Maximum Entropy with Nonlinear Constraints: Physical Examples

Abstract

The Maximum Entropy literature has hitherto recognised only problems with constraints linear in the required probability distribution. However, nonlinear-constraint problems go back unrecognised for over half of the 120-year lifetime of Maximum Entropy. The classic example is calculation of the charge density ρ in plasma in terms of the potential φ where the energy constraint is quadratic in φ In fact the Boltzmann distribution ρ α exp(−β _q φ) traditionally derived under linear constraints, still holds; the condition for this is merely that a conserved quantity exists. The distribution can be combined with Poisson’s equation to give a single equation for φ or ρ. Uniqueness of the entropy maximum must be examined case-by-case in nonlinear problems.