Boltzmann distribution for non-energy models

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 $f_i (x)$: $$ int f_i(x) p(x) = langle f_irangle_{text{obs}} $$ 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: $$ mathcal{L}[p] = -int p(x) log p(x) dx + gamma left(1 - int p(x) dx right) + sum_{i} lambda_i left(langle f_irangle_{text{obs}} - int f_i(x) p(x) dx right) $$ 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