Dynamics of Information Erasure and Extension of Landauer's Bound to Fast Processes

Dynamics of Information Erasure and Extension of Landauer’s Bound to Fast Processes

Salambô Dago and Ludovic Bellon

Phys. Rev. Lett. 128, 070604 – Published 17 February 2022

Abstract

Using a double-well potential as a physical memory, we study with experiments and numerical simulations the energy exchanges during erasure processes, and model quantitatively the cost of fast operation. Within the stochastic thermodynamics framework we find the origins of the overhead to Landauer’s bound required for fast operations: in the overdamped regime this term mainly comes from the dissipation, while in the underdamped regime it stems from the heating of the memory. Indeed, the system is thermalized with its environment at all times during quasistatic protocols, but for fast ones, the inefficient heat transfer to the thermostat is delayed with respect to the work influx, resulting in a transient temperature rise. The warming, quantitatively described by a comprehensive statistical physics description of the erasure process, is noticeable on both the kinetic and potential energy: they no longer comply with equipartition. The mean work and heat to erase the information therefore increase accordingly. They are both bounded by an effective Landauer’s limit $k_{B} T_{eff} \ln 2$ , where $T_{eff}$ is a weighted average of the actual temperature of the memory during the process.

Received 8 October 2021
Accepted 31 January 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.070604

Physics Subject Headings (PhySH)

Brownian motion Entropy production Fluctuations & noise Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics Thermodynamics of computation

Stochastic dynamical systems

Langevin equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Salambô Dago and Ludovic Bellon ^*

Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

^*Corresponding author. ludovic.bellon@ens-lyon.fr

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Issue

Vol. 128, Iss. 7 — 18 February 2022

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Images

Figure 1
Experimental setup and fast erasure cycle. (a) Schematic diagram of the experiment: the underdamped oscillator (resonance frequency $f_{0} = ω_{0} / (2 π) = 1270 Hz$ , quality factor $Q = 10$ ) is a conductive cantilever, sketched in yellow. Its deflection $x$ is measured with a differential interferometer [18]. The potential $U$ is created by the electrostatic force $F$ between the cantilever and the electrode facing it [10]. (b) Measured double-well potential energy $U_{1}$ (blue) when $x_{1} = \sqrt{10} σ_{0}$ ( $5 k_{B} T_{0}$ barrier height), with $T_{0} = 295 K$ and $σ_{0} = \sqrt{k_{B} T_{0} / k} \sim 1 nm$ , the standard deviation of the deflection at equilibrium inside a single well. The potential is inferred from the measured PDF of $x$ during a 10 s acquisition and the Boltzmann distribution. The fit to the $U_{1} (x, x_{1})$ expression is excellent (dashed line). (c) Time recording of the cantilever deflection $x$ on a single trajectory (blue, starting in state 1 in this example), superposed with the centers of two wells: $+ x_{1}$ (black) and $- x_{1}$ (red). Snippets of the potential energy on top of the plot sketch the erasure protocol. Stage 1 (red background) and 2 (green background) both last $τ = 5 ms$ . The equilibrium periods around stages 1 and 2 are chosen freely as long as they allow the cantilever to relax (natural relaxation time: $τ_{r} = 2 Q / ω_{0} \sim 2.5 ms$ ). The gray map corresponds to the PDF of $x$ , computed from $N = 2000$ experimental trajectories of the erasure process.
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Figure 2
Energy evolution during an erasure procedure. (a) In blue, the time evolution of the mean kinetic energy $⟨ K ⟩$ in $k_{B} T_{0}$ units over $N = 2000$ iterations of a quasistatic erasure ( $τ f_{0} ≫ 1$ ): stage 1 and stage 2 (red and green backgrounds) both last $τ = 100 ms$ . The red line corresponds to the theoretical prediction detailed in the Letter. (b) Same, with the mean potential energy $⟨ U ⟩$ . (c),(d) Same for a fast erasure: $τ = 5 ms$ . We add in black line the results of a numerical simulation for step 1 that provides more samples than the experiment $N_{sim} = 5 \times 10^{6}$ and is thus free of statistical uncertainty.
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Figure 3
Model prediction: energy and stochastic work profiles. (a) Time evolution of the mean kinetic energy $⟨ K ⟩$ for different duration $τ$ of the erasure steps computed from Eq. (7c). For small $τ$ , $⟨ K ⟩$ is affected during step 1 (red background) by a transient oscillation due to the dragging, followed by a strong rise in temperature. Only the dragging transient appears during step 2 (green background). (b) Same plot for the potential energy $⟨ U ⟩$ from Eq. (7d). (c) Time evolution of the mean power over 2000 trajectories, following the fast protocol ( $τ = 5 ms$ ) corresponding to Fig. 1 (blue). The red line is computed using Eq. (7b) and closely matches the experimental results. Results of a numerical simulation (black dashed line), corresponding to $5 \times 10^{6}$ trajectories, match the model so well that we cannot distinguish the curves.
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Figure 4
Divergence from the Landauer limit for fast erasures. Erasure cost ( $⟨ W ⟩$ and $⟨ Q ⟩$ in $k_{B} T_{0}$ units) for different operation speeds $X_{1} / (σ_{0} τ)$ . Experimental data (in blue), computed from $N = 2000$ iterations each with $X_{1} \sim 6 σ_{0}$ , are in good agreement with the analytical computation (red line). As a comparison, we plot the empirical description of the divergence from LB as $k_{B} T_{0} (\ln 2 + B / τ)$ used in the existing literature, with $B = (2.6 \pm 0.2) ms$ here (red dotted line). Inset: same considering only the translational motion in step 2.
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