Pierre Resibois

Time dependent correlation functions and mode-mode coupling theories

Abstract We critically discuss the various tools and methods which are used to describe the role of long wave length hydrodynamical processes in the analysis of time-dependent correlation functions. We also review the various physical problems (long time behavior of Green-Kubo integrands, 2 dimensional hydrodynamics, transport properties of the Van der Waals fluid, critical phenomena …) where these methods have received fruitful applications.

H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that∂tH ⩽ 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=− kBH becomes the exact equilibrium entropy of the hard-sphere fluid.

Transport equation of a brownian particle in an external field

Abstract We apply the general theory of irreversible processes developed by I. Prigogine and coworkers to the particular problem of the motion of a heavy charged particle (mass M ) moving in a fluid of light particles ( m; γ = (m/M) 1 2 ⪡ 1 ) under the influence of an external electri field. We recover the results obtained recently by J. Lebowitz and E. Rubin, using a completely different technique: to the lowest order in γ the Brownian particle obeys a Fokker-Planck equation. The higher order corrections are also discussed and special attention is focused on the effect of the field during a collision process between the Brownian particle and the fluid: this effect can, however, be incorporated in a velocity dependent correction to the Fokker-Planck collision term.

Approximate kinetic theory of hard-sphere fluids near equilibrium. I. Formal theory

We propose a kinetic theory of hard-sphere fluids which systematically generalizes the Enskog equation by taking successively into account binary, ternary,... dynamic correlations in a system close to equilibrium. The first approximation, beyond Enskog, is displayed explicitly; it reproduces most of the results previously established in limiting cases (short-and long-time behavior, low-density expansions) and appears to give a good description of hard-sphere dynamics for all times and all densities. This explicit study will be presented in another publication.

Approximate kinetic theory of hard-sphere fluids near equilibrium: II. A quasihydrodynamic approximation for the velocity autocorrelation function

We apply the kinetic theory of hard spheres recently developed by Lebowitz and Résibois to the calculation of the velocity correlation function. To simplify the calculations, a hydrodynamic approximation is made on the non-Markovian kernel of this kinetic equation. The results are in qualitative agreement with computer experiments at all densities.

On the Connection between the Kinetic Approach and the Correlation‐Function Method for Thermal Transport Coefficients

We demonstrate the complete equivalence between the kinetic approach developed by Prigogine and co‐workers and the correlation function formalism for the calculation of linear thermal transport coefficients. We show that in both cases these transport coefficients are determined by the solution of an inhomogeneous integral equation for a one‐particle distribution function which is the generalization to strongly coupled systems of the Chapman—Enskog first approximation of the Boltzmann equation.

New approach to irreversible transport phenomena in plasma dynamics

A factorization theorem is demonstrated for special types of contributions to the time‐dependent collision operator G00(τ) which plays a central role in the classical master equation derived by I. Prigogine and P. Resibois; this theorem expresses the mutual independence of two subgroups of particles when these groups are noninteracting with each other during a given time interval (τi, τj) of the collision process. The theorem is applied to rederive in a very direct way the kinetic equation of an homogeneous stable plasma, obtained first by R. Balescu. Other possible applications of the method are also discussed.

On the equivalence between two generalized master equations

Synopsis The relationship between the master equation derived respectively by L. Van Hove and by I. Prigogine and the author is analyzed. It is shown that the difference between the two approaches essentially lies in a different definition of the irreducible contributions defining the transition probabilities. In the latter formalism, the criterion used is a dynamical one, corresponding to an analysis of the time ordering of the interactions bringing the system toward equilibrium. On the contrary, the former approach defines irreducibility by inspection of the structure of the matrix elements in the product space , irrespective of any dynamical consideration; this has as a consequence that it is not possible to obtain a master-equation for the density matrix itself in contrast with the dynamical formalism.

The three particle collision operator in quantum mechanics

Abstract We analyse the quantum mechanical collision operator for three incident free particles from the point of view of transport theory. Starting from the Liouville-von Neumann equation for the density matrix, in the form introduced previously by Prigogine and coworkers, we obtain the generalization to quantum systems of the well known Choh-Uhlenbeck result for classical genuine triple collisions. This expression, written in terms of the Heisenberg operators of motion exp[—iHt] for two and three particles, is free of the divergence difficulties occurring in standard collision theory when the limits of a large system and of long times are taken in order to define a three particle transition probability. We also discuss the connection of this result with the Lippman-Schwinger theory and show that the scattering operator for three incident particles is not expressible as the square of a scattering matrix, like in the two particle problem.

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...