The concept of generalized unitary transformations of superoperators is introduced and a specific class of transformations studied. For nondissipative systems it is equivalent to the usual unitary transformations that diagonalize the Hamiltonian. The important point is that this class of transformations remains meaningful for dissipative systems, hence a new representation of dynamics that we shall call the "physical particle" representation. It has the following properties: (a) The energy (or an arbitrary function of the energy) is represented by a diagonal matrix. (b) In the ((0))II space (see these Proceedings, 65, 789 (1970)) corresponding to the coherent processes, the evolution can be described in terms of the changes in population of the physical particles. (c) At thermodynamic equilibrium, the physical particles are uncorrelated and behave as independent entities; the entropy has a purely combinatorial meaning.A full description of dynamics in this representation will be given in a forth-coming paper.

Journal Article↗

PubMed1968