Claude George

Subdynamics and correlations

Abstract Prigogine, Henin and the author have introduced the concept of subdynamics to study the behaviour of large systems. The usual thermodynamic description of such systems can be formulated in a subspace οП of the Liouville space. For homogeneous systems, the leading or privileged elements in this subspace are the diagonal or vacuum components of the density matrix. The structure of the subspaces associated with privileged elements which are off-diagonal elements of the density matrix, or correlations, is similarly analyzed. To each type of correlation, one can associate a subspace and a subdynamics. The elements of the transformation theory which extends to dissipative systems the theory of unitary transformations of usual quantum mechanics, are expressed in terms of the operators which characterize the structure of the various subspaces.

Liapounov function for the friedrichs model

The Friedrichs model of an unstable particle provides us with a non-trivial example of a dynamical system with a continuous spectrum. The non-unitary transformation operator which leads in our approach to a semi-group description of the time evolution of the system has been constructed in earlier papers. It is used here to evaluate the Liapounov function which plays the role of a microscopic entropy. The results, obtained without any recourse to perturbation expansions, validate resummation procedures introduced previously in a perturbative treatment. Various types of initial conditions are considered. It is shown explicitly why the distinction between pure states and mixtures is lost in the asymptotic time limit. The result is of relevance for the discussion of the quantum mechanical measurement problem.

Properties de l'operateur de collision pseudo-markovien

Abstract I. Prigogine and P. Resibois have recently introduced, as an intermediate step in their formulation of the approach to equilibrium in a homogeneous system, functionals ωα( Z ) of a given analytical function Ψ( Z ). The purpose of the present article are 1. 1) to analyze, in a more complete way, the sequence of functionals ωα( Z ). 2. 2) to establish the link between two methods to describe the pseudo-markovian evolution for the above-mentioned system 3. 3) to derive, with a strictly mathematical point of view, from the properties of the functionals ωα( Z ), a few remarkable equations, leading when particular choices of function Ψ( Z ) are made to summation relations between binomial coefficients.

On the relation of dynamics to statistical mechanics

A new conceptual framework for the foundations of statistical mechanics starting from dynamics is presented. It is based on the classification and the study of invariants in terms of the concepts of our formulation of non-equilibrium statistical mechanics. A central role is played by thecollision operator. The asymptotic behaviour of a class of states is determined by the collisional invariants independently of the ergodicity of the system. For this class of states we have an approach to thermodynamical equilibrium.We discuss the existence of classes of states which approach equilibrium. The complex microstructure of the phase space, as expressed by the weak stability concept which was introduced by Moser and others, plays here an essential role. The formalism that we develop is meaningful whenever the “dissipativity condition” for the collision operator is satisfied. Assuming the possibility of a weak coupling approximation, this is in fact true whenever Poincarés theorem on the nonexistence of uniform invariants holds. In this respect, our formalism applies to few body problems and no transition to the thermodynamic limit is required.Our approach leads naturally to a ‘classical theory of measurement’. In particular a precise meaning can now be given to ‘thermodynamic variables’ or to ‘macrovariables’ corresponding to a measurement in classical dynamics.

On the time-dependent formulation of analytical continuation in non-equilibrium statistical mechanics

On the basis of the dynamics-of-correlations approach to the solution of the Liouville-von Neumann equation, we develop within the familiar classification of correlations scheme a time-dependent formalism which leads naturally to an analytical continuation procedure, thereby allowing explicit evaluation of each term in the formal series solution. This procedure throws new light on the so-called iϵ-rule, and opens the way to the treatment of the full correlation subdynamics for time-dependent phenomena, which will be discussed in future publications.

Coherence and randomness in quantum theory

Classical ensemble theory is compared to quantum mechanics. The remarkable feature is that classical ensemble theory, when suitably generalized, leads, for a system of one degree of freedom to two independent uncertainty relations. On the contrary classical trajectory theory which deals only with functions of time, contains no uncertainty relations whatsoever.

Single subdynamics for the Friedrichs model

The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self-energy processes (already found in quantum electrodynamics), which allows for the widening of the dynamics through the introduction of specific additional variables leading to the recognition of such processes. A projection technique modeled on that introduced long ago by the Brussels group defines a subdynamics that is shown to contain the starting formulation in its entirety. (It corresponds to a restricted class of initial conditions, namely the reversible ones.) A new kinetic description is obtained in which, e.g., collision processes for privileged components arise from the original interaction. This method is applied successively to various models, and we focus in this paper on the possible presence of temporal behaviour excluded in ordinary quantum mechanics. Among the differences between this framework and that of ordinary quantum mechanics, the possibility appears for the purely exponential decay of a particle, while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.

Vacuum subdynamics in large quantum systems

We present a constructive procedure to deal with large quantum systems in the thermodynamic limit. Starting with a discrete spectrum, we perform a complete decomposition of the evolution into one-dimensional subdynamics. We then go to the limit of a continuous spectrum after collecting them into global subdynamics of given degrees of correlation. Previously obtained results for the vacuum subdynamics are justified. The procedure is applied to the problem of potential scattering.

A generalization of the van Kampen-Case treatment of the Vlasov equation

In the inhomogeneous collisionless system the collective modes are constructed in the physical representation using the method of subdynamics. They correspond to the explicit solutions of the non-linear Vlasov equation and they appear as the generalization of the van Kampen-Case treatment. The absence of entropy production is shown.

The trouble with entropy

An understanding of the mechanisms leading to the symmetry breaking of the dynamical description of a large system with respect to the direction of time is necessary, but not sufficient to ensure the finding of a functional of the state of the system that would satisfy the requirements placed by the Second Law of Thermodynamics upon the non-equilibrium entropy S.