Optimal time estimation and the clock uncertainty relation for stochastic processes
arxiv(2024)
Abstract
Time estimation is a fundamental task that underpins precision measurement,
global navigation systems, financial markets, and the organisation of everyday
life. Many biological processes also depend on time estimation by nanoscale
clocks, whose performance can be significantly impacted by random fluctuations.
In this work, we formulate the problem of optimal time estimation for Markovian
stochastic processes, and present its general solution in the asymptotic
(long-time) limit. Specifically, we obtain a tight upper bound on the precision
of any time estimate constructed from sustained observations of a classical,
Markovian jump process. This bound is controlled by the mean residual time,
i.e. the expected wait before the first jump is observed. As a consequence, we
obtain a universal bound on the signal-to-noise ratio of arbitrary currents and
counting observables in the steady state. This bound is similar in spirit to
the kinetic uncertainty relation but provably tighter, and we explicitly
construct the counting observables that saturate it. Our results establish
ultimate precision limits for an important class of observables in
non-equilibrium systems, and demonstrate that the mean residual time, not the
dynamical activity, is the measure of freneticity that tightly constrains
fluctuations far from equilibrium.
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