Non-quantum uncertainty relations of stochastic dynamics

First we describe briefly an information-action method for the study of stochastic dynamics of Hamiltonian systems perturbed by thermal noise and chaotic instability. It is shown that, for the ensemble of possible paths between two configuration points, the action principle acquires a statistical form 〈δA〉 = 0. The main objective of this paper is to prove that, via this information-action description, some quantum like uncertainty relations such as 〈ΔA〉⩾12η for action, 〈Δx〉〈ΔP〉⩾1η for position and momentum, and 〈ΔH〉〈Δt〉⩾12η for Hamiltonian and time, can arise for stochastic dynamics of classical Hamiltonian systems. A corresponding commutation relation can also be found. These relations describe, through action or its conjugate variables, the fluctuation of stochastic dynamics due to random perturbation characterized by the parameter η.