Thermodynamic uncertainty relation for time-delayed Langevin systems

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. By introducing generalized total dissipation equal to the total entropy production in the absence of time delay, we find that an analogous uncertainty relation exists for such systems. Specifically, we prove that the fluctuation of arbitrary currents at a steady state is constrained by the generalized total dissipation for a finite observation time. The rate of this generalized quantity is always positive and bounded from below by the square of arbitrary currents in the system. This uncertainty relation can be considered generalized one, which reduces to a conventional inequality for non-delayed Langevin systems. We verify the derived uncertainty relation numerically and demonstrate its validity in the Brusselator system, periodic Brownian motion, and a simple two-dimensional model.