Lower bounds on dissipation upon coarse graining

By different coarse-graining procedures, we derive lower bounds on the total mean work dissipated in Brownian systems driven out of equilibrium. With several analytically solvable examples, we illustrate how, when, and where the information on the dissipation is captured.

Figure 1

Sketch of the $n$-slicing procedure in which the full trajectory of the particle is not measured but only its position after time intervals $\Delta t=t_f∕n$, where $t_f$ is the total duration of the experiment.

Figure 2

(a) Plot of $I_n$ (for $n=1,2,3,4$ and $I_{\infty }=⟨W_{\mathrm{diss}}⟩$) as a function of the ratio of characteristic times $k{t}_f$. We have scaled out the prefactor $k{z}_0^2$. Note that $I_n$ is always a lower bound to $⟨W_{\mathrm{diss}}⟩$ and converges to the irreversible instantaneous quench value (dashed line) and to the quasistatic limit (zero value) for $k{t}_f\to 0$ and $k{t}_f\to \infty$, respectively. Inset: the relative error $E_n\equiv (⟨W_{\mathrm{diss}}⟩-I_n)∕⟨W_{\mathrm{diss}}⟩$ increases as a function of $k{t}_f$. (b) Plot of $I_n$ (for different values of $k{t}_f$) as a function of the number of time divisions $n$ of the trajectory.

Figure 3

Schematic representation of the quenching experiment. The only information collected takes place at time $t$ after the quench. Notice that at any time $t$, in the backward process the system is in equilibrium at ${\omega }_1$, $\tilde{\rho }\left(t\right)={\rho }^{\mathrm{eq}}\left({\omega }_1\right)$, whereas in the forward process the system is relaxing precisely toward this equilibrium state.

Figure 4

(a) Relative entropies $D_{x,p}$, $D_x$, and $D_p$ measured at one single time $t$ after the quench. Strongly damped regime $(\sigma <1)$: friction dominates inertia. $D_{x,p}$ and $D_x$ decay monotonically and almost coincide, while $D_p$ is very small and goes through a rise-and-fall. (b) Underdamped regime $(\sigma >1)$: inertia dominates friction, resulting in an out-of-phase oscillatory decay of the relative entropies $D_x$ and $D_p$. Note that in both cases (a) and (b), the position variable $x$ captures at $t=0$ the full information on dissipation, namely $⟨W_{\mathrm{diss}}⟩$.

Figure 5

Scheme of the last study case: the information on dissipation due to a sudden quench in the stiffness of the potential from ${\omega }_0$ to ${\omega }_1$ in one subsystem is transmitted, through a linear coupling of strength $K$, to a second subsystem immersed in a heat bath at temperature $T$, where such information will get irreversibly lost.

Figure 6

(a) Complex behavior of the relative entropies as a function of time for oscillator one. See text for discussion. (b) Same picture for oscillator two.

Figure 7

Time evolution of the relative entropies of the whole system, $D_{x_1,p_1,x_2,p_2}\left(t\right)$, oscillator one, $D_{x_1,p_1}\left(t\right)$, and oscillator two, $D_{x_2,p_2}\left(t\right)$. In each plot, a different coupling constant $K$ between the oscillators is used.

Figure 8

Time evolution of several relative entropies illustrating the effect on correlations between positions and momenta.