Destructive effect of fluctuations on the performance of a Brownian gyrator

TL;DR

This paper analyzes how fluctuations negatively impact the performance of the Brownian gyrator, revealing that its stochastic nature leads to destructive effects on its rotational behavior, supported by theoretical, numerical, and experimental evidence.

Contribution

It provides the first explicit probability density functions for the angular momentum and velocity of a discretized Brownian gyrator, highlighting the destructive influence of fluctuations.

Findings

PDF of angular momentum has exponential tails

PDF of angular velocity exhibits heavy power-law tails

Fluctuations are detrimental to the gyrator's performance

Abstract

The Brownian gyrator (BG) is often called a minimal model of a nano-engine performing a rotational motion, judging solely upon the fact that in non-equilibrium conditions its torque, angular momentum ${\cal L}$ and angular velocity $\cal W$ have non-zero mean values. For a time-discretized model, which is most adapted for the analysis of an essentially discrete-time data garnered in experiments or numerical simulations, we calculate the previously unknown probability density functions (PDFs) of ${\cal L}$ and $\cal W$. For finite time-step $\delta t$, the PDF of ${\cal L}$ has exponential tails and all moments are therefore well-defined, but the noise-to-signal ratio can attain big values for small $\delta t$. Conversely, the PDF of ${\cal W}$ exhibits heavy power-law tails and its mean ${\cal W}$ is the only existing moment. The BG is therefore not an engine in the common sense: it does not exhibit regular rotations on each run and its fluctuations are not only a minor nuisance -- on contrary, their effect is completely destructive for the performance. Our theoretical predictions are confirmed by numerical simulations and experimental data. We discuss some plausible improvements