Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon

TL;DR

This paper investigates the behavior of particles in unstable cubic potentials, introducing measurable local characteristics to understand their dynamics and the role of Maxwell's demon as a feedback mechanism stabilizing the system.

Contribution

It proposes and analyzes local measurable features like the most probable position and local uncertainty, revealing stabilization mechanisms via measurement-feedback in unstable stochastic systems.

Findings

Most probable position shifts against force over time

Local uncertainty does not outpace the position shift

Quasi-stationary probability density forms long-term

Abstract

Trajectories of an overdamped particle in a highly unstable potential diverge so rapidly, that the variance of position grows much faster than its mean. Description of the dynamics by moments is therefore not informative. Instead, we propose and analyze local directly measurable characteristics, which overcome this limitation. We discuss the most probable particle position (position of the maximum of the probability density) and the local uncertainty in an unstable cubic potential, both in the transient regime and in the long-time limit. The maximum shifts against the acting force as a function of time and temperature. Simultaneously, the local uncertainty does not increase faster than the observable shift. In the long-time limit, the probability density naturally attains a quasi-stationary form. We explain this process as a stabilization via the measurement-feedback mechanism, the Maxwell demon, which works as an entropy pump. Rules for measurement and feedback naturally arise from basic properties of the unstable dynamics. Observed thermally induced effects are inherent in unstable systems. Their detailed understanding will stimulate the development of stochastic engines and amplifiers and later, their quantum counterparts.