Equilibrium in Gibbsian Statistical Mechanics

Consider a macroscopic system such as a gas or a solid, and assume that this system is described by macroscopic variables such as temperature, pressure, volume and magnetization. Intuitively, a system has reached equilibrium when all change has come to a halt and the values of the macroscopic variables remain constant over time. How can equilibrium be characterized exactly, and why and how do systems that are not initially in an equilibrium state approach equilibrium? It is the aim of statistical mechanics (SM) to answer these questions in terms of the mechanical properties of the micro-constituents of these systems and the dynamical laws that govern their time evolution. Gibbsian SM (GSM) offers answers to these questions by associating an ensemble with each physical system. Let X be the state space of a system of interest. In a mechanical n-particle system, for instance, the state space has 6n dimensions, three dimensions for the position of each particle and three dimensions for the corresponding momenta. An ensemble is specifed by a probability density ρ(x, t) over the state space of a system, where t is time (which can be either continuous or discrete) and x ∈ X . Physical observables are associated with real-valued functions f : X → R. These functions represent physical quantities, such as internal energy, magnetization, and polarization. The ensemble average (or sometimes “phase average”) of an observable is: Z hf i = f (x)ρ(x, t)dx (28.1) X