Uniform distribution of initial states: The physical basis of probability.

Abstract

For repetitive experiments performed on a deterministic system with initial states restricted to a certain region in phase space, the relative frequency of an event has a definite value insensitive to the preparation of the experiments only if the initial states leading to that event are distributed uniformly in the prescribed region. Mechanical models of coin tossing and roulette spinning and equal a priori probability hypothesis in statistical mechanics are considered in the light of this principle. Probabilities that have arisen from uniform distributions of initial states do not necessarily submit to Kolmogorov's axioms of probability. In the finite-dimensional case, a uniform distribution in phase space either in the coarse-grained sense or in the limit sense can be formulated in a unified way.