I. Collective Behavior, From Particles to Fields

The object of the first part of this course was to introduce the principles of statistical mechanics which provide a bridge between the fundamental laws of microscopic physics, and observed phenomena at macroscopic scales. Microscopic Physics is characterized by large numbers of degrees of freedom; for example, the set of positions and momenta {~ pi, ~qi}, of particles in a gas, configurations of spins {~si}, in a magnet, or occupation numbers {ni}, in a grand canonical ensemble. The evolution of these degrees of freedom is governed by an underlying Hamiltonian H. Macroscopic Physics is usually described by a few equilibrium state variables such as pressure P , volume V , temperature T , internal energy E, entropy S, etc., which obey the laws of thermodynamics. Statistical Mechanics provides a probabilistic connection between the two realms. For example, in a canonical ensemble of temperature T , each micro-state μ, of the system occurs with a probability p(μ) = exp ( −βH(μ) ) /Z, where β = (kBT ) . To insure that the total probability is normalized to unity, the partition function Z(T ) must equal