Excess and loss of entropy production for different levels of coarse graining

We investigate the effect of coarse graining on the thermodynamic properties of a system, focusing on entropy production. As a case of study, we consider a one-dimensional colloidal particle in contact with a thermal bath, moving in a sinusoidal potential and driven out of equilibrium by a small constant force. Different levels of coarse graining are evaluated: At first, we compare the results in the underdamped dynamics with those in the overdamped one (first coarse graining). For large values of the friction coefficient, the two dynamics have the same thermodynamics properties, while, for smaller friction values, the overdamped approximation produces an excess of entropy production with respect to that of the underdamped dynamics. Moreover, for further smaller values of the drag coefficient, the excess of entropy production turns into a loss. These regimes are explained by evaluating the jump statistics, observing that the inertia is able to induce multiple jumps and affect the average jump rate. The periodic shape of the potential allows us to approximate the continuous dynamics via a Markov chain after the introduction of a suitable time and space discretization (second level of coarse graining). This discretization procedure is implemented starting both from the underdamped and the overdamped evolution and is analyzed for different values of the friction coefficient.

Figure 1

Periodic potential. In panel (a), a three-dimensional representation of the periodic potential $U\left(x\right)$ is shown while, in panel (b), the whole potential profile $U\left(x\right)-Fx$ is reported, where $U\left(x\right)=Acos(2\pi x/L)$. Vertical lines in panel (b) draw the heights of the potential barriers for forward and backward jumps, ${\mathrm{\Delta }}^{+}$ and ${\mathrm{\Delta }}^{-}$, respectively. The parameters of the potential are $A=4, F=1$, and $L=2\pi$.

Figure 2

Single-particle trajectories. Panels (a), (b), and (c) show the time-trajectory of the position $x\left(t\right)$ for three different values of $\gamma =10,3\times {10}^{-1},3\times {10}^{-2}$, respectively. In each panels, the underdamped [simulating Eq. (1)] and the overdamped cases [simulating Eq. (3)] are shown, while the two cases are separately plotted in two different graphs, namely (c.1) and (c.2), for $\gamma =3\times {10}^{-2}$ because of presentation reasons. The other parameters of the numerical study are $L=2\pi , F=1, A=4, T=1$, and $m=1$.

Figure 3

Coarse graining (i): underdamped $\to$ overdamped. Panel (a): Entropy productions, $\dot{\sigma }$ (red upper triangles) and ${\dot{\sigma }}_o$ (yellow lower triangles), as a function of the friction coefficient $\gamma$. $\dot{\sigma }$ is calculated numerically by using Eq. (2) while ${\dot{\sigma }}_o$ by Eq. (5), corresponding to the underdamped and the overdamped cases, respectively. Solid colored lines are obtained by plotting $2\pi /\mathcal{T}$ and $2\pi /{\mathcal{T}}_o$ for the underdamped and the overdamped cases. Finally, the prediction (8) is reported as a blue dashed line, while the curve $FT/\gamma$ as a dashed-dotted line. Panel (b) plots the average jump time from, ${\tau }_{+}$ (underdamped) and ${\tau }_o^{+}$ (overdamped) as a function of $\gamma$, while panel (c) plots the average length in units of $L$ run in each jump process, namely $\ell$ (underdamped) and ${\ell }_o$ (overdamped). The other parameters of the simulations are $L=2\pi , F=1, A=4, T=1$, and $m=1$.

Figure 4

Coarse graining (ii): continuous $\to$ discrete. Entropy production defined by Eq. (10) as a function of the Markov chain time step $\mathrm{\Delta }t$. Panels (a) and (b) show ${\dot{\sigma }}^M$ (underdamped) and ${\dot{\sigma }}_o^M$ (overdamped) where each curve is obtained for different values of $\gamma$, namely the friction coefficient. We remark that the larger value of the friction, $\gamma =10$, belongs to regime (i) while the other two values belong to regime (ii). The Markov chain (or the master equation) transition rate probabilities (to calculate ${\dot{\sigma }}^M$ and ${\dot{\sigma }}_o^M$) are obtained from the underdamped [Eq. (2)] and overdamped [Eq. (5)] continuous dynamics in panels (a) and (b), respectively. Finally, the dashed-dotted lines mark the value of $\dot{\sigma }$ and ${\dot{\sigma }}_o$ in each case, maintaining the same color legend. The first $\mathrm{\Delta }t$ value of each panel corresponds to the master equation approximation, so that $\mathrm{\Delta }t$ coincides with the time step of the continous-time simulation. The other paramters of the simulations are $L=2\pi , F=1, A=4, T=1$, and $m=1$.