文档介绍：I. Collective Behavior, From Particles to Fields
Introduction
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 ex­
ample, the set of positions and momenta p∂, ∂ q , of particles in a gas, configurations of
{ i i}
spins ∂s , in a , or occupation numbers n , in a grand canonical ensemble. The
{ i} { i}
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
−1
occurs with a probability p(µ) = exp κ(µ) /Z , where κ= (k T ) . To insure
− H B
that the total probability is normalized to unity, the partition function Z (T ) must equal
exp κ(µ) . Thermodynamic information about the macroscopic state of the sys-
µ − H
tem is then extracted from the free energy F = k T ln Z .
− B
The above program can in fact be fully carried out only for a limited number of sim­
ple systems; mostly describing non–interacting collections of particles where the partition
function can be calculated exactly. Some effects of interactions can be included by per­
turbative treatments around such exact solutions. However, even for the relatively simple
case of an imperfect gas, the perturbative approach breaks down close to the condensa­
tion point. One the other hand, it is precisely the multitude of new phases and prperties
resulting from interactions that renders mac