Probability and Statistical Equilibrium

If we were to create macrostates by assembling particles by randomly throwing them into various microstates, then the macrostate most likely to occur is the one with the greatest number of microstates. That is why a bridge hand consisting of 13 spades occurs so rarely compared to a hand with four spades and three hearts, three diamonds, or three clubs. If we consider a system where the particles are continually moving from one phase space volume to another, say, by collisions, then the most likely macrostate is the one with the largest number of associated microstates. There is an implicit assumption here that all microstates are equally probable. Is this reasonable?

Imagine a case where all the molecules in a room are gathered in one corner. This represents a particular microstate; a particularly unlikely one, we would think. Through random motions, it would take an extremely long time for the particles to return to that microstate. However, given the position and velocity of each particle in

an ordinary room of gas, is this any more unlikely than each particle to returning to that specific position with the same velocity? The answer is no. Thus, if each microstate is equally probable, then the associated macrostates are not equally probable and it makes sense to search for the most probable macrostate of a system. In a system which is continually rearranging itself by collisions, the most probable macrostate becomes the most likely state in which to find the system. A system which is in its most probable macrostate is said to be in statistical equilibrium.

Many things can determine the most probable macrostate. Certainly the total number of particles allowed in each microstate and the total number of particles available to distribute will be important in determining the total number of microstates in a given macrostate. In addition, quantum mechanics places some conditions on our ability to distinguish particles and even limits how many of certain kinds of particles can be placed in a given volume of phase space. But, for the moment, let us put aside these considerations and concentrate on calculating the number of microstates in a particular macrostate.

Figure 1.1 Shows a phase space composed of only two cells in which four particles reside. All possible macrostates are illustrated.

Consider a simple system consisting of only two phase space volumes and four particles (see Figure 1.1). There are precisely five different ways that the four particles can be arranged in the two volumes. Thus there are five macrostates of the system. But which is the most probable? Consider the second macrostate in Figure 1.1 (that is, N₁ = 3, N₂ = 1). Here we have three particles in one volume and one particle in the other volume. If we regard the four particles as individuals, then there are four different ways in which we can place those four particles in the two volumes so that one volume has three and the other volume has only one (see Figure 1.1). Since the order in which the particles are placed in the volume does not matter, all permutations of the particles in any volume must be viewed as constituting the same microstate.

Now if we consider the total number of particles N to be arranged sequentially among m volumes, then the total number of sequences is simply N!. However, within each volume (say, the ith volume), N_i particles yield N_i! indistinguishable sequences which must be removed when the allowed number of microstates is counted. Thus the total number of allowed microstates in a given macrostate is

(1.1)

Figure 1.2 Consider one of the macrostates in figure 1.1, specifically the state where N₁ = 3, and N₂ = 1. All the allowed microstates for distinguishable particles are shown. For the five macrostates shown in Figure 1.1, the number of possible microstates is

(1.2)

Clearly W_{2, 2} is the most probable macrostate of the five. The particle distribution of the most probable macro state is unique and is known as the equilibrium macrostate.

In a physical system where particle interactions are restricted to those between particles which make up the system, the number of microstates within the system changes after each interaction and, in general, increases, so that the macrostate of the system tends toward that with the largest number of microstates - the equilibrium macrostate. In this argument we assume that the interactions are uncorrelated and random. Under these conditions, a system which has reached its equilibrium macrostate is said to be in strict thermodynamic equilibrium. Note that interactions among particles which are not in strict thermodynamic equilibrium will tend to drive the system away from strict thermodynamic equilibrium and toward a different statistical equilibrium distribution. This is the case for stars near their surfaces.

The statistical distribution of microstates versus macrostates given by equation (1.1) is known as Maxwell-Boltzmann statistics and it gives excellent results for a classical gas in which the particles can be regarded as distinguishable. In

a classical world, the position and momentum of a particle are sufficient to make it distinguishable from all other particles. However, the quantum mechanical picture of the physical world is quite different. So far, we have neglected both the Heisenberg uncertainty principle and the Pauli Exclusion Principle.