Thermodynamic cost for an (almost) periodic sequence of events

A mechanism that produces a precise sequence of events, or “ticks”, is a crucial element for the design of any modern clock. Examples for such processes from biology include heart beats and neural spike trains. In a thermal environment, the periodicity of such a sequence will be affected by noise. We ask how much energy needs to be provided to overcome this noise. At small damping, almost periodic oscillations of a pendulum arise naturally even in equilibrium, which can be exploited in a model for a clock that breaks the bound on cost and precision set by the the thermodynamic uncertainty relation. However, this loophole cannot be used by biological mechanisms, which are subject to strong damping. We analytically derive new universal bounds that describe the trade-off between cost and precision in a sequence of events in such systems system. The precision is quantified by the fluctuations in either the number of events counted over time or the times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

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