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:
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:
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
A typical question to be answered by applying statistical
methods to such
an ensemble is this: what is the average of the squared particle
over all - a priori equally
probable - states on this surface?
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''.
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
Table 3.1 presents an overview of the state spaces and
energy functions for various model systems.
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.)