Tight thermodynamic uncertainty relation for systems with inertia
In plain words
For small systems that jump between states or move without inertia, the relative noise in any current is bounded below by an amount set by the energy dissipated. Systems with inertia can beat that bound, and the correct tight bound for them is unknown.
Precise statement
The thermodynamic uncertainty relation $\operatorname{Var}(J)/\langle J\rangle^{2} \ge 2 k_{B}/\Sigma$ ($\Sigma$ = total entropy production in the observation time) holds for steady-state Markov jump and overdamped Langevin dynamics. For underdamped Langevin dynamics $m\, \mathrm{d}v = (F(x) - \gamma v)\, \mathrm{d}t + \sqrt{2 \gamma k_{B} T}\, \mathrm{d}W$ in a steady state, find the optimal lower bound on $\operatorname{Var}(J)/\langle J\rangle^{2}$ in terms of $\Sigma$ and other measurable quantities (dynamical activity, mean kinetic energy, $m/\gamma$), and identify systems that saturate it.
What would settle it
A proof of a bound together with a family of underdamped systems that saturates it.
Status in the literature
Several valid but non-tight bounds for inertial dynamics exist; the optimal bound is not known.