Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

Abstract

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. In this study, we present a more generalized TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [preprint arXiv:1901.05715]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Classical pendulum clocks break the thermodynamic uncertainty relation

“…Here, transitions in the configuration space appear irreversible without the simultaneous reversal of the momentum degrees of freedom, which then leads to a formally divergent entropy production, despite the actual entropy production being finite [10]. Bounds on the precision of irreversible currents in underdamped systems have been derived [10][11][12], however, they are weaker than the TUR or require additional information about the system (beyond the entropy production rate). Violations of the TUR have been observed where time-reversal symmetry is broken through the presence of an external magnetic field [13,14].…”

Quantitative analysis of non-equilibrium systems from short-time experimental data

“…As we discuss in the supplemental material however, there is another component of the entropy production, related to the work that the flow does against the confining potential [37,62]. This component, which does indeed depend on the value of u d , is not estimated by our inference scheme, due to the fact that u d is a field (corresponding to the velocities of the molecules of the thermal bath) which is odd under time reversal, for which the TUR does not hold [46,52,[63][64][65]]. Hence we expect that the values of σ we find close to the bubble are underestimates of the true value.…”

Estimating entropy production in a stochastic system with odd-parity variables

“…A scatter plot between ∆S tot and ∆ Ŝtot [Eq. (25)] at c = 10 with a fitted linear regression line (red line) is shown in the Fig. 3 inset.…”

“…To estimate the EP of such systems, existing methodologies require additional information because the definition of EP in these systems differs from that in systems without odd-parity variables. Indeed, TURs for underdamped Langevin systems require detailed information about the systems [20,24,25], and therefore we cannot exactly measure EP from only trajectory data using TURs. Due to these difficulties, to our knowledge, methods for estimating EP in a stochastic system with odd-parity state variables have yet to be developed.…”