III. Kinetic Theory of Gases

• Kinetic theory studies the macroscopic properties of large numbers of particles, start­ ing from their (classical) equations of motion. Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work, heat, and entropy. The phenomenological laws of thermody­ namics tell us how these quantities are constrained as a system approaches its equilibrium. At the microscopic level, we know that these systems are composed of particles (atoms, molecules), whose interactions and dynamics are reasonably well understood in terms of more fundamental theories. If these microscopic descriptions are complete, we should be able to account for the macroscopic behavior, i.e. derive the laws governing the macro­ scopic state functions in equilibrium. Kinetic theory attempts to achieve this objective. In particular, we shall try to answer the following questions: (1) How can we define “equilibrium” for a system of moving particles? (2) Do all systems naturally evolve towards an equilibrium state? (3) What is the time evolution of a system that is not quite in equilibrium? The simplest system to study, the veritable work–horse of thermodynamics, is the dilute (nearly ideal) gas. A typical volume of gas contains of the order of 10 particles. Kinetic theory attempts to deduce the macroscopic properties of the gas from the time evolution of the individual atomic coordinates. At any time t, the microstate of a system of N particles is described by specifying the positions ~qi(t), and momenta p~i(t), of all particles. The microstate thus corresponds to a point μ(t), in the 6N dimensional phase