# Even Boltzmann had trouble with probability

Boltzmann was one of the genius founders of statistical thermodynamics, and yet the subtleties of probability tripped him up:

From “Compendium of the foundations of classical statistical physics” by Jos Uffink:

He introduced the probability distribution as follows:

“Let (v)dv be the sum of all the instants of time during which the velocity of a disc in the course of a very long time lies between v and v + dv, and let N be the number of discs which on average are located in a unit surface area, then

N ϕ(v)dv

is the number of discs per unit surface whose velocities lie between v and v + dv”

Thus, ϕ(v)dv is introduced as the relative time during which a (given) disc has a particular velocity. But, in the same breath, this is identified with the relative number of discs with this velocity. This remarkable quote shows how he identified two different meanings for the same function.*

*This is not to say that he always conflated these two interpretations of probability. Some papers employ a clear and consistent choice for one interpretation only. But then that choice differs between papers, or even in different sections of a single paper. In fact, in (Boltzmann 1871c) he even multiplied probabilities with different interpretations into one equation to obtain a joint probability. But then in (1872) he conflates them again. Even in his last paper (Boltzmann & Nabl 1904), Boltzmann identifies two meanings of probability with a simple-minded argument.

This, from the section that leads op to a discussion of the controversial ergodic hypothesis. ## Author: Mike White

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