TABLE OF CONTENTS We avoid the gravest difficulties when, giving up the attempt to frame hypotheses concerning the constitution of matter, we pursue statistical inquiries as a branch of rational mechanics.
J.Willard Gibbs
We are now in a position to quantify the relationship between temperature and the probability of occupation of energy levels. We consider a microsystem in equilibrium with a heat bath at temperature T. Our purpose is to find a mathematical expression for the probability that a state of the microsystem with energy E is occupied, as a function of E and T.
First, we must make the concept of heat bath precise. Suppose that we have two systems, ? 0 and ?, where ? 0 is extremely large while ? is much smaller (Figure 5.1(a)). We assume that both have fixed volume and are isolated from the surroundings. Initially, ? 0 is at temperature T, while ? is at a different temperature. The two systems are then brought into contact. They come into equilibrium by transferring a certain amount of energy, in the form of heat, in one direction or the other. However, because ? 0 is much larger than ?, the effect of this energy transfer on the properties of the large system ? 0 is very small compared with its effect on the small system ?. It follows that in equilibrium, both systems are at...
