Kinetic uncertainty relation

Abstract

Relative fluctuations of observables in discrete stochastic systems are bounded at all times by the mean dynamical activity in the system, quantified by the mean number of jumps. This constitutes a kinetic uncertainty relation that is fundamentally different from the thermodynamic uncertainty relation recently discussed in the literature. The thermodynamic constraint is more relevant close to equilibrium while the kinetic constraint is the limiting factor of the precision of a observables in regimes far from equilibrium. This is visualized for paradigmatic simple systems and with an example of molecular motor dynamics. Our approach is based on the recent fluctuation response inequality by Dechant and Sasa [arXiv:1804.08250] and can be applied to generic Markov jump systems, which describe a wide class of phenomena and observables, including the irreversible predator-prey dynamics that we use as an illustration.

Unifying thermodynamic uncertainty relations

“…We now prove the activity bound [31]: the precision of an observable that is the weighted sum of the transitions undergone by a finite-state continuous time Markov chain during an arbitrary time interval is bounded by the mean number of transitions.…”

“…It yields an upper bound for the response of f , which reduces to (1) when the chosen perturbation results in a time rescaling of the dynamics. The entropic [23] as well as the activity bound [31] have thus been extended to both current-like and counting observables. This approach, which makes contact with inequalities originally derived by Kullback [25], has sparked much interest in the application of information theoretic results and concepts.More recently [28], the exponential bound p max = (exp σ − 1)/2 has been derived for Langevin dynamics with feedback, under the condition of validity of the detailed (joint) fluctuation theorem for σ and f .…”

“…The activity bound turns out to be another avatar of the Periodic Uncertainty Relation, where the bound can be derived on an infinitesimal interval and then extended to arbitrary times (see SI). Example-We illustrate the different bounds for stationary Markovian dynamics on a benchmark example [31] which provides a minimal model for the molecular motor kinesin moving under load along a microtubule. Kinesin is either in a low energy state (1) with both heads on the microtubule or in a high energy state (2) with only one head attached.…”

Frenesy: Time-symmetric dynamical activity in nonequilibria

“…where the friction is increased by 1 + α without touching the diffusion constant D. Then, the second order in the action is the kinetic energy: s α = α 2 γ 2 v 2 /(4D). , and with · α the expectation in the perturbed process; see [70,71] for details. Observe that the (unperturbed) expectation A ′′ 0 is related to the frenesy as the above examples illustrate.…”

Thermodynamic uncertainty relations in a linear system

“…A connection between discrete and continuous time uncertainty relations is shown in [34]. One can also see similar uncertainty relations in the context of discrete processes [35], multidimensional systems [36], Brownian motion in the tilted periodic potential [37], general Langevin systems [38], molecular motors [39], run and tumble processes [40], biochemical oscillations [41], interacting oscillators [42], effect of magnetic field [43], linear response [44], measurement and feedback control [45], information [46], underdamped Langevin dynamics [47], timedelayed Langevin systems [48], various systems [49], etc.. Recently, Hasegawa et al [50] found an uncertainty relation for the time-asymmetric observable for the system driven by a time-symmetric driving protocol using the steady state fluctuation theorem.…”