Equilibrium Statistical Mechanics

Classical Statistical Mechanics studies properties of macroscopic aggregates of particles, atoms and molecules, based on the assumption that they are point masses subject to the laws of classical mechanics. Distinction between macroscopic and microscopic is evanescent and in fact the foundations of statistical mechanics have been laid on properties, proved or assumed, of few particles systems. Macroscopic systems are often considered in stationary states: which means that their microscopic configurations follow each other as time evolves while looking the same macroscopically. Observing time evolution is the same as sampling (“not too closely” time-wise) independent copies of the system prepared in the same way. A basic distinction is necessary: a stationary state can be either in equilibrium or not. The first case arises when the particles are enclosed in a container Ω and are subject only to their mutual conservative interactions and, possibly, to external conservative forces: typical example is a gas in a container subject forces due to the walls of Ω and to gravity, besides the internal interactions. This is a very restricted class of systems and states. A more general case is when the system is in a stationary state but it is also subject to non conservative forces: a typical example is a gas or fluid in which a wheel rotates, as in the Joule experiment, with some device acting to keep the temperature constant. The device is called a thermostat and in statistical mechanics it has to be modeled by forces, non conservative as well, which prevent an indefinite energy transfer from the external forcing to the system: such transfer would impede reaching stationary states. For instance the thermostat could be simply a constant friction force (as in stirred incompressible liquids or as in electric wires in which current circulates because of an electromotive force). A more fundamental approach would be to imagine that the thermostatting device is not a phenomenologically introduced non conservative force (like a friction force) but is due to the interaction with an external infinite system which is in “equilibrium at infinity”. In any event non equilibrium stationary states are intrinsically more complex than equilibrium states. Here attention will be confined to equilibrium statistical mechanics of systems of N identical point particles Q = (q1, . . . ,qN ) enclosed in a cubic box Ω, with volume V and side L, usually supposed with perfectly reflecting walls. Particles of mass m located at q,q′ will be supposed to interact via a pair potential φ(q− q′). Microscopic motion will follow the equations