3.1 Microscopic variables

The microscopic state of a model system is uniquely determined by the specification of the complete set of microscopic variables. The number of such variables is of the same order as the number of particles. In contrast, the macroscopic state is specified by a small number of measurable quantities such as net mass, energy, or volume. Generally a huge number of different microstates are compatible with one single macrostate; and it is a primary task of statistical mechanics to find out just how many microstates there are for a given set of macroscopic conditions.

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 $\vec{\Gamma}$.

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:

$$\vec{\Gamma} \equiv \{ \vec{r}_{1}, \dots, \vec{r}_{N} ; \ve... ... \vec{r}_{i} \epsilon V; \; v_{i,\alpha} \epsilon (\pm\infty)$$ (3.1)

The number of degrees of freedom (d.o.f.) is $n = 6 N$ (or, in 2 dimensions, $4N$); the number of velocity d.o.f. is $3N$ (or $2N$). It is often allowed - and always advantageous - to treat the subspaces $\vec{\Gamma}_{r} \equiv \{\vec{r}_{i} \}$ (position space) and $\vec{\Gamma}_{v} \equiv \{\vec{v}_{i} \}$ (velocity space) separately.

The representation of a microstate of the entire system by a single point in $6N$-dimensional Gibbs space must not be mixed up with the treatment of Chapter 2, introduced by Boltzmann. The $\mu$-space defined there had only $6$ dimensions $\{\vec{r},\vec{v} \}$, and each particle in the system had its own representative point. Thus the microstate of a system of $N$ particles corresponded to a swarm of $N$ points $\{\vec{r}_{i}, \vec{v}_{i} \}$.

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

$$\vec{\Gamma} \equiv \{ (n_{ix}, n_{iy}, n_{iz}); \; i=1, \dots N\}$$ (3.2)

Figure 3.2: Phase space of the 2D quantum gas

Model systems made up of $N$ spins are specified by

$$\vec{\Gamma} \equiv \{ \sigma_{i}; \; i=1, \dots N\}$$ (3.3)

with $\sigma_{i} = \pm 1$.

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

The assumption that all microstates that are compatible with the condition $E = E_{0}$ 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

$$\rho(\vec{r},\vec{v}) = \left\{ \begin{array}{ll} \rho_{0} \... ...vec{v}) = E_{0} \\ 0 \; & {\rm elsewhere} \end{array} \right.$$ (3.4)

The condition $E=const$ defines an $(n-1)$-dimensional ``surface'' within the $n$-dimensional phase space of the system. The set of all points upon that ``energy surface'' is named microcanonical ensemble. To take an example, in the $3N$-dimensional velocity space of a classical ideal gas the equation

$$E_{kin} \equiv \frac{m}{2} \sum_{i}v_{i}^{2} = const =E_{0}$$ (3.5)

defines the $(3N-1)$-dimensional surface of a $3N$-sphere of radius $r = \sqrt{2E_{0}/m}$.

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 $\langle v_{i}^{2} \rangle$ 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 $(N-1)$-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 $t_{0}$ needed for a sufficiently thorough perambulation of the energy surface is in the range of $10^{-9}-10^{-6}$ 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 $\Gamma$ 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

ensemble average = time average

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 $N$-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 $E = E_{0}$ we will generally require the weaker condition $E_{0}-\Delta E$ $\leq E$ $\leq E_{0}$ to hold. In other words, the permitted states of the system are to be restricted to a thin ``shell'' at $E \approx E_{0}$. 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

$$\rho(\vec{r},\vec{v}) = \left\{ \begin{array}{ll} \rho_{0} \... ...Delta E \right] \\ 0 \; & {\rm elsewhere} \end{array} \right.$$ (3.6)

For the classical ideal gas this requirement reads^3.1

$$E_{0}-\Delta E \leq E_{kin} \equiv (m/2) \sum_{i}v_{i}^{2} \leq E_{0}$$ (3.7)

For the ideal quantum gas we require

$$E_{0}-\Delta E \leq E_{N,\vec{n}} \equiv \frac{h^{2}}{8mL^{2}} \sum_{i}\vec{n}_{i}^{2} \leq E_{0}$$ (3.8)

In a system of $N$ non-interacting spins the energy shell is defined by $- H \sum_{i} \sigma_{i}$ $\epsilon [E_{0}, \Delta E]$.

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

Table 3.1: State spaces of some model systems

Model	Microvariables ($i=1,\dots ,N$)	Dimension of phase space	Energy
Classical ideal gas	$\vec{r}_{i}, \vec{v}_{i}$	$6N$	$E_{kin} \equiv \frac{m}{2}\sum v_{i}^{2}$
Hard spheres in a box	$\vec{r}_{i}, \vec{v}_{i}$	$6N$	$E_{kin} \equiv \frac{m}{2}\sum v_{i}^{2}$
Hard discs in a box	$\vec{r}_{i}, \vec{v}_{i}$	$4N$	$E_{kin} \equiv \frac{m}{2}\sum v_{i}^{2}$
LJ molecules, periodic boundary conditions	$\vec{r}_{i}, \vec{v}_{i}$	$6N$ (or. $6N-3$)	$E_{kin} + E_{pot} \equiv$ $\; \frac{m}{2}\sum v_{i}^{2} + \sum_{i,j>i} U_{LJ}(r_{ij})$
Linear or nonlinear molecules	$\vec{r}_{i}, \vec{v}_{i},\vec{e}_{i}, \vec{\omega}_{i}$	$10N$ or. $12N$	$E_{kin} + E_{pot} \equiv \newline \frac{m}{2}\sum v_{i}^{2}+ \frac{I}{2} \sum \omega_{i}^{2}+ E_{pot}$
Harmonic crystal	$\vec{q}_{i}, \dot{\vec{q}}_{i}$	$6N$	$E_{kin} + E_{pot} \equiv \newline \frac{m}{2}\sum \dot{\vec{q}}_{i}^{2} + \frac{f}{2}\sum \left\vert\vec{q}_{i}-\vec{q}_{i-1} \right\vert^{2}$
Ideal quantum gas	$\vec{n}_{i}$	$3N$	$\frac{h^{2}}{8mL^{2}}\sum_{i}\left\vert \vec{n}_{i}\right\vert^{2}$
Spins, non-interacting	$\sigma_{i}$	$N$	$- H \sum_{i} \sigma_{i}$

PHASE SPACE VOLUME AND THERMODYNAMIC PROBABILITY
Let us now recall the fundamental assumption that all microstates having the same energy are equally probable. Thus they represent elementary events as defined in Section 1.3.

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 $[E, \Delta E]$; 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 $(1/2^{N})$ and may thus be neglected - in accordance with experience and intuition.)