Statistical Physics

Abstract

From the analysis given in the first two chapters the reader may have noticed the fundamental difference between the mechanical and the thermodynamic description of a system. In the first case, the mechanical state of N material points at time t is determined by 3N generalized coordinates and their 3N conjugate momenta (classical mechanics) or by the wave function (quantum mechanics). In the second instance, the equilibrium state of a simple closed system is specified by only two independent variables, such as the volume and the entropy. It is well known that the laws of mechanics provide an adequate description of the motion in systems with a small number of degrees of freedom. This is indeed the case of the two-body problem whose most representative examples are the planetary motion in classical mechanics and the hydrogen atom in quantum mechanics. The integrability of mechanical systems, namely the possibility of predicting their time evolution given some prescribed initial conditions, is, however, the exception rather than the rule. The most evident manifestation of the complexity of mechanical motions is the chaotic behavior of their evolution even for systems with only a few degrees of freedom. The origin of chaos lies in the nonlinear character of the equations of motion which, as a consequence, implies that a small change of the initial condition completely alters the time evolution. Assume that an experiment is performed in which one measures the final pressure of a gas initially occupying one half of a cubic box of volume V, at the temperature T which is held constant throughout the experiment. Once the wall separating the two halves of the box is removed, the gas, which starts from a non-equilibrium initial condition, expands all over the box until, after a certain time, the pressure reaches a stationary equilibrium value. When the system is initially prepared, its mechanical state is defined, within classical mechanics, by some generalized coordinates and their conjugate momenta. If the experiment is repeated, the result of the new measurement, except for small fluctuations, will be the same as that of the first experiment. On the other hand, it is clear that the initial mechanical state of the second experiment will, in general, be different from the one that the gas had in the first experiment. It should be noted that even in the case where both initial mechanical states are rather close to each other, their time evolution will, in general, be totally different due to the instability of the solutions to Hamilton’s equations. The question, thus, arises as to why the same macroscopic result is reproduced by the different experiments.