The Boltzmann-Gibbs Distribution

Abstract

The preceding two chapters helped us to set up the formalism of statistical mechanics. We introduced in Chap.2 the density operators $\hat D$, and their classical limit, the densities in phase. They sum up our knowledge about the system and enable us to make predictions of a statistical nature about physical quantities, the expectation values of which we can calculate, starting from $\hat D$. In Chap.3 we defined the statistical entropy ($\hat D$) which measures the random nature, or disorder, of a density operator. In those two chapters we assumed that the latter was given. However, in order actually to be able to calculate the properties of a system which has been prepared in some given way we must know how to assign to it a density operator representing the physical situation that we want to describe. This problem of the choice of $\hat D$ will be solved in the present chapter for thermodynamic equilibrium states. In order to find the general form, the so-called Boltzmann-Gibbs distribution, of the density operators, or the densities in phase, describing these states, we shall use a postulate of a statistical nature which is similar to the criteria used in statistics to find the unbiased probability law for a set of random events. We introduce in this way a general prediction method (§4.1.3). This method leads us to represent a system in thermodynamic equilibrium by the most disordered macro-state compatible with the macroscopic data (§4.1).

“L’équilibre est la loi suprême et mystérieuse du grand Tout.”

V. Hugo, Post-scriptum de ma Vie

“En remontant chez moi pour y passer la soirée à travailler de mon mieux, je me disais que le monde n’est pas construit pour l’équilibre. Le monde est désordre. L’équilibre n’est pas la règle, c’est l’exception.”

G. Duhamel, Maîtres, 1937