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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?

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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

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