General criterion of equally probable states.

In the kinetic theory of gases, and in the theory of radiation, one of the fundamental problems is to compare the relative probabilities of different states of a given system, of which not enough is known to determine the actual state. Various criteria of what may be taken to be equally probable states have been suggested, mostly based on what are conceived to be fundamental or intuitive conceptions.

For example if, in the kinetic theory of gases, nothing is known of a given particle save that it is within a certain region of space, it is assumed that all positions within that region are equally likely to be the actual position of the particle; or again it is assumed that for a given particle of which nothing is known to restrict its velocity, all velocities are equally probable, no matter how great.

Now in the case of the velocity it is obvious that the principle of relativity cannot admit this as a reasonable assumption, since the continual addition of velocities never leads to a velocity greater than that of light; and so the question may be asked: ‘What criteria of equal probability are consistent with the principle?’

A general criterion may be laid down applying to all cases.

Any two states of a self-contained system which can be transformed into one another by a Lorentz transformation are to be considered as equally probable.