Multidimensional entropic bound: Estimator of entropy production for Langevin dynamics with an arbitrary time-dependent protocol

Entropy production (EP) is a key quantity in thermodynamics, and yet measuring EP has remained a challenging task. Here we introduce an EP estimator, called multidimensional entropic bound (MEB), utilizing an ensemble of trajectories. The MEB can accurately estimate the EP of overdamped Langevin systems with an arbitrary time-dependent protocol. Moreover, it provides a unified platform to accurately estimate the EP of underdamped Langevin systems under certain conditions. In addition, the MEB is computationally efficient because optimization is unnecessary. We apply our developed estimator to three physical systems driven by time-dependent protocols pertaining to experiments using optical tweezers: A dragged Brownian particle, a pulling process of a harmonic chain, and an unfolding process of an RNA hairpin. Numerical simulations confirm the validity and efficiency of our method.

Figure 1

Plot for the rate EP estimator $\tilde{\sigma }$ normalized with respect to the total EP rate ${\sigma }^{\mathrm{tot}}\left(t\right)$ as a function of time $t$. The green solid and red dotted line denote the MEB results of the overdamped and underdamped dynamics, respectively. The orange dashed line represents the result of the modified TUR by Koyuk and Seifert, ${\sigma }^{\mathrm{KS}}\left(t\right)/{\sigma }^{\mathrm{tot}}\left(t\right)$. The parameters used for this plot are $k=\mu =\omega =T=1$. The inset shows a schematic of the Brownian particle dragged by optical tweezers. Note that these plots are obtained from the analytic expressions.

Figure 2

(a) Schematic of the harmonic chain pulled by optical tweezers. (b) Estimated EP via the MEB method (black) and the NEEP method (green) as a function of time $t$. The inset shows the EP of the $i\mathrm{th}$ bead. $◯, \times , +, \square$, and $\diamond$ represent the estimated EPs for $x_1, x_2, x_3, x_4$, and $x_5$ beads, respectively. Solid lines denote the analytic results. Four Gaussian weight vectors are used for the MEB estimation. (c) Plot of $\mathrm{\Delta }{S}^{\mathrm{tot}}$ at $t=1.0$ against the number of weight vectors $\ell$. The green dashed line represents the NEEP result, while the red solid lines in (b) and (c) denote the analytic results. The parameters used for these plots are $k=5, \mu =1, \omega =5$, and $T=1$.

Figure 3

EP estimation results for the RNA unfolding process. (a) Histogram of the distance $x$ between the two ends of P5GA at initial time $0\text{ns}$ (light blue) and final time $7.19\text{ns}$ (orange). The inset shows a schematic of the RNA pulled by optical tweezers. (b) EP estimated via MEB with four Gaussian weight vectors (black) and NEEP (green) as a function of time $t$. The red solid line denotes the results obtained from Eq. (52). (c) Estimated EP via MEB at $t=7.19\text{ns}$ as a function of the number of weight vectors $\ell$ (black). The green dashed line and the red solid line denote the results of the NEEP and the EP obtained from Eq. (52), respectively.

Figure 4

Estimated EP of the RNA unfolding process as a function of the number of trajectory samples. Open circles represent the MEB results with four Gaussian bases ($\ell =4$). Green filled circles denote the NEEP results. The red solid line is the estimated EP via the direct method, which is the same line as in Fig. 3. The other parameters are the same as those used to plot Fig. 3. The error bars represent the standard deviation of the EP estimations from five different trajectory sets.

Figure 5

Estimated EP as a function of time resolution $dt$. Open circles represent the MEB results with four Gaussian bases ($\ell =4$). Green circles and red crosses denote the results of the NEEP and the direct method, respectively. The other parameters are the same as those used to plot Fig. 3.