Cost and Precision of Brownian Clocks

TL;DR

This paper explores the thermodynamic costs and precision limits of Brownian clocks, revealing that periodic driving allows for high precision at low cost, unlike constant-force driven clocks.

Contribution

It demonstrates that periodically driven Brownian clocks can achieve arbitrary precision with minimal thermodynamic cost, contrasting with fixed-force clocks.

Findings

Periodic driving enables high-precision clocks at low thermodynamic costs.

Constant thermodynamic force clocks require diverging costs for high precision.

Mapping to bipartite Markov processes simplifies uncertainty calculations.

Abstract

Brownian clocks are biomolecular networks that can count time. A paradigmatic example are proteins that go through a cycle thus regulating some oscillatory behaviour in a living system. Typically, such a cycle requires free energy often provided by ATP hydrolysis. We investigate the relation between the precision of such a clock and its thermodynamic costs. For clocks driven by a constant thermodynamic force, a given precision requires a minimal cost that diverges as the uncertainty of the clock vanishes. In marked contrast, we show that a clock driven by a periodic variation of an external protocol can achieve arbitrary precision at arbitrarily low cost. This result constitutes a fundamental difference between processes driven by a fixed thermodynamic force and those driven periodically. As a main technical tool, we map a periodically driven system with a deterministic protocol to one subject to an external protocol that changes in stochastic time intervals, which simplifies calculations significantly. In the non-equilibrium steady state of the resulting bipartite Markov process, the uncertainty of the clock can be deduced from the calculable dispersion of a corresponding current.