Thermodynamic uncertainty relation for time-delayed Langevin systems

Understand

The thermodynamic uncertainty relation, which establishes a universal trade-off between the relative fluctuation of arbitrary currents and the dissipation, has been found for various Markovian systems.

• However, this relation has not been revealed for non-Markovian systems.
• Thus, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems.
• First, we demonstrate numerically that the conventional uncertainty relation does not hold in certain delayed systems, which suggests that the lower bound should be refined to sustain the validity of the inequality.