Boltzmann distribution for non-energy models My question is rather general and it regards the possibility to associate the Boltzmann probability distribution to some energy model.
What are the general assumptions that an Energy-based model needs to have, in order to be described by a Boltzmann probability distribution?
Furthermore, is it possible to extend this to other scalar variables, other than energy? What are the requirements to do this extension?
 A: In the most general setting the probability distribution can be derived with the help of MaxEntropy method : you search for the distribution, that maximizes the entropy, subject to constrains (conservation laws).
For instance, you have some probability function $p(x)$, it has to obey the normalization condition :
$$
\int p(x) dx
$$
Also, the can be several conservation laws - we fixed expectation value of some observables $f_i (x)$:
$$
\int f_i(x) p(x) = \langle f_i\rangle_{\text{obs}}
$$
The goal is to minimize the Shannon entropy, subject to the above constraints. In order to do this, one introduces Lagrange multipliers, and solves following optimization problem:
$$
\mathcal{L}[p] = -\int p(x) \log p(x) dx + \gamma \left(1 - \int p(x) dx \right) + \sum_{i} \lambda_i \left(\langle f_i\rangle_{\text{obs}} - \int f_i(x)  p(x) dx \right)
$$
The general solution of the MaxEnt distribution is:
$$
p(x) = \frac{1}{Z} e^{\sum_i \lambda_i f_i(x)} 
$$
These ideas are from the following papers:
Jaynes, Edwin T (1957a), “Information theory and statistical
mechanics,” Physical review 106 (4), 620.
Jaynes, Edwin T (1957b), “Information theory and statistical
mechanics. ii,” Physical review 108 (2), 171
