Chapter 1 - Fisher Variational Principle and Thermodynamics

Abstract

This chapter illustrates that thermo statistics, usually derived microscopically from Shannon's information measure via Jaynes' MaxEnt procedure, can equally be obtained from a constrained extremization of Fisher's information measure. The new procedure has the advantage of dealing with both equilibrium and off equilibrium processes, as illustrated by two significant examples. The chapter shows that the essential thermo dynamical features of equilibrium and non-equilibrium problems can be derived from a constrained Fisher extremization process that results in a Schrodinger-like stationary equation (SSE). Equilibrium corresponds to the ground-state (gs) solution. Non-equilibrium corresponds to super-positions of gas with excited states. With reference to dilute gases, two typical illustrative examples, viscosity and electrical conductivity are discussed. The entire Legendre-transform structure of thermo-dynamics can be obtained using Fisher information in place of Boltzmann's entropy. The Legendre structure constitutes an essential ingredient that allows construction of a statistical mechanics. Fisher's information I then allow such a construction. The desired concavity property obeyed by I further demonstrates its utility as a statistical-mechanics “generator”. A thermodynamics can be developed that is able to treat equally well both equilibrium and non-equilibrium solutions, as illustrated with reference to Boltzmann's transport equation.