Soshichi Uchii, Kyoto University

3. A Quick Review of Thermodynamics

Maybe some words of explanation are in order here, for those of you who are unfamiliar with thermodynamics. Thermodynamics, or the theory of heat, was developed by many people, but probably Clausius is most responsible for its systematization. Roughly speaking, Clausius constructed his theory on two basic laws. The first law is that heat and mechanical work are interchangeable and there is a definite quantitative relation between the two [W=JQ, where J is Joule's constant]. The second law is that heat cannot move from cold body into hot body without some other change; in other words, we need to add some work in order to make such a heat transfer possible. Clausius made this second law more precise and gave it a mathematical formulation in terms of entropy: in any process of a closed system, its entropy either stays the same or increases.

As you already know from your experience, this second law makes many heat processes irreversible. For instance, the hot water in a kettle is bound to cool down, and unless we heat, it can never boil again by itself. In terms of entropy, we can describe it in this way: Suppose that the kettle is in a room and they form a closed system (no heat coming in or going out). Then the initial state of the room with the boiling water has a low entropy; a certain amount of heat is condenced in the kettle. As the time goes, the kettle radiates heat around the room, cooling itself but raising the room temperature a little bit; thus the entropy of the whole system increases. And when its temperature becomes the same as that of the room, the heat process ends, reaching an equilibrium state, which is the state with highest entropy. In other words, the distribution of heat in the room becomes uniform.

You can imagine a similar process in terms of a deck of cards, initially separated into two parts, the one all red and the other all black; as you shuffle the deck, reds and blacks are distributed more and more uniformly. Now, anticipating the kinetic theory, suppose a red card corresponds to a molecule with high velocity (thus with a larger kinetic energy), a black card to a molecule with low velocity. Then the initial state of the deck has a low entropy (energy or temperature unevenly distributed), the uniform distribution a highest entropy.

4. Mechanics and Probability: Maxwell's Adventure

2. What is the problem with the Kinetic Theory?

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March 1, 1999; last modified March 4. (c) Soshichi Uchii