Dissipativity and the analytical structure of the collision operator in a model characterizing a long-range force

One important objective of the non-equilibrium statistical mechanics is to establish a theoretical connection between microscopic laws of evolution and the phenomenological laws of thermodynamics. Much progress has been made toward this goal since Prigogine and his coworkers discovered the transformation theory and subdynamics. In this paper, we present a formalism of this theory in terms of the spectrum of the collision operator and apply it to a two-dimensional model of the collision operator which has a cut characterizing the long range nature of the interaction. It is shown that, in spite of the failure of Boltzmann's kinetic theory due to the singularity of the collision operator at z = 0, one may still define a functional of the density operator which can be identified with the thermo-dynamical entropy. Moreover, this definition of entropy includes the correlations as well as the vacuum of correlations.Guo Théodore C., Guo Wendy W. Dissipativity and the analytical structure of the collision operator in a model characterizing a long-range force. In: Bulletin de la Classe des sciences, tome 60, 1974. pp. 1490-1511