What Could a “Natural Measure” Be?

The invocation of probabilistic considerations in physics often involves, implicitly or explicitly, some notion of relative sizes, or measures, of sets of possibilities. In equilibrium statistical mechanics, certain standard measures are introduced explicitly. It is often said that these measures are “natural,” in some sense. This chapter explores what that could mean. It does so by means of a toy example, a fictitious machine that I call the parabola gadget. The dynamics of the parabola gadget pick out a measure on the space of states of the gadget that other measures converge towards. In this sense, that measure is a natural one to use for systems that have been evolving freely long enough for the requisite washing-out of disagreements among input distributions to have taken place. We have good reason to think that the standard measures evoked in equilibrium statistical mechanics are of this sort One upshot of this is that this notion of standard measure is of no use for making judgments about probability or improbability of conditions in the early universe.