Non quantum uncertainty relations of stochastic dynamics
Résumé
After a recapitulation of an information-action method for the study of stochastic dynamics of hamiltonian systems perturbed by thermal noise and chaotic instability, we show that, for the ensemble of all the possible paths between two state points, the action principle acquires a statistical form $\langle\delta A\rangle=0$. The main objective of this paper is to prove that, via this information-action description, some uncertainty relations such as $\langle\Delta A\rangle\geq\frac{1}{2\eta}$ for action, $\langle\Delta x\rangle\langle\Delta P\rangle\geq\frac{1}{\eta}$ for position and momentum, and $\langle\Delta H\rangle\langle\Delta t\rangle\geq\frac{1}{2\eta}$ for hamiltonian and time, exist for stochastic dynamics of hamiltonian systems. These relations describe, through action or its conjugate variables, the fluctuation of stochastic dynamics due to random perturbation characterized by the parameter $\eta$.