Fluctuating systems under cyclic perturbations: Relation between energy dissipation and intrinsic relaxation processes

This study focuses on fluctuating classical systems in contact with a thermal bath, and whose configurational energetics undergoes cyclic transformations due to interaction with external perturbing agents. Under the assumptions that the configurational dynamics is a stochastic Markov process in the overdamped regime and that the nonequilibrium configurational distribution remains close to the underlying equilibrium one, we derived an analytic approximation of the average dissipated energy per cycle in the asymptotic limit (i.e., after many cycles of perturbation). The energy dissipation is then readily translated into average entropy production, per cycle, in the environment. The accuracy of the approximation was tested by comparing the outcomes with the exact values obtained by stochastic simulations of a model case: a “particle on a ring” that fluctuates in a bistable potential perturbed in two different ways. As pointed out in previous studies on the stochastic resonance phenomenon, the dependence of the average dissipation on the perturbation period may unveil the inner spectrum of the system's fluctuation rates. In this respect, the analytical approximation presented here makes it possible to unveil the connection between average dissipation, intrinsic rates and modes of fluctuation of the system at the unperturbed equilibrium, and features of the perturbation itself (namely, the period of the cycle and the projections of the energy perturbation over the system's modes). The possibilities of employing the analytical results as a guide to devising and rationalizing a sort of “spectroscopic calorimetry” experiment, and of employing them in strategies aiming to optimize the system's features on the basis of a target average dissipation, are briefly discussed.

Figure 1

(a) A pictorial representation of the fluctuating system initially at equilibrium and then cyclically perturbed by an external agent. (b) The interrelations investigated in the present study.

Figure 2

Energetics for the unidimensional case model. (a) Energy profile of the unperturbed system; (b) contour plots showing the time modulation of the energy profile under perturbation in Cases 1 and 2.

Figure 3

Examples of Langevin trajectories for the unperturbed system (a) and for the perturbed system in Case 1 (b) and Case 2 (c) with period $\tau =5$.

Figure 4

Dependence of the average dissipated energy per cycle, ${\overline{w}}_{\mathrm{diss},n_c}$, vs the number of performed cycles for the unidimensional system in Cases 1 and 2. The profiles refer to the period $\tau =5$.

Figure 5

Dependence of the asymptotic average energy dissipation per cycle, ${\overline{w}}_{\mathrm{diss}}^{\infty }$, vs the frequency $\omega$ for the unidimensional system in Cases 1 and 2. Solid lines refer to the approximate solution Eq. (21), while circles refer to the exact values from the numerical solution of the nonstationary Fokker-Planck-Smoluchowski equation. The bars on the top axes correspond to the first 30 eigenvalues of the Smoluchowski operator for the unperturbed system. For Case 2, the dashed line in the low-frequency range indicates the linear growth on $\omega$.

Figure 6

Distribution functions of the dissipated energy (work) per cycle, in the asymptotic limit, for the unidimensional system in Cases 1 and 2. The panels refer to different frequencies $\omega$, as indicated. The circles display the distributions constructed by histograms from a long Langevin simulation. The solid lines are the Gaussian distributions parametrized by ${\overline{w}}_{\mathrm{diss}}^{\infty }$ obtained from the approximate expression in Eq. (21) (see the text for details).

Figure 7

Same as for Fig. 6, but here for Case 2.

Figure 8

Plot of $lnp\left(w_{\mathrm{diss}}^{\infty }\right)/p(-w_{\mathrm{diss}}^{\infty })$ vs $w_{\mathrm{diss}}^{\infty }$ for the unidimensional system in Case 1. The circles correspond to points obtained from the distributions constructed by histograms (see Fig. 6). The dashed straight lines have a unitary slope.

Figure 9

Plot of $lnp\left(w_{\mathrm{diss}}^{\infty }\right)/p(-w_{\mathrm{diss}}^{\infty })$ vs. $w_{\mathrm{diss}}^{\infty }$ for the uni-dimensional system in Case 2. The circles correspond to points obtained from the distributions constructed by histograms (see Fig. 7). The dashed straight lines have a unitary slope.