# 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?

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: $$p(x) = \frac{1}{Z} e^{\sum_i \lambda_i f_i(x)}$$