M. De Haan

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.

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.

Derivation of a kinetic equation for an inhomogeneous gas of quantum spinless particles obeying Boltzmann statistics

Abstract The kinetic equation for an inhomogeneous dilute gas of quantum spinless particles obeying Boltzmann statistics is derived from a general method based on the correlation patterns. The equation obtained for the one-body Wigner distribution function takes into account the full two-body collision contributions, leads to the usual conservation laws for density, momentum and energy and can thus be used to derive density corrections to transport coefficients. Its generalization to more general systems can be obtained in a straightforward way by using the same formalism.

Derivation of a kinetic equation for an inhomogeneous gas of particles bearing a spin and obeying quantum statistics

The kinetic equation for an inhomogeneous dilute gas of particles bearing a spin and obeying quantum statistics is derived from a general method based on correlation patterns. This equation takes into account the delocalization of the collision process and we consider the case where spin polarization of the gas is present. The equation obtained for the one-body Wigner density and spin polarization function can be used for the computation of transport coefficients of the gas since we show that it leads to the local conservation laws for the matter density, spin polarization density, momentum and energy, including potential energy contributions. The two-body Wigner distribution function is displayed as a functional, generated by the dynamics, of the one-body distribution without resourse to any molecular chaos hypothesis.

Full quantum treatment of field-matter interaction: A reduced description

Abstract Reduced distribution functions are well known useful tools in the realm of statistical mechanics of particles. The concept is extended to deal with the electromagnetic field so that mean values of field observables then become simple vacuum expectation values instead of a trace involving all the possible occupation numbers for all the field modes. The reduction procedure is illustrated first on simple cases with a single field mode and is shown to lead to a convinient framework for the description of the whole system (initial condition, time evolution, observation). For a massless scalar field coupled with a two level system, a kinetic equation is proposed for the (field) reduced density matrix and a compared with the usual procedures. We show that it incorporates nicely their results while making more transparent the approximations involved. For the coupling of particles (electrons and nuclei) with the electromagnetic field, equations analog to the BBGKY hierarchy are written for the whole system in terms of (combined) reduced Wigner distribution functions for the particles and reduced density matrix for the field.

Kinetic derivation of the second virial coefficient for a quantum gas

A kinetic equation has been proposed earlier for an inhomogeneous quantum dilute gas obeying Boltzmann statistics. We show here that it leads to the correct equilibrium properties of the gas by computing the second virial coefficient as it can be deduced from the trace of the pressure tensor and by checking the result with the value deduced from various approaches.

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.

A derivation of quantum kinetic equations for a gas of composite particles I

Abstract The formalism of non-equilibrium quantum statistical mechanics developed by R. Balescu provides a convenient framework for deriving kinetic equations. It requires the definition of concepts such as vacuum of correlation and free motion operator. We propose a realization of such concepts for bound states in the concrete case of quantum gases starting from a description in terms of electrons and nuclei. Polarized hydrogen and unpolarized helium gases are considered.

Field theory reformulated without self-energy parts: The dressing operator

Abstract The reformulation of field theory for avoiding self-energy parts in the dynamical evolution has been applied successfully in the framework of the Lee model [Ann. Phys. 311 (2004) 314], enabling a kinetic extension of the description. The basic ingredient is the recognition of these self-energy parts [Trends Stat. Phys. 3 (2000) 115]. The original reversible description is embedded in the new one and appears now as a restricted class of initial conditions [Progr. Theor. Phys. 109 (2003) 881]. This program is realized here in the reduced formalism for a scalar field, interacting with a two-level atom, beyond the usual rotating wave approximation. The kinetic evolution operator, previously surmised [Physica A 171 (1991) 159], is here derived from first principles, justifying the usual practice in optics where the common use of the so-called pole approximation [Atoms in Electromagnetic Fields, 1994, 119] should no longer be viewed as an approximation but as an alternative description in the appropriate formalism. That model illustrates how some dressing of the atomic levels (and vertices), through an appropriate operator, finds its place naturally into the new formalism since the bare and dressed ground states do no longer coincide. Moreover, finite velocity for field propagation is now possible in all cases, without the presence of precursors for multiple detections.

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.