Optimal finite-time erasure of a classical bit

Information erasure inevitably leads to the generation of heat. Minimizing this dissipation will be crucial for developing small-scale information processing systems, but little is known about the optimal procedures required. We have obtained closed-form expressions for maximally efficient erasure cycles for deletion of a classical bit of information stored by the position of a particle diffusing in a double-well potential. We find that the extra heat generated beyond the Landauer bound is proportional to the square of the Hellinger distance between the initial and final states divided by the cycle duration, which quantifies how far out of equilibrium the system is driven. Finally, we demonstrate close agreement between the exact optimal cycle and the protocol found using a linear response framework.

Figure 1

Double-well potential for storage of a single classical bit. (a) and (b) The system begins in thermal equilibrium with equipotential wells and a potential barrier of height much larger than the thermal fluctuation scale. Observing the particle (black dot) to the left (right) of the potential barrier corresponds to memory value 1 (0). The width $2w$ of the central barrier and the width $l$ of each well satisfy $2w/l\ll 1$. (c) Optimally efficient erasure protocols are sought in which the tilt $V_l$ (orange, left) and the barrier height $V_b$ (blue, center) are control parameters. After the erasure step, the particle is much more likely to be in the right well, regardless of where it originated.

Figure 2

Optimally efficient finite-time erasure cycles. (a) The optimal cycle consists of two parts: the erasure stage and the reset stage. The erasure stage begins at $(ln\{w/l\gamma \},0)$ (red star) and then jumps to the initial point of the erasure protocol (brown square). The erasure protocol (blue curve) proceeds for time $t_f$ until reaching its terminus (green triangle). The reset stage consists of the jump from this terminus to the parameter values defining the original equilibrium state (red star). Blue dots indicate points separated by equal times along the erasure stage. For these parameters [$\overline{t_f}\equiv (2D/l^2)t_f=50$, $\delta =0.01$, $\gamma =exp(-10)$, and $w/l=0.01$] the efficiency of the optimal cycle is $94.01\%$. (b) An approximate optimal efficiency erasure cycle determined by the inverse diffusion tensor framework is nearly identical to the exact solution shown in (a).

Figure 3

The Landauer bound and the added dissipation necessary for finite-time cycles exhibit different dependences on the spatial distribution of the particle. (a) If the particle is initially distributed equally between the two wells, $p_l\left(0\right)=0.5$, where $p_l\left(0\right)$ is the probability of being found in the left well at $t=0$, then the Landauer bound, proportional to $-\Delta S$ (dashed blue curve), is zero for no change in the likelihood of finding the particle on the left [$p_l\left(t_f\right)=0.5$] and positive for any other final distribution. This is always true for the second term in the full dissipation (solid red curve), which is proportional to $K$ [Eq. (79)]. In all panels, $p_b\left(0\right)=p_b\left(t_f\right)=0$, where $p_b\left(t\right)$ is the probability of finding the particle at the central barrier at time $t$. (b) and (c) For other initial conditions, the Landauer bound can be positive, zero, or negative depending on how the particle's spatial distribution changes.