Universal Bound on Energy Cost of Bit Reset in Finite Time

We consider how the energy cost of bit reset scales with the time duration of the protocol. Bit reset necessarily takes place in finite time, where there is an extra penalty on top of the quasistatic work cost derived by Landauer. This extra energy is dissipated as heat in the computer, inducing a fundamental limit on the speed of irreversible computers. We formulate a hardware-independent expression for this limit in the framework of stochastic processes. We derive a closed-form lower bound on the work penalty as a function of the time taken for the protocol and bit reset error. It holds for discrete as well as continuous systems, assuming only that the master equation respects detailed balance.

Figure 1

Our analysis applies to a wide range of bit-reset scenarios, including: (a) a double-well potential $V(x,t)$ which is gradually turned into a single well, (b) a system with two energy levels where one level is lifted gradually. (c) During the finite-time bit reset, the system state $p(t)$ lags behind the thermal state $\gamma (t)$, and we will show that two distances are closely related to the minimal work cost.

Figure 2

Time scaling of work penalty, relative entropy, and entropy production for the discrete level-shifting protocol. In the simulation, $N=100$, $\mu =0.1$, $\beta =1$, $E_{\max }=10$ (left), and $\epsilon =0.25$ (right).

Figure 3

Domination of $D_{\epsilon }(\tau )$ or $\mathrm{\Sigma }(\tau )$ for the work penalty. Region I: $D_{\epsilon }(\tau )>\mathrm{\Sigma }(\tau )$. Region II: $D_{\epsilon }(\tau )<\mathrm{\Sigma }(\tau )$. Region III: the protocol fails to reset the bit given the error and energy. Solid line: $D_{\epsilon }(\tau )=\mathrm{\Sigma }(\tau )$. Dash-dotted line: $\epsilon ={\gamma }_1(E_{\max })$. In the simulation, $N=100$, $\mu =0.1$, and $\beta =1$.