Non quantum uncertainty relations of stochastic dynamics

TL;DR

This paper demonstrates that classical stochastic Hamiltonian systems exhibit quantum-like uncertainty relations and commutation relations due to thermal noise and chaos, linking fluctuations to a parameter $\eta$.

Contribution

It introduces an information-action framework showing how quantum-like uncertainty relations naturally emerge in classical stochastic dynamics.

Findings

Derivation of quantum-like uncertainty relations for action, position-momentum, and energy-time in stochastic systems.

Establishment of a commutation relation analogous to quantum mechanics.

Fluctuations characterized by the parameter $\eta$ in classical stochastic Hamiltonian systems.

Abstract

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 $<\delta A>=0$. The main objective of this paper is to prove that, via this information-action description, some quantum like uncertainty relations such as $<\Delta A>\geq\frac{1}{\sqrt{2}\eta}$ for action, $<\Delta x><\Delta P>\geq\frac{1}{\eta}$ for position and momentum, and $<\Delta H><\Delta t>\geq\frac{1}{\sqrt{2}\eta}$ 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 $\eta$.