We ask: how large is the probability of the last-mentioned state relative to the original state? Or, what is the probability that at some point in time all *n* independently moving points in a volume *v*_{0} have by chance ended up in the volume *v*?

For this probability, which is a "statistical probability" one obtains the value:

one derives from this, applying Boltzmann's principle:

It's noteworthy that for this derivation, from which the Boyle-Gay-Lussac law and the identical law of osmotic pressure can be easily derived thermodynamically ^{[1]}, there is no need to make any assumption regarding the way the molecules move.

## Interpretation of the Volume Dependence of the Entropy of Monochromatic Radiation using Boltzmann's Principle

In paragraph 4 we found for the dependence of Entropy of the monochromatic radiation on volume the expression:

This formula can be recast as follows:

- ↑ If
*E*is the energy of the system, then one obtains:- ;