Diffusing Up the Hill: Dynamics and Equipartition in Highly Unstable Systems

TL;DR

This paper experimentally verifies a new approach that uses local characteristics of particle motion to analyze highly unstable systems where traditional statistical methods fail, revealing stable local behavior despite diverging trajectories.

Contribution

It introduces and experimentally confirms a local analysis method for unstable systems, enabling stable position detection and understanding of their energetics.

Findings

Most-likely particle position shifts against the force.

Local uncertainty saturates despite strong diffusion.

Particle distribution converges quickly to a quasi-stationary state.

Abstract

Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard statistical analysis of unstable mechanical processes and their applications. A newly proposed approach takes advantage of the local characteristics of the most probable particle motion instead of the divergent averages. We experimentally verify its theoretical predictions for a Brownian particle moving near an inflection in a highly unstable cubic optical potential. The most-likely position of the particle atypically shifts against the force despite the trajectories diverge in the opposite direction. The local uncertainty around the most-likely position saturates even for strong diffusion and enables well-resolved position detection. Remarkably, the measured particle distribution quickly converges to the quasi-stationary one with the same atypical shift for different initial particle positions. The demonstrated experimental confirmation of the theoretical predictions approves the utility of local characteristics for highly unstable systems which can be exploited in thermodynamic processes to uncover energetics of unstable systems.