Scientists have discovered a set of mathematical equations which describe how any sequence of random events, no matter the timescale or regularity, can be used to estimate time.
The findings are useful for individuals stranded on desert islands without a watch and researchers trying to uncover the limits of nanoscale clocks in quantum, electronic and biomolecular systems.
“Our goal was to find out the minimum ingredients you need to build a clock,” says Dr Mark Mitchison from the Department of Physics at King’s College London, UK.
“For example, could you still measure time precisely even when stranded on a desert island? We found equations that tell you how to create a ‘clock’ by counting random events around you, like waves lapping on the shore or your heartbeats.”
(a) Successful cooking (among other important tasks) requires an accurate and precise time estimator to be constructed from an observer’s record of events. Stochastic (random) phenomena can be used for this purpose but have varying degrees of frequency, regularity, and interdependency. (b) The study focuses on Markovian jump processes described by a master equation on a discrete network of 𝑑 states, where 𝑝𝜎 is the probability to find the system in state 𝜎, 𝑅𝜇𝜎 is the rate for a jump 𝜎 →𝜇, and Γ𝜎 is the escape rate from 𝜎. An observer lacking a clock may record the stochastic sequence of states visited by the system, 𝝈 =𝜎0 →𝜎1 →⋯. An asymptotically optimal time estimator is the sum of mean dwell times Γ−1
𝜎 for each state observed in the sequence. Credit: Prech et al 2025, Physical Review X (CC BY 4.0)
Lots of physical and biological processes progress via random ‘jumps’ which happen at irregular intervals.
If each jump only depends on the previous one, then the sequence of events can be described by a general mathematical model called a ‘classical Markovian jump process’.
The newly discovered equations show that the timekeeping accuracy of these Markovian systems is limited by the ‘mean residual time’ or average time until the first observed event.
In other words, the less time before a system makes its first jump, the faster and more frequently it undergoes jumps in general. This means it fluctuates less and can be better used to estimate how much time has passed.
Illustration of the residual and waiting times, with jumps depicted as blue crosses on the time axis. The mean residual time 𝒯 is the expected value of 𝜏0, the time interval until the first jump after observations begin at 𝑡0. Paradoxically, this exceeds the mean (average) of the waiting time between jumps 𝜏𝑗 because the jumps tend to cluster in bunches and the system spends more time in long-lived states. Credit: Prech et al 2025, Physical Review X (CC BY 4.0)
“This turns out to be the best possible clock you can build by counting Markovian events in a system governed by classical physics,” says Mitchison.
“So, if you find a system that doesn’t follow the expected pattern, you can be sure something else is going on.”
These hidden features could include unseen states, memory effects or even quantum behaviour. The researchers say their equations open new paths for detecting these complex dynamics from limited data.
They also hope the work can be used to study how biological systems, like molecular motors and oscillators, operate efficiently in the presence of random fluctuations.
“Thinking about molecular machines as ‘clocks’ gives us insight into how some natural processes spontaneously generate order from chaos,” says Mitchison.
“We see this occurring at many different scales in our universe, from biological organisms and ecosystems down to the microscopic world. By establishing a fundamental limit on how well clocks can operate in the realm of classical physics, we also gain a better understanding of what makes quantum clocks different.
“Time lies at the heart of many unsolved mysteries in quantum physics. Why does time seem to flow in only one direction? Why do we only remember the past and not the future? Is time quantised in discrete chunks, in the same way as energy? By thinking about what clocks can do, we ultimately hope to answer some of these questions about the nature of time itself.”
The equations are presented in a paper published in the journal Physical Review X.