Every distribution (that agrees with the base measure about null sets) is a Boltzmann distribution. Simply define E(x):=−kBTlnP[x], and presto, P[x]=e−1kBTE(x).
This is a very useful/important/underrated fact, but it does somewhat trivialize “Boltzmann” and “maximum entropy” as classes of distributions, rather than as certain ways of looking at distributions.
A related important fact is that temperature is not really a physical quantity, but 1kBT is: it’s known as inverse temperature or β. (The nonexistence of zero-temperature systems, the existence of negative-temperature systems, and the fact that negative-temperature systems intuitively seem extremely high energy bear this out.)
I am a little confused about this. It was my understanding that exponential families are distinguished class of families of distributions.
For instance, they are regular (rather than singular).
The family of mixed Gaussians is not an exponential family I believe.
So my conclusion would be that the while “being Boltzmann” for a distribution is trivial as you point out, “being Boltzmann” (= exponential) for a family is nontrivial.
Every distribution (that agrees with the base measure about null sets) is a Boltzmann distribution. Simply define E(x):=−kBTlnP[x], and presto, P[x]=e−1kBTE(x).
This is a very useful/important/underrated fact, but it does somewhat trivialize “Boltzmann” and “maximum entropy” as classes of distributions, rather than as certain ways of looking at distributions.
A related important fact is that temperature is not really a physical quantity, but 1kBT is: it’s known as inverse temperature or β. (The nonexistence of zero-temperature systems, the existence of negative-temperature systems, and the fact that negative-temperature systems intuitively seem extremely high energy bear this out.)
I am a little confused about this. It was my understanding that exponential families are distinguished class of families of distributions. For instance, they are regular (rather than singular).
The family of mixed Gaussians is not an exponential family I believe.
So my conclusion would be that the while “being Boltzmann” for a distribution is trivial as you point out, “being Boltzmann” (= exponential) for a family is nontrivial.