ABSTRACT
We are now in a position to approach the problem that we have been postponing for so long: how to deal with a many-particle system in which the individual particles cannot be distinguished from each other. The simplest and most useful example of such a system is a “perfect” gas, of which the “ideal” gas is a special case. Note the distinction, the need for which will become clear later in this chapter.1 A perfect gas is a gas in which the energy of any one molecule is independent of the presence of the others; that is, interactions between the molecules have a negligible e&ect on their energy states. An ideal gas is a perfect gas whose density is su^ciently low that quantum e&ects can be neglected. Because this distinction is not always observed in the literature, some authors prefer to call a perfect gas a noninteracting gas. However, this terminology is somewhat misleading. Except at extremely low density, gas molecules collide frequently (see Chapter 16) and hence interact; if they did not the gas could not reach
internal equilibrium. Thus even in a perfect gas there are intermolecular interactions, but they are not strong enough to a&ect the energy states. Our purpose is to hnd the distribution function of a perfect gas. In
the present context, this term has a specialized meaning, and is dehned as follows. Suppose that we know all the possible states of a single particle in the system (for example, the states of a particle in a box, calculated in Chapter 6). We call these single particle states, and here we dehne occupancy n as the number of particles in a given single particle state. In a perfect gas, the energy of any one particle is not a&ected by the presence of the other particles, so that the single particle states are independent of the occupancies. It follows that a macrostate of a perfect gas with a given particle density is completely dehned if we know the mean occupancy n(E)X for all E. The mean occupancy is called the distribution function and is written f(E):
f(E) = n(E)X. (8.1)
While f(E) is often less than unity (much less in the case of an ideal gas), it is not a probability. As we shall see, it can exceed unity in a Bose gas; furthermore, the sum of the occupancies over all possible states is f(E) = N , the number of particles in the system, whereas probability
P is normalized so that P = 1. In general, f(E) depends parametrically
on temperature and chemical potential.
