Page 21 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

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The classical distribution function

Historically, the first and most successful case in which statistical mechanics has
made the connection between microscopic and macroscopic description is that
in which the system can be said to be in equilibrium. We define this carefully
later but, to proceed, may think of the equilibrium state as the one in which the
values of the macroscopic variables do not drift in time. The macroscopic vari-
ables may have an obvious relation to the underlying microscopic description
(as for example in the case of the volume of the system) or a more subtle rela-
tionship (as for temperature and entropy). The macroscopic variables of a system
in equilibrium are found experimentally (and in simulations) to obey historically
empirical laws of thermodynamics and equations of state which relate them to
one another. For systems at or near equilibrium, statistical mechanics provides
the means of relating these relationships to the underlying microscopic physical
description.
We begin by discussing the details of this relation between the microscopic and
macroscopic physical description in the case in which the system may be described
classically. Later we run over the same ground in the quantum mechanical case.
Finally we discuss how thermodynamics emerges from the description and how the
classical description emerges from the quantum mechanical one in the appropriate
limit.

Foundations of equilibrium statistical mechanics
Here we will suppose that the systems with which we deal are nonrelativistic and can
be described fundamentally by 3N time dependent coordinates labelled q i (t) and
their time derivatives ˙ q i (t)(i = 1,..., 3N). A model for the dynamics of the system
is specified through a Lagrangian L({q i }, {˙ q i }) (not explicitly time dependent) from
which the dynamical behavior of the system is given by the principle of least

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