Probabilistic Description of Systems

Abstract

In the present chapter we start the study of the general formalism of statistical physics by showing how one can mathematically represent a system which is not well known on the microscopic scale. We deliberately use quantum mechanics for several reasons. On the one hand, microscopic physics is basically quantal; classically there would exist neither atoms nor molecules with discrete bound states. On the other hand, we shall see that, notwithstanding a few conceptual difficulties to begin with, quantization brings about simplifications by replacing integrals by discrete sums. Last and not least, many important phenomena, such as the very existence of solids or magnetic substances, the properties of black-body radiation, or even the extensivity of matter, can only be explained by a quantum-mechanical approach. Even in the case of gases or liquids, classical statistical mechanics is insufficient; it does not enable one to elucidate the Gibbs paradox or to understand the values of the specific heats.

“Une cause très petite qui nous échappe détermine un effet considérable que nous ne pouvons pas ne pas voir, et alors nous disons que cet effet est dû au hasard.”

Henri Poincaré, Calcul des Probabilités, 1912

“Probabilitatem esse deducendam.”

Letter from Einstein to Pauli, 1932

“We meet here in a new light the old truth that in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience.”

Niels Bohr