Diffusion laws, information and action principle

Abstract

This is an attempt to address diffusion phenomena from the point of view of information theory. We imagine a regular hamiltonian system under the random perturbation of thermal (molecular) noise and chaotic instability. The irregularity of the random process produced in this way is taken into account via the dynamic uncertainty measured by a path information associated with different transition paths between two points in phase space. According to the result of our previous work, this dynamic system maximizes this uncertainty in order to follow the action principle of mechanics. In this work, this methodology is applied to particle diffusion in external potential field. By using the exponential probability distribution of action (least action distribution) yielded by maximum path information, a derivation of Fokker-Planck equation, Fick's laws and Ohm's law for normal diffusion is given without additional assumptions about the nature of the process. This result suggests that, for irregular dynamics, the method of maximum path information, instead of the least action principle for regular dynamics, should be used in order to obtain the correct occurring probability of different paths of transport. Nevertheless, the action principle is present in this formalism of stochastic mechanics because the average action has a stationary associated with the dynamic uncertainty. The limits of validity of this work is discussed.