Reformulation of a stochastic action principle for irregular dynamics

A stochastic action principle for random dynamics is revisited. Numerical diffusion experiments are carried out to show that the diffusion path probability depends exponentially on the Lagrangian action A=∫abLdt. This result is then used to derive the Shannon measure for path uncertainty. It is shown that the maximum entropy principle and the least action principle of classical mechanics can be unified into δA¯=0 where the average is calculated over all possible paths of the stochastic motion between two configuration points a and b. It is argued that this action principle and the maximum entropy principle are a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.