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We illustrate the use of the « master equation » of weakly coupled gases, derived by Brout and Prigogine, by the calculation of the dynamical friction coefficient in a stellar system. The method used is very direct and can easily be generalized to the calculation of other moments.

Conditions are established, under which a system of slightly anharmonic oscillators does not admit of uniform and analytic integrals of the motion. The results are applied to a discussion of Peierls' system of oscillators on a lattice.

The effect of small dipolar interactions in solutions is examined. Prigogine, Bellemans and Englert's theory of solutions is applied quite generally to interpret this effect in solutions of molecules of different sizes and different central interactions. Pure dipolar interactions and inductive forces in binary solutions containing one polar constituent are discussed. The theory is in good agreement with experimental data on the excess functions of the system CHCl3 — CCl4.

The diffusion of charged particles in a gaussian stochastic magnetic field characterized by finite correlation lengths λI and λ[formula] and gaussian coloured collisions with a finite correlation time τc is studied through the method of expansion with respect to a parameter ε which is introduced to denote the order of smallness. The mean square displacement in the perpendicular direction, Γ(t), is derived to the order of ε2. Several limiting cases are discussed and
shown to recover the previous results. To the order of α5 [formula], the
result is the same as that of Balescu, Misguich and Nakach, which was obtained after " Corrsin approximation It proves that the " =Corrsin approximation" is also reasonable in the short time regime.

Thermodynamic systems described by antisymmetric phenomenological laws are studied in the neighbourhood of a stationary non-equilibrium state. The differential law dXP ≤ 0 associates to such systems a fixed sense of rotation about the Stationary state (which can never be reached). This sense only depends upon the nature of the system. An example of such systems is provided by Volterra's-model of interacting biological populations.

A differential property of the entropy production, established in a preceding paper, is used to study variational principles associated to stationary non-equilibrium situations when the theorem of minimum entropy production is no longer valid. The case of one and two variables is readily solved in a general way. Restrictions which exist for more than two independent variables are discussed.

Abstract. — The solution of a general evolution equation
[formule]
— describing the time evolution of a function ƒ in terms of a general time-dependent differential operator C(0 — is given in terms of time evolution operators.
The explicit solution for the direct and inverse propagators are shown to satisfy separately a semi-group property. A simple prescription concerning the time-ordering allows us to define a general operator, describing both direct and inverse time propagations, satisfying the group property.
An interaction representation is given to derive useful expressions involving new operators which are dressed either by an external force, or by a free propagation. As an application, useful compact expressions are obtained for the free propagator in a constant magnetic field and for the propagator of Vlasov plasma turbulence. On the other hand, two examples are given to show how this interaction representation can be used to define two new approximations which could be useful to go beyond the usual weak coupling approximation of plasma turbulence.

A closed formula is obtained for the long living correlations in an inhomogeneous plasma ; it is expressed in terms of the one particle distribution function. This forms an appropriate starting point for a rigorous theory of transport phenomena in plasmas, including the effect of molecular correlations. An expression is obtained for the thermal conductivity.

The following questions are discussed.
A. The statistical description of the electromagnetic field and the concept of a distribution functional.
B. The canonical realization of the Poincaré group for a free field. It is shown in particular that the known generators are valid only in an infinite volume, and that the microscopic potential transforms as a 4-vector only in an infinite volume.
C. The choice of the variables which lead to the simplest properties of the Liou ville operators.
D. The explicit expressions for the Liouville operators both in phase space and in Fourier space are derived and listed in great detail.

The correlation pattern formalism is used to derive a kinetic equation for a relativistic plasma and radiation. Landau and Thomson scattering are taken into account to compute the collision operator for the homogeneous weakly coupled case. An H-theorem is proved. The corresponding linearized kinetic equation is a two dimensional matrix equation, the collision operator of which is hermitian. Its spectrum is shown to be definite negative and its collisional invariants are computed.

A simple relation is shown to exist at all times between the exact evolution operator U(t) and the asymptotic evolution operator Σ(t) introduced by the Brussels school. Furthermore, it is shown that all invariants under U(t) are invariant under Σ(t) and that under certain conditions the converse is also true.