# Dispersion of the time spent in a state: General expression for unicyclic model and dissipation-less precision

## TL;DR

This paper derives an exact expression for the dispersion of the time spent in states within a unicyclic network and demonstrates a dissipation-less precision in a stochastic clock, challenging traditional thermodynamic limits.

## Contribution

It introduces a new random variable related to state residence time and provides an exact dispersion formula for unicyclic models, revealing dissipation-less precision possibilities.

## Key findings

- Exact dispersion expression for state residence time in unicyclic networks
- Demonstration of precision with arbitrarily low energy dissipation
- Contrast with thermodynamic uncertainty relation for current

## Abstract

We compare the relation between dispersion and dissipation for two random variables that can be used to characterize the precision of a Brownian clock. The first random variable is the current between states. In this case, a certain precision requires a minimal energetic cost determined by a known thermodynamic uncertainty relation. We introduce a second random variable that is a certain linear combination of two random variables, each of which is the time a stochastic trajectory spends in a state. Whereas the first moment of this random variable is equal to the average probability current, its dispersion is generally different from the dispersion associated with the current. Remarkably, for this second random variable a certain precision can be obtained with an arbitrarily low energy dissipation, in contrast to the thermodynamic uncertainty relation for the current. As a main technical achievement, we provide an exact expression for the dispersion related to the time that a stochastic trajectory spends in a cluster of states for a general unicyclic network.