12: Temperature

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Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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In previous chapters of these notes we introduced the Principle of Maximum Entropy as a technique for estimating probability distributions consistent with constraints.

In Chapter 8 we discussed the simple case that can be done analytically, in which there are three probabilities, one constraint in the form of an average value, and the fact that the probabilities add up to one. There are, then, two equations and three unknowns, and it is straightforward to express the entropy in terms of one of the unknowns, eliminating the others, and find the maximum. This approach also works if there are four probabilities and two average-value constraints, in which case there is again one fewer equation than unknown.

In Chapter 9 we discussed a general case in which there are many probabilities but only one average constraint, so that the entropy cannot be expressed in terms of a single probability. The result previously derived using the method of Lagrange multipliers was given.

In Chapter 11 we looked at the implications of the Principle of Maximum Entropy for physical systems that adhere to the multi-state model motivated by quantum mechanics, as outlined in Chapter 10.

We found that the dual variable \(\beta\) plays a central role. Its value indicates whether states with high or low energy are occupied (or have a higher probability of being occupied). From it all the other quantities, including the expected value of energy and the entropy, can be calculated.

In this chapter, we will interpret \(\beta\) further, and will define its reciprocal as (to within a scale factor) the temperature of the material. Then we will see that there are constraints on the efficiency of energy conversion that can be expressed naturally in terms of temperature.