Linear stochastic system with delay: Energy balance and entropy production

We study the energy balance in a linear stochastic dynamics with delay under the impact of an external periodic force. The linearity of the model, in combination with a response function method, enables us to perform detailed analytic calculations of each term in the energy balance equation. From this, we discuss thermodynamics and entropy production rate sigma. With use of the delay time tau and strength of the external force A(0), sigma is simply expressed as sigma=sigma(D,1)(tau)+A(0)(2)eta(tau), with both sigma(D,1)(tau) and eta(tau) positive definite. We thus conclude that even when there is no external force (A(0)=0), the entropy production rate sigma=sigma(D,1)(tau) is positive, meaning that the delay force produces work, which is dissipated into a reservoir. Numerical experiments are performed to confirm theoretical results.