Finite-Time Landauer Principle

We study the thermodynamic cost associated with the erasure of one bit of information over a finite amount of time. We present a general framework for minimizing the average work required when full control of a system’s microstates is possible. In addition to exact numerical results, we find simple bounds proportional to the variance of the microscopic distribution associated with the state of the bit. In the short-time limit, we get a closed expression for the minimum average amount of work needed to erase a bit. The average work associated with the optimal protocol can be up to a factor of 4 smaller relative to protocols constrained to end in local equilibrium. Assessing prior experimental and numerical results based on heuristic protocols, we find that our bounds often dissipate an order of magnitude less energy.

Figure 1

Optimal protocols in the short- and long-time limits. Starting from the equilibrium distribution $p_0(x)$ for a symmetric potential $V_0(x)$ at time $t=0$ (left distribution), the system state is transformed to $p_{\tau }(x)$ at time $\tau$. For $\tau \ll 1$ (upper right distribution), the final distribution is approximately the sum of the initial probability density of the left well plus a sharp peak composed of probability transported from the right well. For $\tau \gg 1$ (lower right distribution), the final distribution is in local equilibrium in the left well.

Figure 2

Erasure protocol for $\tau =0.2$ and $E_b=4$. The original and final double-well potentials $V_0(x)$ are shown in black at the top and bottom of the left column. Red potentials $V(x,t)$ denote the control. The control is carried out for $0^{+}<t/\tau <1^{-}$, and the potential changes discontinuously at $t=0$ and $t/\tau =1$. At $t/\tau =1$, the probability distribution is peaked near $x=0$. Therefore, an infinitesimally narrow, extra barrier $\delta (0)$ is added to $V_0(x)$ to prevent probability from leaking back into the right well. It is removed at a later time $t/\tau \gg 1$, after the system has relaxed to local equilibrium and the peak in the probability distribution has disappeared. The right-hand column shows corresponding probability distributions.

Figure 3

Minimum entropy production in excess of the Landauer bound. The shaded regions show upper ($2{\tau }^{-1}$) and lower (${\tau }^{-1}/2$) bounds. The inset shows $a\equiv \tau (W_{\min }/k_{B}T-\ln 2)$ relative to its lower bound. Red curves are plotted for $E_b=\{0,2,4,6,8\}$. Heavier lines denote the $E_b=\{0,8\}$ cases.