Classical Distribution Function and Transport Equation

Abstract

When a system made of a large number of molecules is considered, the description of the dynamics of each individual member of the system is practically impossible, and it is necessary to resort to the methods of Statistical Mechanics. The chapter introduces the concept of distribution function in the phase space and provides the definition of statistical average (over the phase space and momentum space) of a dynamical variable. The derivation of the equilibrium distribution in the classical case follows, leading to the Maxwell-Boltzmann distribution. The analysis proceeds with the derivation of the continuity equation in the phase space: the collisionless case is treated first, followed by the more general case where the collisions are present, this leading to the Boltzmann Transport Equation. In the Complements, after a discussion about the condition of a vanishing total momentum and angular momentum in the equilibrium case, and the derivation of statistical averages based on the Maxwell-Boltzmann distribution, the Boltzmann H-theorem is introduced. This is followed by an illustration of the apparent paradoxes brought about by Boltzmann’s Transport Equation and H-theorem: the violation of the symmetry of the laws of mechanics with respect to time reversal, and the violation of Poincaré’s time recurrence. The illustration is carried out basing on Kac’s ring model. The chapter is completed by the derivation of the equilibrium limit of the Boltzmann Transport Equation.

Problems

• 1. [6.1] Calculate the average energy like in (6.37) assuming that energy, instead of being a continuous variable, has the discrete form $E_n = n h \nu$, $n = 0,1,2,\ldots$, $h\nu = \mbox{const}$. This is the hypothesis from which Planck deduced the black-body’s spectral energy density (Sect. 7.4.1).