# Fluctuation theorem uncertainty relation

## TL;DR

This paper introduces the fluctuation theorem uncertainty relation, a new fundamental bound linking fluctuations and entropy production in nonequilibrium thermodynamics, applicable to various dynamics and observables.

## Contribution

It derives a general uncertainty relation from the fluctuation theorem valid for both stochastic and deterministic systems, expanding the theoretical framework of thermodynamic bounds.

## Key findings

- The relation applies to anti-symmetric observables in Langevin dynamics.
- It differs from existing relations for current-type observables in Markov chains.
- It incorporates systems with time-symmetric external protocols, relating bounds to work exerted.

## Abstract

The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty relation was discovered, which states that the fluctuation of observables is lower bounded by the entropy production. In the present Letter, we derive a thermodynamic uncertainty relation from the fluctuation theorem. We refer to the obtained relation as the fluctuation theorem uncertainty relation, and it is valid for arbitrary dynamics, stochastic as well as deterministic, and for arbitrary anti-symmetric observables for which a fluctuation theorem holds. We apply the fluctuation theorem uncertainty relation to an overdamped Langevin dynamics for an anti-symmetric observable. We demonstrate that the anti-symmetric observable satisfies the fluctuation theorem uncertainty relation, but does not satisfy the relation reported for current-type observables in continuous-time Markov chains. Moreover, we show that the fluctuation theorem uncertainty relation can handle systems controlled by time-symmetric external protocols, in which the lower bound is given by the work exerted on the systems.