Françoise Henin

Collision operator in the physical representation, the friedrichs model

Recent work by the Brussels group has shown the importance of parity properties of the dynamical operators for large systems with respect to the inversion L → -L, L being the Liouville-Von Neumann operator. This has led to a causal formulation of dynamics. The transformation Λ(L) leading from the initial representation (Liouville-Von Neumann equation) to the causal (physical) representation is a nonunitary, called starunitary, transformation. We first summarize the general theory and recall the properties of the evolution operator for a dynamical dissipative system in the physical representation. The Friedrichs model is then studied in detail. We restrict ourselves to the discussion of the diagonal elements of the density operator in the physical representation and their evolution. This is possible because these elements obey independent equations. We start with a finite system and keep only dominant terms with respect to the volume L3. We then show that the Friedrichs model belongs to the class of dynamical dissipative systems, for which an 1–197 theorem can be established. We also discuss, for this model, the relation between the causal formulation of dynamics and probability theory.

On the General Theory of the Approach to Equilibrium. II. Interacting Particles

The general method described in a recent paper by Prigogine and Henin [J. Math. Phys. 1, 349 (1960), hereafter referred to as I] is applied to a system of interacting particles. A full use is made of the diagram technique due to Prigogine and Balescu. The distribution function is Fourier analyzed and each Fourier coefficient is decomposed into two parts: one (ρ′) whose evolution results from scattering processes and which obeys a diagonal differential equation; the second one (ρ″), whose evolution is due to direct mechanical interactions which build the correlation described by the Fourier coefficient. The ρ″ can be expressed in terms of functions ρ′ corresponding to lower correlations. We study first the velocity distribution function. Only scattering processes contribute to the evolution of this function. The equations obtained ensure evolution of this function to the correct equilibrium value at any order in the concentration C and the coupling constant λ. We then study the asymptotic behavior of the F...

A theorem in dynamics of correlations

Abstract In a homogeneous system, n-correlations can be defined as correlations which require at least n steps to be reached from the vacuum of correlations. We show that, in the thermodynamic limit, successive appearances of a given monocorrelation in the formal perturbative solution of the Liouville equation are possible only under well defined conditions.

A presentation of pulsed nuclear magnetic resonance with full quantization of the radio frequency magnetic field

We show that there is excellent quantitative agreement between the usual semiclassical predictions for nuclear magnetic resonance experiments (quantized spins interacting with a classical radio frequency magnetic field, including radiation damping), and the predictions of a fully quantized description of the same experiment. We also propose a pictorial description of the relevant dynamical quantum states, which helps to lead the intuition towards the same conclusion. Our model ignores spin relaxation, and simulates the equivalent damping resistor of the sample circuit by a very long piece of lossless coaxial cable. The classical rf field (pulses and free induction decay) is described quantum mechanically by means of Glauber’s “coherent” or “quasiclassical” states.

Entropy, dynamics and molecular chaos: The McKean's model

Abstract We briefly recall how the developments of the theory of irreversible processes have led to a dynamical H theorem, requiring only very general assumptions about the initial conditions and a rather simple property of the collision operator in the physical representation. Our aim is to investigate the connection between this theorem and Boltzmanns theory. Instead of the rather complicated case of a dilute gas, we use a simple probabilistic model due to McKean. This enables us to see that the conditions imposed in Boltzmanns formulation on the initial preparation of the system (molecular chaos) are quire restrictive but nevertheless essential for obtaining Boltzmanns H theorem. We also show how, asymptotically, for an aged system, one can recover Boltzmanns results.

LIOUVILLE EQUATION FOR A RELATIVISTIC CHARGED PARTICLE

Synopsis The Liouville equation for a relativistic charged particle interacting with its radiation field in the presence of an external electromagnetic field is derived. This equation is solved formally by an iteration procedure and the integration over the field variables is carried on. One then obtains the evolution equation for the distribution function ρ(t) of the particle alone; it involves perturbed propagators which are solutions of anon, linear integral equation and which properties are briefly discussed. An exact solution of the equation for ρ(t) is then given; it is the distribution function for a particle whose motion is governed by the equations of classical electrodynamics i.e. a particle acted upon by the Lorentz force due to the external field and the self retarded field. The point character of the charge of course introduces divergent contributions to the Liouville equation. Within the Born approximation, we show how the regularization procedure used previously by I. Prigogine and the author for the equations of motion can be introduced in the Liouville formalism.

Kinetic equations and light-matter interaction

Abstract The evolution of strongly coupled systems is much clarified by the introduction of quasiparticles in the Boltzmann representation. These quasiparticles are described by a distribution function which is obtained by a new transformation from the asymptotic bare particle distribution function. They are entities which from the points of view of both their dynamical (kinetic equation) and thermodynamical (entropy) properties, behave like weakly coupled objects. The aim of the present work is to discuss the application of these concepts to the problem of interaction between radiation and bound electrons. The kinetic equation for quasiparticles, i.e. electron in perturbed levels, is derived up to order λ 4 . Whereas the Pauli equation (Born approximation) only describes one-photon processes, two-photon processes are now included. The meaning of the perturbed states, from the points of view of equilibrium properties and comparison with Rayleigh Schrodinger perturbation expansion, is briefly discussed. The kinetic equation is then used to discuss the problem of spontaneous decay of an excited state, taking into account the contributions of one- and two-photon processes. First, a rather general model is used to exhibit the main differences between the bare and quasiparticles pictures. Then, we restrict ourselves to a generalized Wigner-Weisskopf model and discuss mainly the sequential decay. A brief comparison with the T -matrix formalism is sketched; a more detailed comparison is left out for another paper.

Interaction between radiation and a two-level system: Kinetic equation up to order λ6 and applications

Abstract This paper is one of a series devoted to the discussion, by means of kinetic equations, of the problem of interaction between radiation and matter. The model used for matter is the Wigner-Weisskopf two level model. The kinetic equation for quasiparticles (i.e. atom in perturbed levels) is given, without proof, up to order λ 6 (The derivation may be found in another paper). As λ 2 and λ 4 contributions have already been discussed in another work, we concentrate ourselves on the discussion of the λ 6 (3-photon) scattering operator. The kinetic equation is then applied to a discussion of spontaneous emission and resonance light scattering. The connection with the usual Lorentz formula is then discussed.

On the dynamical derivation of equilibrium statistical mechanics

Work on nonequilibrium statistical mechanics, which allows an extension of the kinetic proof to all results of equilibrium statistical mechanics involving a finite number of degrees of freedom, is summarized. As an introduction to the general N-body problem, the scattering theory in classical mechanics is considered. The general N-body problem is considered for the case of classical mechanics, quantum mechanics with Boltzmann statistics, and quantum mechanics including quantum statistics. Six basic diagrams, which describe the elementary processes of the dynamics of correlations, were obtained. (M.C.G.)

On the entropy of strongly interacting systems

Abstract The entropy may be expressed, both in and out of equilibrium in terms of the usual H -functional, when a quasi-particle representation exists. It is shown that there exists a remarkable, simple relation with the Gibbs canonical entropy.