Abstract :
A dynamical system of infinite volume and of infinite number of identical interacting particles occupying energy levels has been constructed as the limit of an infinite sequence of finite, equivalent systems of increasing size and particle number. Systems both in equilibrium and in non-equilibrium state (designated S∞=limSk, , respectively, k=1,2,…) were investigated. The main results are:
(i) The values in the T-limit (thermodynamic limit) of the physical quantities characterizing these systems are determined.
(ii) The time evolution process both in and in systems is governed by the non-linear rate equations with common initial conditions pi(t0), where pi(t)=ni(t)/N are the occupation probabilities at time t. The time evolution process in the and systems is the same. The asymptotic approach to the equilibrium state is proved.
(iii) For the case of the equilibrium state, the Boltzmann probability distribution pi is given by the equation −lnpi+a+eib=0 common to Sk and S∞ systems with the same value of a and b. The term a=β−1ae, where ae is the free energy per particle, and .
(iv) The conditions for the equivalence of the systems being in equilibrium and also of the ones in non-equilibrium are stated.
