Macroscopic and microscopic approaches concerning irreversible processes are paradoxically different. As the experiments show, macroscopic systems irreversibly evolve towards the thermodynamical equilibrium, maximizing the entropy, independently of their initial conditions. In contradistinction, fundamental equations of classical and quantum mechanics are reversible.
In order to solve this contradiction, Boltzmann introduced a semi-phenomenological integro-differencial irreversible equation for the one-particle velocity distribution function [1]. His famous H-theorem, statistical mechanics analog of entropy, unfortunately holds only for dilute gases.
A decisive step in the understanding of irreversibility was made by Ilya Prigogine and Brussels group, by introducing SUBDYNAMICS approach [2]. Using such a SUBDYNAMICS of correlations, I obtained ab initio a general Boltzmann-like equation without dilute gases restrictions [3]. The perturbative formal solution of Liouville equation is decomposed into two SUBDYNAMICS. The C-SUBDYNAMICS contains all collisions making that the system forgets all initial condition, hence, it cannot go back. The remaining T-SUBDYNAMICS vanishes in the thermodynamic limit when the number of particles and the volume of the system tend to infinity so that their ratio, i.e., density remains constant. I demonstrated the factorization of the two-particles distribution function into a nonlinear product of two one-particle distribution functions, necessary to establish the Lyapounov functional, which is the general statistical mechanics analog of entropy [4]. The dynamics of correlations is conveniently represented in terms of diagrams composed of elementary vertices, i.e., crossings of two lines representing interaction of two particles correlations. I established a topological theorem proving this factorization in the thermodynamical limit [5]. Diagrams are a compact representation of heavy formulas, which are however, necessary for the final demonstration of the factorization [4].
References
[1] L. Boltzmann, Wien, Ber, 66, 275 (1872).
[2] I. Prigogine, C. George, F. Henin, and L. Rosenfeld, Chemica Scripta, 4, 5-32 (1973).
[3] V. Skarka, Physica, 129A, 62-80 (1984).
[4] V. Skarka, Contribution to the statistical mechanics of irreversible processes in inhomogeneous gases through the subdynamics approach, Thesis, U.L.B. Brussels (1981).
[5] V. Skarka, Bull. Acad. Roy.Belg. Cl. Sci. 64, 578 and 795 (1978).