Continuity equations, H-theorems, and maximum entropy

Abstract

Linear partial differential evolution equations that have the form of continuity equations are investigated with reference to the temporal behavior of the Gibbs-Shannon-Jaynes entropy functional S( t) = − ∫ f (ln f) d x associated with their solutions (regarded as probability distributions f( x, t)). Many attempts have been made to approximately solve these type of equations, based upon Jaynes' maximum entropy principle. We show that, within this maximum entropy approach, the functional relation d S[ f ME]/d t = F [ f ME(x, t)], giving the time derivative of the entropy in terms of the approximate maximum entropy solution, has exactly the same form as the corresponding relation d S[ f exact]/d t = F [ f exact(x, t)] that hold in the case of the exact solutions. From this it follows that any H-theorem (involving the Gibbs-Shannon-Jaynes entropy) satisfied by the exact solutions of the differential equations holds for the maximum entropy approximations f ME( x, t) as well. Some illustrative examples are discussed.