Justification in Statistical Mechanics

According to a standard view of the second law of thermodynamics, our belief in the second law can be justified by pointing out that low entropy macrostates are less probable than high entropy macrostates, and then noting that a system in an improbable state will tend to evolve toward a more probable state. I would like to argue that this justification of the second law of thermodynamics is fundamentally flawed, and will show that some puzzles sometimes associated with the second law are merely artifacts of this incorrect justification. 1 The Standard Story. If we squirt some colored ink into a closed container of moving water, we are justified in expecting that the ink will eventually disperse itself evenly through the water. Indeed, if we prepare any system in a state of low entropy, and then isolate the system, we are justified in expecting that the system will experience an increase in entropy. In virtue of what are these beliefs justified? Such beliefs can, of course, be justified on purely inductive grounds. But one might also think that such beliefs ought to be able to be justified on the basis of more fundamental physical and mathematical principles. In this paper, I would like to discuss and criticize a common justification of this later sort. The challenge posed in the last paragraph is to explain why we are justified in believing the second law of thermodynamics, given only more fundamental physical and mathematical principles. For our purposes, the second law of thermodynamics states that the entropy of an isolated system will increase until the system reaches its equilibrium state, which is a state of maximum entropy. Once in this equilibrium state, the system will remain there (or at least will remain there for a very long time.) A standard (though somewhat schematic) argument for the second law is as follows: