Fisher information, canonical ensemble, and Hamiltonian systems

We show that the mathematical form of the information measure of Fisher's I[f]I[f], if expressed as a functional of Gibbs' canonical probability distribution f (the most important one in statistical mechanics), incorporates important features of the intrinsic structure of classical mechanics and has a universal form in terms of “forces” (derivatives of generalized momenta with respect to time) and “generalized velocities”, i.e., one that is valid whenever Hamilton's equations of motion hold. Additionally, if the system of differential equations associated with Hamilton's canonical equations of motion is linear (a very important special instance), we show that there is an equal amount of Fisher information per degree of freedom, proportional to the inverse temperature. This gives temperature an “information” meaning. This “equipartition” of I is also shown to hold in a simple but relevant (magnetic) example involving a non-linear system of differential equations, where the Fisher measure turns out to be proportional to Langevin's function.