Article révisé par les pairs
| Résumé : | This paper is devoted to the study of statistical mechanics of irreversible processes in linear assemblies of harmonic oscillators where nearest neighbours only are interacting. The equations of motion are solved in terms of the initial conditions of the system and these are subjected to probability distributions. The change of the distribution functions with time is described in some detail and the asymptotic behaviour after large times is investigated. Particularly simple asymptotic results can be obtained for infinite systems and by restricting the initial distribution function to a product of uncorrelated distribution function of the different variates (velocities, relative distances). The physically significance of such initial factorized distribution functions in terms of normal modes is indicated. It is shown that such distribution functions correspond to initial states for which equipartition of energy between the different normal modes is already achieved. On the other hand the distribution of phases is initially not the equilibrium random distribution. In this case it can be shown that each finite part of the system tends asymptotically to statistical equilibrium owing to the mechanical interaction with its surroundings. In particular, the velocity distribution of each particle and the distribution of distance between any two neighbouring particles tend to the appropriate gaussian equilibrium distributions; equipartition of kinetic and potential energy is demonstrated. This last result holds also under more general conditions. The limiting process involved in passing from the infinite to the finite system is considered and it is shown that for sufficiently short time the behaviour of a finite system is the same as that of an infinite one. The asymptotic distribution functions evaluated furnish an illustration of the central limit theorem of statistics: owing to the dependence of phase velocity on wave length, after a long time a larger and larger number of the random variables which describe the initial conditions will contribute to the distribution function of any finite part. Thus the conditions of the theorem become satisfied - this is in sharp contrast to the case of a continuous system in which there is only a single velocity of propagation. © 1953. |
