Capturing the Landauer bound through the application of a detailed Jarzynski equality for entropic memory erasure

The states of an overdamped Brownian particle confined in a two-dimensional bilobal enclosure are considered to correspond to two binary values: 0 (left lobe) and 1 (right lobe). An ensemble of such particles represents bits of entropic information. An external bias is applied on the particles, equally distributed in two lobes, to drive them to a particular lobe erasing one kind of bit of information. It has been shown that the average work done for the entropic memory erasure process approaches the Landauer bound for a very slow erasure cycle. Furthermore, the detailed Jarzynski equality holds to a very good extent for the erasure protocol, so that the Landauer bound may be calculated irrespective of the time period of the erasure cycle in terms of the effective free-energy change for the process. The detailed Jarzynski equality applied to two subprocesses, namely the transition from entropic memory state 0 to state 1 and the transition from entropic memory state 1 to state 1, connects the work done on the system to the probability to occupy the two states under a time-reversed process. In the entire treatment, the work appears as a boundary effect of the physical confinement of the system not having a conventional potential energy barrier. Finally, an analytical derivation of the detailed and classical Jarzynski equality for Brownian movement in confined space with varying width has been proposed. Our analytical scheme supports the numerical simulations presented in this paper.

Figure 1

(a) The bilobal enclosure with its geometric parameters, (b) time series plot of the external bias $B\left(t\right)$ with the parameter set $B_{\text{max}}=1.5,t_{\text{therm}}=400,{\tau }_1=1425$, and ${\tau }_2=75$.

Figure 2

Plot of $A_{01},A_{11}$, and $A_{01}+A_{11}$ against $\tau$ (a) for three different temperatures: $D=0.08 \left(B_{\text{max}}^{\text{opt}}=0.9\right),D=0.1 \left(B_{\text{max}}^{\text{opt}}=1.2\right)$, and $D=0.12 \left(B_{\text{max}}^{\text{opt}}=1.4\right)$; and (b) for three different amplitudes of the external bias: $B_{\text{max}}=$ 1.0, 1.2, and 1.4 at $D=0.1$. The parameter set used is $a=0.25,b=0.5,c=0.02$ and $t_{\text{therm}}=400,{\tau }_1:{\tau }_2=19:1$, and $G=0.0001$. The half-filled symbols correspond to $A_{01}+A_{11}$, open symbols to $A_{01}$, and completely filled symbols to $A_{11}$.

Figure 3

Plot of $\langle W\rangle$ and $\Delta F_{\text{effective}}$ against $\tau$ (a) for three different temperatures: $D=0.08 \left(B_{\text{max}}^{\text{opt}}=0.9\right),D=0.1 \left(B_{\text{max}}^{\text{opt}}=1.2\right)$, and $D=0.12 \left(B_{\text{max}}^{\text{opt}}=1.4\right)$ (the large symbols denote $\langle W\rangle$, the similar small symbols represent corresponding $\Delta F_{\text{effective}}$, and the dotted lines present the respective Landauer bounds) and (b) for three different forcing amplitudes: $B_{\text{max}}=1.5$ (over forced), $B_{\text{max}}=1.2$ (optimized), and $B_{\text{max}}=1.0$ (under forced) at temperature $D=0.1$. The fitting with the function $k_{B}T\text{ln}2+\frac{C_1}{\tau }$ has been shown, where $C_1=90$. The parameter set used is $a=0.25,b=0.5,c=0.02$ and $t_{\text{therm}}=400,{\tau }_1:{\tau }_2=19:1$, and $G=0.0001$.

Figure 4

Time series plot of (a) $B\left(t\right)$ with $B_{\text{max}}=1.2$ and (b) $x\left(t\right)$ for 100 trajectories up to the completion of the erasure process. The parameter set used is $a=0.25,b=0.5,c=0.02$ and $t_{\text{therm}}=400,{\tau }_1=1900,{\tau }_2=100$, and $G=0.0001$.

Figure 5

Time-series plot of (a) $B\left(t\right)$ with $B_{\text{max}}=1.2$ and (b) $x\left(t\right)$ for 100 trajectories tending to reach equilibrium. The parameter set used is $a=0.25,b=0.5,c=0.02$ and $t_{\text{therm}}=400,{\tau }_1=1900,{\tau }_2=100$, and $G=0.0001$.

Figure 6

Variation of $A_{01},A_{11},{\overline{P}}_{10}$, and ${\overline{P}}_{11}$ with $\tau$. The parameter set used is $a=0.25,b=0.5,c=0.02$ and $D=0.1,B_{\text{max}}=1.2,t_{\text{therm}}=400,{\tau }_1=1900,{\tau }_2=100$, and $G=0.0001$.