Boltzmann distribution for non-energy models

Question

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?

Answer 1

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