3.1 Microscopic variables

Interpreting the microscopic variables as coordinates in a high-dimensional
space we may represent a particular microstate as a **point** or
**vector** in that space. The space itself is called
**Gibbs phase space**; the state vector is often symbolized as
.

Consider a simple, classical many-particle system - say, an ideal gas,
or a fluid made up of hard spheres or Lennard-Jones molecules. The
state vector is then defined by all position and velocity coordinates:

(3.1) |

The representation of a microstate of the entire system by a single point
in -dimensional Gibbs space must not be mixed up with the treatment
of Chapter 2, introduced by Boltzmann. The -space
defined there had only dimensions
, and
each **particle** in the system had its own representative point.
Thus the microstate of a system of particles corresponded to a
swarm of points
.

In the case of an ideal quantum gas it suffices to specify all
quantum numbers to define a state:

(3.2) |

Model systems made up of spins are specified by

(3.3) |

We consider a collection of systems, all having the same energy , the same number of particles , and - in the case of a gas or fluid - the same volume . All microstates compatible with these macroscopic conditions are then equally probable, that is, they have the same relative frequency within the ensemble of systems. The size of the ensemble is assumed to be very large - indeed, to approach infinity.

The assumption that all microstates that are compatible with the
condition have the same probability is one of the solid
foundations Statistical Mechanics is built upon. It is called
the ``postulate of equal a priori probability''.
For a mathematically exact formulation of this axiom we use the
**phase space density** which is supposed to have the property

(3.4) |

(3.5) |

A typical question to be answered by applying statistical
methods to such
an ensemble is this: what is the average of the squared particle
velocity
over all - a priori equally
probable - states on this surface?

**ERGODICITY**

Instead of considering an ensemble of systems let us now watch just
one single system as it evolves in time according to the laws of
mechanics. In a closed system the total energy will remain constant;
the microstates visited by the system must therefore lie on the
-dimensional energy surface defining the microcanonical ensemble.

The **ergodic hypothesis** states that in the course of such a
``natural evolution'' of the system any permitted microstate
will be reached (or closely approximated) with the same relative
frequency.

This hypothesis cannot be proven in general; in fact, it does not always hold. However, for many relevant systems such as gases or fluids under normal conditions it is quite true. In such cases the time needed for a sufficiently thorough perambulation of the energy surface is in the range of seconds, i.e. safely below the typical observation time in an experiment. Among those systems which we may characterize as ``barely ergodic'' or non-ergodic we have supercooled liquids and glasses. In such systems the state vector remains trapped for long times in a limited region of the energy surface; it may then take seconds, days, or even centuries before other parts of the microcanonical surface are reached.

The ergodic hypothesis, if true, has an important practical consequence: for the calculation of mean values over the microstates on the energy surface it does not matter if we take the average over states randomly picked from a microcanonical ensemble, or over the successive states of one single, isolated system. This corrolary of the ergodic hypothesis is often succinctly stated as

The assumption of ergodicity enables us to support our theoretical
arguments by ``molecular dynamics'' computer experiments. These
are deterministic simulations reproducing the temporal evolution of
a single isolated -particle system. We will later touch upon
another kind of computer simulation, in which the state space is
perambulated in a *stochastic* manner; it bears the suggestive
name ``Monte Carlo simulation''.

**ENERGY SHELL**

In place of the strict condition we will generally require
the weaker condition
to hold.
In other words, the permitted states of the system are to be restricted
to a thin ``shell'' at
. This more pragmatic
requirement - which is made to keep the mathematics simpler - agrees
well with the experimental fact that the exchange of energy between a
system and its environment may be kept small, but can never be completely
suppressed.

Thus we assume for the density in phase space that

(3.6) |

For the ideal quantum gas we require

In a system of non-interacting spins the energy shell is defined by .

Table 3.1 presents an overview of the state spaces and
energy functions for various model systems.

Let us now recall the fundamental assumption that all

It follows then that a *macrostate* that allows for more
*microstates* than others will occur more often - and thus will be
more probable.

As an example, we may ask for the probability to find all molecules of
an ideal gas in the left half of the vessel, with no other restrictions
on position or speed. All such microstates are located in a small
part of the total permitted phase space shell
; they make up a *sub-ensemble* of the complete
microcanonical ensemble. Now, the probability of the macrostate
``all particles in the left half of the container'' is obviously
equal to the size ratio of the subensemble and the total ensemble.
We will have to compare the volume (in phase space) of the partial
shell pertaining to the subensemble to the volume of the total
shell. (We will see later that the ratio of the two volumes
is and may thus be neglected - in accordance with
experience and intuition.)

2005-01-25