Sur la mécanique statistique des phénomènes irréversibles IV

This paper continues G. Klein and I. Prigogine's study of irreversible processes in linear chains of harmonic oscillators and constitutes its generalized application to three-dimensional crystals. The equations of motion are first solved in terms of the initial positions and velocities. Thus, “influence functions” have to be defined, of which some asymptotic properties can be established.It is shown by means of these influence functions that the local distribution functions all tend to Gaussian distributions and the crystal to local homogeneity, whatever the initial distribution functions. The only assumption made concerns the absence of correlation at distances of the same order of magnitude as the dimensions of the crystal. Distribution functions are thus found, which are already very near to the equilibrium distribution and only depend on the second order cumulants (the first order cumulants being zero). It can similarly be shown that the second moments are quasi-invariant: after a more or less complicated variation, they tend to an asymptotic value which is not generally that of a crystal in thermodynamic equilibrium.These asymptotic values of the second moments correspond to statistical equilibrium only if initially the equipartition between the normal modes is realized without any correlation between their phases. No other condition is imposed; in particular the amplitudes of the normal modes need not have a Gaussian distribution.The behaviour of the local distribution functions is characteristic of a kind of local ergodicity, in contrast with the non-ergodicity of the system as a whole. The thermodynamic consequences of this behaviour will be analyzed in a subsequent paper.SCOPUS: ar.j