Dissipation, lag, and drift in driven fluctuating systems

This work deals with thermostated fluctuating systems subjected to driven transformations of the internal energetics. The main focus is on generally multidimensional systems with continuous configurational degrees of freedom over which overdamped Markovian fluctuations take place (diffusive regime of the motion). Mutual bounds are established between the average energy dissipation, the deviation between nonequilibrium probability density and underlying equilibrium distribution due to the system's lag, and the statistical properties of the components of the directed flow induced by the transformation itself. The directed flow is here expressed in terms of time-dependent “drift velocity” associated with the probability current in a advection-like formulation of the nonstationary Fokker-Planck equation. Consideration of the drift makes that the bounds achieved here extend the inequality derived by Vaikuntanathan and Jarzynski [Europhys. Lett. 87, 60005 (2009)] involving only dissipation and lag. The key relations are then specified for the so-called stochastic pumps, i.e., systems that reach a periodic steady state in response of cyclic transformations and that are prototypes of nonautonomous dissipative converters of input energy into directed motion; a one-dimensional case model is adopted to illustrate the main features. Complementary results concerning bounds between the evolution rates of dissipation and lag, valid for both overdamped and underdamped dynamics, are also presented.

Figure 1

Energetics of the one-dimensional case model subjected to cyclic energy perturbation. (a) Contour plot of the system's energy $V_t\left(q\right)$ as function of coordinate $q$ and time $t$ over one period of the perturbation (the quantities are scaled as indicated); (b) contour plot of the underlying equilibrium distribution $p_{\mathrm{eq},t}\left(q\right)$; (c) contour plot of the nonequilibrium distribution at the periodic steady state for $\tau =1$.

Figure 2

Trajectories in the plane of the variables $\beta \mathrm{\Delta }A\left(t\right)$ and $\mathcal{D}\left(t\right)$ for the one-dimensional case model under the cyclic energy perturbation with $\tau =1$ starting from thermal equilibrium (thin black line). The medium-thickness blue line corresponds to the first cycle. The thicker red line corresponds to a cycle when the periodic steady state is considered to be reached (see the text for details).

Figure 3

Profiles of the probability current (a) and of the related drift velocity (b), versus the coordinate $q$, for the one-dimensional case model at the periodic steady state under cyclic energy perturbation with $\tau =1$. The panels show profiles at various advancements over one cycle.

Figure 4

$\tau$ dependence, on the logarithmic scale, of the steady state probability current integrated over one cycle of perturbation for the one-dimensional case model (filled circles). The dashed line is the fit with a Gaussian function.

Figure 5

$\tau$ dependence of the average energy dissipation per cycle, $\beta {\overline{w}}_{\mathrm{diss}}^{\infty }$, and of the provided upper and lower bounds, for the one-dimensional case model at the periodic steady state under cyclic perturbation. The inset shows the same profiles on a wider range.