Title: Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials,
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 $2k_BT\mu_q$.