🌀 nonlinear sciences

Information Thermodynamics of Agents: The Work Capacity of Channels with Memory

This paper introduces a framework for the information thermodynamics of percept-action loops, defining "work capacity" to demonstrate that energy-efficient agents must balance prediction with forgetting, revealing a fundamental trade-off between predictive accuracy and energy efficiency in active systems.

Original authors: Lukas J. Fiderer, Paul C. Barth, Isaac D. Smith, Hans J. Briegel

Published 2026-10-01

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CC BY 4.0

Original authors: Lukas J. Fiderer, Paul C. Barth, Isaac D. Smith, Hans J. Briegel

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

Every living thing, from a single bacterium to a human being, exists in a constant loop of sensing and acting. It sees the world, processes that information, and then moves to change the world, which in turn changes what it sees next. This cycle, known as a percept-action loop, is the engine of survival and learning. For decades, scientists have understood that the ability to predict the future is a powerful tool in this loop. If you can guess what will happen next, you can prepare for it, saving energy and avoiding danger. This idea is so central to biology and artificial intelligence that many assume the best way to be efficient is to be a perfect predictor, remembering every detail of the past to forecast the future with total accuracy.

However, a new study from a team of physicists in Innsbruck challenges this long-held belief. They asked a fundamental question: what is the absolute limit of energy efficiency for an agent that interacts with its environment? To answer this, they looked at the thermodynamics of information. In simple terms, thermodynamics is the study of how energy moves and changes form. The second law of thermodynamics tells us that energy cannot be created or destroyed, only transformed, and that every process generates some waste heat. When an agent processes information—like a brain remembering a pattern or a computer calculating a move—it must pay an energy cost. The researchers wanted to know how much useful work an agent could extract from its environment while obeying these strict physical laws.

The team developed a mathematical framework to model agents and their environments as channels that pass information back and forth. They defined a new concept called "work capacity," which represents the maximum rate at which an agent can expect to pull energy out of its surroundings. In their model, the agent is not just a passive observer; it actively influences the environment, and the environment responds. This creates a feedback loop where the agent's actions change the very data it receives. By running simulations with different types of agents, they discovered a surprising trade-off. In scenarios where the agent's actions directly influence future observations, the most energy-efficient strategy is not to remember everything.

The study found that in these interactive loops, agents that try to be perfect predictors can actually perform worse than those that forget. To operate at peak energy efficiency, an agent must balance prediction and forgetting. This is because holding onto every detail of the past to make a perfect prediction can lock the agent into a rigid pattern that prevents it from generating the randomness needed to extract energy. The researchers showed that for certain environments, the set of agents that are best at predicting the future and the set of agents that are best at harvesting energy are completely different. In fact, they proved that in these specific conditions, an agent cannot be both maximally predictive and maximally efficient at the same time.

This finding marks a significant departure from previous theories, which were often based on a "tape setting" where an agent simply reads a pre-existing stream of data without influencing it. In that passive scenario, being a perfect predictor is indeed the most efficient path. But in the real world, where actions change the environment, the rules are different. The study demonstrates that there is a fundamental tension between the drive to predict and the drive to be energy-efficient. To be efficient, an agent must balance its memory, knowing exactly when to let go of the past. This suggests that the behaviors we see in nature or in artificial intelligence, which might look like they are "forgetting" or acting unpredictably, could actually be a sophisticated strategy to save energy. The work provides a new lens for understanding how biological systems and future machines might be designed to operate within the strict energetic limits of our universe.

Technical Summary: Information Thermodynamics of Agents: The Work Capacity of Channels with Memory

Problem Statement

Predicting future observations is a central paradigm in machine learning, biology, and economics, often grounded in principles like the variational free energy principle. While the thermodynamic limits of passive observation (e.g., the "tape setting" where an agent processes a pre-existing sequence of symbols) are well-studied, the thermodynamic implications of active agents that interact with their environment through percept–action loops remain less understood. In these loops, an agent's actions influence future percepts, creating a feedback mechanism.

The central question addressed is: What are the energetic limits of adaptive information processing in percept–action loops, and how should agents be designed to operate efficiently under these constraints? Specifically, the authors investigate whether the design principles that optimize work extraction in passive observation (where prediction is key) hold true when the agent actively influences the environment.

Methodology

The authors develop a framework combining stochastic thermodynamics and the information theory of hidden Markov models.

Modeling Framework:
- Agents and Environments: Both are modeled as hidden Markov channels (finite-state channels with memory). The environment channel ( $\nu_{S|A}^{env}$ ) maps actions ( $A$ ) to percepts ( $S$ ), while the agent channel ( $\eta_{A|S}^{agt}$ ) maps percepts to actions.
- Percept–Action Loops: The interaction is defined as a tuple of an agent and an environment, forming a global stochastic process. The authors prove that the joint process of agent memory, actions, percepts, and environment hidden states forms a finite-state homogeneous Markov chain, ensuring well-behaved asymptotic dynamics (convergence to a cyclostationary process).
Thermodynamic Assumptions (Isothermal Information Processing - IIP):
- Information processing is modeled as a physical protocol acting on an information carrier coupled to a thermal reservoir at temperature $T$ .
- The protocol is isothermal and cyclic regarding the potential landscape, ensuring that work extraction is bounded solely by the intrinsic cost of information processing (changes in entropy), not by changes in internal energy or external control apparatus.
- Under these assumptions, the expected work $W$ extracted by a logical operation is bounded by the change in nonequilibrium free energy, which reduces to an entropy bound: $W \leq H(X_{out}) - H(X_{in})$ .
Key Metric: Work Capacity ( $C_{work}$ ):
- The authors define work rate as the asymptotic mean of the extractable work per round: $W(agt_M, env) = \langle H(A_t|M_t) - H(S_t|M_t) \rangle_t$ .
- Work Capacity is defined as the supremum of this work rate over all possible agent models (with finite memory) compatible with a given environment channel. It represents the maximum rate at which an agent can extract work from the environment.
Predictive Gap:
- To analyze the role of prediction, the authors define the predictive gap as the conditional mutual information $I[A_{0:t+1}S_{0:t}; S_t | M_t]$ . A zero predictive gap implies the agent's memory contains all relevant information from the past to predict the current percept.

Key Contributions and Results

1. Work Capacity Properties

The authors establish fundamental properties of work capacity for any environment channel:

Existence: The work capacity exists due to the convergence properties of the underlying finite-state Markov chains.
Bounds: $0 \leq C_{work}(env) \leq \log_2 |S|$ .
Subadditivity: For a cascade of two channels, the work capacity of the cascade is less than or equal to the sum of the individual work capacities ( $C_{work}(env_2 \circ env_1) \leq C_{work}(env_1) + C_{work}(env_2)$ ).

2. Divergence of Prediction and Efficiency in Active Loops

The most significant finding is a fundamental departure from the thermodynamics of passive observation (the tape setting):

Tape Setting: In passive settings, work-efficient agents must be predictive (minimizing the predictive gap) and randomize their actions.
Percept–Action Loops: In active settings with genuine feedback, prediction and work efficiency can be mutually exclusive.
- The authors prove (Theorem 3) that there exist environments where the set of work-efficient agents ( $A_{eff}$ ), the set of maximally predictive agents ( $A_{pred}$ ), and the set of maximum entropy action agents ( $A_{mea}$ ) are mutually exclusive.
- Specifically, a maximally predictive agent in these environments retains complete information about its past actions to predict the future. However, this retention of information reduces the entropy of the agent's actions ( $H(A_t|M_t)$ ), which is a positive term in the work rate equation. Consequently, the predictive agent achieves a lower (or even non-positive) work rate compared to an agent that "forgets" aspects of its past actions.

3. Trade-off Between Prediction and Forgetting

In percept–action loops, work-efficient agents must balance prediction (minimizing $H(S_t|M_t)$ ) and forgetting (maximizing $H(A_t|M_t)$ ).

Retaining information to predict future percepts often requires the agent to condition its actions on past history, thereby reducing action entropy.
Conversely, maximizing action entropy (to extract work) may prevent the agent from learning the statistical structure of the environment, increasing percept uncertainty.
The optimal strategy often involves a "predictive gap" that is non-zero, meaning the agent deliberately forgets certain past details to maintain high action entropy.

4. Illustrative Example

The authors analyze a specific environment with a feedback parameter $r \in [0, 1]$ :

$r=0$ (Tape Setting): The environment ignores actions. Here, predictive agents are work-efficient.
$0 < r < 1$ (Active Loop): Feedback exists. Numerical optimization shows that as $r$ increases, the optimal work-efficient agents deviate from maximally predictive agents. The predictive gap of the optimal agent exhibits a discontinuous jump, indicating a qualitative change in the memory mechanism (switching between different "ansatz" designs) to balance the trade-off.
$r=1$ (Deterministic): The environment becomes fully deterministic based on actions. Predictive agents regain optimality.

Significance and Claims

The paper claims to provide a fundamental framework for analyzing the thermodynamics of information processing in percept–action loops. Its primary significance lies in challenging the assumption that "prediction is always optimal" for adaptive systems.

Theoretical Shift: The work demonstrates that in active learning systems, the goals of prediction (minimizing uncertainty about the future) and energy efficiency (maximizing work extraction) may be at odds.
Design Principles: It suggests that for agents operating in feedback-rich environments, optimal design may require strategic forgetting rather than maximal memory retention.
Context: The authors situate these findings within the broader context of biological and artificial intelligence, suggesting that behaviors appearing suboptimal from a predictive standpoint (e.g., deliberate forgetting) may actually reflect underlying thermodynamic constraints aimed at maximizing energetic efficiency.

The paper does not propose specific experimental setups or immediate applications but rather establishes the theoretical limits and design principles for agents interacting with their environments under the laws of thermodynamics.