The work sits at the intersection of stochastic thermodynamics and information theory. Nearby are studies that examine neural network training through the lens of the Ising model, the entropy of climate systems, synchronization of genes with chaos, and robot learning in turbulent flows. These works share an interest in how systems process information under non-equilibrium conditions, but most focus on stationary regimes or fixed statistics. This paper stands out by explicitly considering a continuously non-stationary environment where parameters drift over time, and by analyzing transient rather than steady-state characteristics.
A minimal stochastic model of three coupled variables was constructed: particle position, an adaptive control parameter, and an external environmental signal. The dynamics are described by overdamped Langevin equations with independent Gaussian noises. The potential has a double-well form whose asymmetry depends on the control parameter. The environment is modeled as an Ornstein-Uhlenbeck process with a slowly drifting mean, creating non-stationarity. Stochastic energetics and the Fokker-Planck formalism are used for analysis. Expressions for entropy production, the rate of mutual information, and instantaneous efficiency are derived. Simulations were performed using the Euler-Maruyama method with a step of 0.001, averaging over 30 realizations of 100,000 steps each, with Numba acceleration.
Sharp transient peaks in instantaneous learning efficiency were found when the environment changes rapidly and the adaptive parameter is forced to readjust. At these moments, entropy production and the rate of information gain increase, but the efficiency peaks do not exactly coincide with the dissipation peaks. This means that maximum dissipation does not equal maximum learning; what matters is the temporal synchronization of the system's response with environmental changes. 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 numerically: it decays as the square root of the product of adaptation rate and environmental drift time.
Unlike neighboring works that analyze stationary or quasi-stationary regimes, this study explicitly shows for the first time that maximum thermodynamic efficiency of learning is achieved during transient moments, not in a steady state. This shifts the focus from averaged characteristics to trajectory analysis and emphasizes the importance of temporal correlation between the system's internal dynamics and the external signal. Such an approach can be extended to biological sensory networks and adaptive algorithms where rapid responses to changes matter.
The topic could develop toward studying more complex environments with multiple time scales, as well as feedback systems where the adaptive parameter influences the environment. It would be interesting to investigate how transient efficiency peaks depend on the architecture of the adaptive mechanism and whether they can be predicted in advance. Applications to neural network training under data drift are also possible.
The results could influence the design of energy-efficient adaptive systems, such as sensors or online learning algorithms, where maximizing information per unit of energy expended during critical moments is important.
The next step could be an analytical explanation of the empirical dependence of maximum efficiency on parameters and verification of the universality of the found scaling over a wider range. It is also important to study the role of noise and finite observation time in real systems.
In stochastic thermodynamics, the question of how to define and measure information processing efficiency in systems with memory and complex correlations remains open. This work contributes by showing that transient regimes can be key and offers a specific metric for their analysis.
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.