Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Abstract

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the nonMarkovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The new relation includes a contribution arising from the acausal character of the reverse process. This in turn leads to another second-law-like inequality. We illustrate the general formalism by a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Heat fluctuations for underdamped Langevin dynamics

“…Fig. 2 shows the influence of the delay τ on µ(λ) for a feedback-cooled resonator operating in its second stability lobe (see [36,38] for details). The quality factor Q 0 corresponds to the AFM micro-cantilever used in the experiments of [39].…”

“…As a second example, we consider a non-Markovian dynamics governed by the time-delayed Langevin equation [35,36] …”

Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations

“…We can then compute the integral over ω by using Cauchy's residue theorem, which requires to locate the poles of χ(ω) in the complex frequency plane (they are not restricted to be in the lower half plane, in contrast with the poles of the causal response function χ(ω)). Fortunately, this nontrivial task has already been accomplished in I in order to calculate the quantitẏ S J ≡ lim t→∞ (1/t) ln J t / J t involved in the second-lawlike inequality (26) obtained from time reversal (we recall that the Jacobian J [X] becomes a path-independent quantity J t when the dynamics is linear [6]). Specifically, it was shown in I [Eq.…”

“…One reason is the non-Markovian nature of the dynamics which makes the theoretical description more challenging (for instance, one cannot resort to a spectral approach using Fokker-Planck operators). This is not an impossible task, though, and in a previous work [6], hereafter referred to as I, we have initiated a theoretical study of an underdamped Langevin equation that models the motion of a nanomechanical resonator in contact with a thermal reservoir and subjected to a time-delayed, position-dependent force. The role of the control force is to damp thermal fluctuations * Electronic address: mlr@lptmc.jussieu.fr and to maintain the resonator in a nonequilibrium steady state (NESS) where its average (configurational or kinetic) temperature is much smaller than the temperature of the environment.…”

Two coupled, driven Ising spin systems working as an engine

“…Many studies have also looked at full distributions of efficiency [13,14] and also the large deviation functions [20]. Models with feedback control both instantaneous and delayed have also been investigated [21][22][23][24]. Most of the earlier studies both theoretical and experimental were based on applying the thermodynamic engine protocols like Carnot or Stirling to a colloidal particle placed…”