A surfer lies on his board far from shore. Waves come and go, but he doesn't waste energy until he sees the one that's quickly building. In that brief moment, he pops up, paddles with all his might, and glides down the slope. His efficiency peaks exactly when he syncs with the wave, not when he's bobbing calmly on the water. The same goes for adaptive systems, from biological sensors to online learning algorithms: they hit their peak energy-to-information conversion not in steady rest, but in transient moments when the environment shifts sharply.
Picture a particle that can sit in one of two wells of an energy landscape, like a ball in one of two hollows. The depth of the hollows depends on a control parameter that, in turn, tries to follow an external signal. That signal doesn't stay put; it drifts slowly, like the water level in a lagoon fed by tides of varying strength. When the environment changes, the parameter has to readjust, and the system leaves equilibrium. In those moments, it produces entropy, meaning it dissipates energy, and simultaneously gains information about the environment. The ratio of the rate of information gain to the rate of energy dissipation is the instantaneous learning efficiency.
Numerical simulations, run with the Euler-Maruyama method with a step of 0.001 and averaged over 30 realizations of 100,000 steps each, showed: when the environment changes rapidly, entropy production and information gain rate increase, but efficiency peaks don't align exactly with dissipation peaks. This means a system can burn a lot of energy without learning if its internal rhythm doesn't match the rhythm of the external signal. When averaged over the ensemble, the peaks smooth out, and the average efficiency decays to zero, confirming the purely transient nature of optimal information-to-energy conversion. An empirical dependence of maximum efficiency on parameters was found: it decays as the square root of the product of adaptation rate and environmental drift time.
Before this work, it was unclear how the thermodynamic efficiency of learning behaves in a continuously changing environment where no steady state exists. Now it's clear that maximum energy-to-information conversion efficiency is achieved in short transient moments when the system rapidly adapts to environmental changes, not in a steady regime. A minimal model of an adaptive particle in a potential tracking a drifting signal was built, and instantaneous thermodynamic quantities were studied numerically. This approach shifts focus from averaged characteristics to trajectory analysis and highlights the importance of temporal correlation between the system's internal dynamics and the external signal.
The surfer doesn't waste energy on every wave in a row. He waits for the right moment, when the wave is steep and fast, and then puts all his energy into a few powerful paddles. Adaptive systems in a changing world do the same: their thermodynamic performance peaks not in calm paddling, but in a short sprint when they sync with the shift. This knowledge could be useful for designing energy-efficient sensors and algorithms that need to learn quickly during moments of change, rather than spending resources on constantly maintaining equilibrium.
In the model, efficiency can exceed unity or be negative at certain moments, which does not violate the second law of thermodynamics but reflects a temporary mismatch between dissipation and information gain.