free energy

In physics, a 'free energy' measures our ability to extract work from a system in a given macrostate. We can represent a macrostate by a distribution $q(x)$ over microstates $x$, which we might think of as representing a constrained subset of all the microstates a system might be in. (for example, if all the gas molecules in a box happen to be in the left half of the box). Work can be extracted as the system relaxes towards the maximum-entropy distribution over microstates. The 'distance from', or missing entropy relative to, this distribution, measures how much work can be extracted.

As Jaynes points out, ability to do work is subjective: work can only be extracted from observable quantities of a system (doing work is observable, so anything that can't be observed can't do work!). Some observers may have the ability to measure and extract work from distinctions in a system that other observers can't.

A free energy typically has the form

where $E$ is an energy function over microstates, $U$ is the average (or total) energy of the macrostate, and T is a temperature.

Free energy is minimized at the Boltzmann distribution $\frac{e^{- U(x) /T} }{ \int e^{-U(x)/T} dx}$ . Minimizing the free energy can equivalently be seen as maximizing entropy conditioned on the average energy U, where temperature acts as the Lagrange multiplier imposing the constraint.