A central result of non-equilibrium statistical physics

Here is a helpful quote from William Bialek. It is a footnote in a nice article, Perspectives on theory at the interface of physics and biology.

The Boltzmann distribution is the maximum entropy distribution consistent with knowing the mean energy, and this sometimes leads to confusion about maximum entropy methods as being equivalent to some sort of equilibrium assumption (which would be obviously wrong). But we can build maximum entropy models that hold many different expectation values fixed, and it is only when we fix the expectation value of the Hamiltonian that we are describing thermal equilibrium. What is useful is that maximum entropy models are equivalent to the Boltzmann distribution for some hypothetical system, and often this is a source of both intuition and calculational tools.

This type of approach features in the statistical mechanics of income distributions.

Examples where Bialek has applied this includes voting patterns of the USA Supreme Court, flocking of birds, and antibody diversity.

For a gentler introduction to this profound idea [which I still struggle with] see
*James Sethna's textbook, Entropy, Order parameters, and Complexity.
* review articles on large deviation theory by Hugo Touchette, such as this and this.
I thank David Limmer for bringing the latter to my attention.