# Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials

## TL;DR

This paper provides physical insights into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials, showing the relation's bounds are attained at various nonequilibrium conditions and linking heat distribution deviations to the inequality.

## Contribution

The study demonstrates that the thermodynamic uncertainty relation bounds are valid beyond near-equilibrium, especially when heat distribution is Gaussian, and introduces a new bound relating heat variance and mean.

## Key findings

- Bound is attained at both near- and far-from-equilibrium conditions.
- Maximum energetic cost occurs near the critical force where potential barriers vanish.
- Deviation from Gaussian heat distribution influences the uncertainty relation.

## Abstract

Using Brownian motion in periodic potentials $V(x)$ tilted by a force $f$, we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation $q$, and in order to limit the output fluctuation within a relative uncertainty $\epsilon$, at least $2k_BT/\epsilon^2$ of heat must be dissipated. Our model shows that this bound is attained not only at near-equilibrium ($f\ll V'(x)$) but also at far-from-equilibrium $(f\gg V'(x))$, more generally when the dissipated heat is normally distributed. Furthermore, the energetic cost is maximized near the critical force when the barrier separating the potential wells is about to vanish and the fluctuation of Brownian particle is maximized. These findings indicate that the deviation of heat distribution from Gaussianity gives rise to the inequality of the uncertainty relation, further clarifying the meaning of the uncertainty relation. Our derivation of the uncertainty relation also recognizes a new bound of nonequilibrium fluctuations that the variance of dissipated heat ($\sigma_q^2$) increases with its mean ($\mu_q$) and cannot be smaller than $2k_BT\mu_q$.