🔬 condensed matter

Work as a function of protocol duration for the efficient erasure of an underdamped memory: isothermal to adiabatic transition

Using evolutionary reinforcement learning, this study reveals that the mean work required to erase an underdamped memory bit transitions from an isothermal regime (scaling as $1/\tau$ above the relaxation time) to a distinct adiabatic regime (scaling more slowly than $1/\tau$ below the relaxation time), demonstrating that learned protocols can outperform those optimized solely for equilibrium boundary conditions.

Original authors: Nicolas Barros, Stephen Whitelam, Sergio Ciliberto, Ludovic Bellon

Published 2026-09-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicolas Barros, Stephen Whitelam, Sergio Ciliberto, Ludovic Bellon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of microscopic physics, there is a fundamental rule about information and energy. It states that if you want to delete a single piece of information, such as a simple yes or no, you must spend a tiny, unavoidable amount of energy to do it. This is not just a theoretical idea; it is a law of nature. When a computer erases a bit of data, it heats up slightly, and that heat is the physical cost of the deletion. Scientists have long known that if you perform this erasure very slowly, you can get as close as possible to this minimum energy cost. However, real-world devices need to work quickly. The big question has been what happens when you try to erase information fast. Does the energy cost shoot up dramatically, or is there a smarter way to do it? This question matters because as our technology shrinks and speeds up, understanding the limits of energy efficiency becomes crucial for building better machines.

A team of researchers in Lyon, France, and Berkeley, California, set out to answer this by building a tiny, physical memory device and teaching a computer how to erase it efficiently. They used a microscopic metal beam, called a cantilever, which vibrates like a guitar string. This beam acts as a one-bit memory: it can rest in one of two positions, representing a zero or a one. The researchers wanted to force this beam to move from a random position into a specific target position, effectively erasing its previous state. To do this, they applied an electric force to the beam, shaping an invisible energy landscape that guided the beam's movement. The challenge was that the beam is "underdamped," meaning it is light and bouncy; once it starts moving, it tends to keep oscillating rather than stopping immediately. This makes it difficult to control quickly without wasting energy or missing the target.

To find the best way to erase the bit, the scientists did not rely on human intuition or standard formulas. Instead, they used a method called neuroevolution, a type of artificial intelligence that mimics natural selection. They created a digital twin of their physical experiment on a computer and let an algorithm evolve thousands of different ways to move the beam. The computer tested these strategies, keeping the ones that used the least energy and successfully moved the beam to the right spot, while discarding the failures. Over many generations, the algorithm discovered a set of instructions—a protocol—that was far more efficient than any human-designed method. The researchers then took these computer-generated instructions and applied them to their real, physical metal beam in the laboratory to see if the digital lessons held up in the real world.

The results revealed a surprising split in how the system behaves, depending entirely on how much time is allowed for the erasure. When the researchers gave the system plenty of time, much longer than the time it takes for the beam to naturally settle down, the energy cost followed a predictable pattern. The work required was just slightly higher than the theoretical minimum, and it decreased as the process got slower. This matched what scientists had seen in heavier, slower systems where friction dominates. However, when they tried to erase the bit quickly, in a time shorter than the beam's natural settling time, the rules changed completely. In this fast regime, the energy cost did not skyrocket as one might expect. Instead, the learned protocols found a way to keep the energy cost rising very slowly, even as the time was cut down.

The key to this efficiency lies in how the beam moves. In a slow erasure, the beam stays in balance with the surrounding air, and the process is smooth and steady. In a fast erasure, the beam is pushed so hard and so quickly that it cannot stay in balance; it heats up and gains kinetic energy, vibrating wildly. The researchers found that the smartest protocols did not try to fight this heating or stop the vibrations immediately. Instead, they embraced the speed. They moved the beam in a way that accepted this temporary, high-energy state, effectively treating the system as if it were isolated and adiabatic, rather than trying to force it to stay cool. This allowed them to reach the target with far less work than if they had tried to keep the system in a calm, equilibrium state.

The study showed that these computer-learned protocols were not just slightly better; they were significantly superior to the basic, hand-designed methods used in previous experiments. For fast erasures, the basic method often failed, leaving the beam in the wrong position, while the learned method succeeded almost every time. The researchers also compared their findings to what happens in slower, heavier systems. They discovered that in the fast, underdamped regime, the most efficient way to erase a bit is to allow the system to end up in a state that is not perfectly balanced with its surroundings. This is a crucial distinction because many traditional theories assume that a process must end in a calm, balanced state to be considered optimal. The researchers found that by relaxing this requirement, they could achieve much lower energy costs.

Ultimately, the team established a new lower limit for how much energy is needed to erase a bit quickly. They proposed a theoretical "thought experiment" involving a perfect, ideal controller that could read the state of the memory and act accordingly. This theoretical limit showed that the energy cost for fast erasure should grow very slowly as the time decreases, scaling in a specific way that is much more favorable than the old rules suggested. The experimental data and the computer simulations both hovered just above this theoretical limit, confirming that the learned protocols were nearly as good as physics allows. The work demonstrates that by using machine learning to explore the full range of possible movements, rather than sticking to simple, intuitive steps, we can push the boundaries of energy efficiency in computing. It suggests that the future of ultra-fast, low-power computing may depend on learning to let systems move freely and quickly, rather than trying to force them to move slowly and carefully.

Technical Summary: Work as a Function of Protocol Duration for Efficient Erasure of an Underdamped Memory

Problem Statement
Landauer's principle establishes a minimal thermodynamic cost, $k_B T \ln 2$ , for erasing one bit of information in a quasi-static process. While the scaling of the mean work $\langle W \rangle$ required for finite-time erasure in overdamped systems is well-understood—following a $1/\tau$ scaling law where $\tau$ is the protocol duration—this relationship lacks a general proof and clear characterization in the underdamped regime. In underdamped systems (e.g., mechanical oscillators with high quality factors), the interplay between inertia, thermal fluctuations, and finite-time driving creates complex dynamics. Specifically, it is unclear how the work cost scales when the erasure time $\tau$ becomes comparable to or shorter than the system's relaxation time $t_r$ . Furthermore, while optimal transport theory provides bounds for overdamped systems, no known applicable optimal protocol exists for underdamped oscillators with small viscous damping ( $Q > 1$ ).

Methodology
The authors employ a combination of experimental realization and machine learning to investigate this problem:

Experimental System: The study utilizes a micromechanical cantilever acting as an underdamped harmonic oscillator ( $Q \approx 7$ , $\omega_0 \approx 1090$ Hz) in air. The cantilever is trapped in a tunable double-well potential created via electrostatic feedback. Logical states (0 and 1) correspond to the cantilever residing in the left or right well, respectively.
Neuroevolutionary Learning: To find efficient erasure protocols, the authors use "neuroevolution," a form of reinforcement learning. They encode the time-dependent control parameters of the potential ( $z_0(t)$ and $z_1(t)$ ) as a deep neural network. This network is trained using a genetic algorithm on a "digital twin" of the experiment (a Langevin equation model) to minimize an order parameter $\phi = P_f + \langle W \rangle/100$ , where $P_f$ is the failure rate (probability of ending in the wrong well) and $\langle W \rangle$ is the mean work.
Validation: The learned protocols are implemented on the physical cantilever. The performance (work and failure rate) is compared against a "basic" linear ramp protocol and numerical simulations across a range of erasure times ( $0.5 t_0 \le \tau \le 20 t_0$ ).
Theoretical Bound: The authors propose a "gedanken" (thought) optimal protocol to establish a theoretical lower bound on the work. This protocol involves a "demon" that reads the initial state and applies an optimal translation only if a shift is required, followed by a cost-free reconstruction of the potential.

Key Results
The investigation reveals two distinct regimes for the mean work $\langle W \rangle$ as a function of protocol duration $\tau$ , separated by the system's relaxation time $t_r$ :

Quasistatic Regime ( $\tau \gg t_r$ ):
- In this limit, the system remains close to thermal equilibrium.
- Both the basic and learned protocols yield a mean work that scales as $\langle W \rangle \approx k_B T (\ln 2 + B/\tau)$ .
- This recovers the standard Landauer bound plus an overhead consistent with overdamped systems, confirming that the $1/\tau$ scaling holds for slow erasures in underdamped memories.
Adiabatic/Non-Equilibrium Regime ( $\tau \lesssim t_r$ ):
- For fast erasures, the basic protocol fails significantly (high $P_f$ ) and requires rapidly increasing work due to residual kinetic energy and heating effects that cannot be dissipated to the bath.
- Learned Protocols: The neuroevolutionary approach discovers protocols that significantly outperform the basic protocol. The mean work grows much more slowly than the $1/\tau$ extrapolation from the quasistatic regime.
- Scaling Behavior: The work in the fast regime is bounded from below by a $1/\tau$ scaling, but with a different intercept and coefficient. The learned protocols maintain a near-zero failure rate while keeping the final total energy low.
- Gedanken Bound: The authors derive a lower bound $\langle W_g \rangle$ based on their thought experiment. This bound transitions from an isothermal value ( $\ln 2$ ) to an adiabatic value ($1$ in units of $k_B T$ ) as $\tau$ decreases. The experimental and numerical results for the learned protocols lie close to this adiabatic bound for short $\tau$ .

Comparison with Overdamped Systems
The paper compares the learned underdamped protocols with those in the overdamped regime. It finds that, similar to the overdamped case, learned protocols that do not enforce an equilibrium end state can achieve lower work costs than protocols optimized under the constraint of a final equilibrium state. This suggests that allowing the system to end in a non-equilibrium state (specifically, one with lower stored energy) is a key mechanism for efficiency in fast erasure.

Significance and Claims
The authors claim that their work provides new insight into the thermodynamics of information processing in underdamped systems, a regime where general proofs for optimal work scaling were previously lacking.

Regime Separation: They demonstrate that the universal $1/\tau$ scaling of overdamped systems is only valid for $\tau \gg t_r$ . For $\tau \lesssim t_r$ , the physics shifts to an adiabatic-like regime where work scales differently.
Optimization Capability: They show that neuroevolution can reliably produce protocols that are nearly optimal for rapid driving, even in the presence of inertia, outperforming hand-designed "basic" protocols.
Theoretical Bound: The proposed gedanken protocol establishes a rigorous lower bound for the work of fast erasure, which the experimental data approaches but does not strictly reach (likely due to the limited class of potentials used in the experiment).
Modesty: The authors explicitly state that they do not claim the learned protocols are globally optimal, as the search space was restricted to a specific subset of energy landscapes (bi-quadratic potentials). However, they assert that the protocols provide a clear, experimentally verified bound for future studies and demonstrate that machine learning is a robust tool for exploring non-equilibrium thermodynamics in complex mechanical systems.

Technical Summary: Work as a Function of Protocol Duration for Efficient Erasure of an Underdamped Memory

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