# Uncertainty relations for time-delayed Langevin systems

## TL;DR

This paper extends the thermodynamic uncertainty relation to non-Markovian, time-delayed Langevin systems, establishing bounds on fluctuations based on path distribution divergences and symmetry considerations.

## Contribution

It introduces a generalized thermodynamic uncertainty relation for time-delayed Langevin systems, applicable to non-Markovian dynamics without detailed underlying models.

## Key findings

- Fluctuations are constrained by the Kullback-Leibler divergence between forward and reversed path distributions.
- Bounds are derived for observables antisymmetric under time reversal, involving a generalized entropy production.
- Numerical verification confirms the validity of the derived uncertainty relations in various time-delay systems.

## Abstract

The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback--Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmetric under time reversal, the fluctuation is bounded from below by a function of a quantity that can be identified as a generalization of the total entropy production in Markovian systems. We also provide a lower bound for arbitrary observables that are odd under position reversal. The term in this bound reflects the extent to which the position symmetry has been broken in the system and can be positive even in equilibrium. Our results hold for finite observation times and a large class of time-delayed systems because detailed underlying dynamics are not required for the derivation. We numerically verify the derived uncertainty relations using two single time-delay systems and one distributed time-delay system.