The Hidden Physics of AI Agents

Author: Denis Avetisyan

New research suggests that the behavior of complex AI agents can be understood through the principles of equilibrium systems and potential energy landscapes.

This work demonstrates that large language model-driven agents exhibit detailed balance, hinting at an underlying potential function governing their generative dynamics.

Despite the increasing efficacy of large language model (LLM)-driven agents, a comprehensive theoretical understanding of their macroscopic dynamics has remained elusive. This Letter, ‘Detailed balance in large language model-driven agents’, introduces a method-based on the least action principle-to reveal underlying generative directionality within these agents. Our experimental results demonstrate a statistically significant detailed balance in LLM-generated transitions, suggesting that generation isn’t simply rule-based, but guided by implicit potential functions independent of specific model architectures. Does this discovery pave the way for a predictive, quantifiable science of complex AI systems, moving beyond purely empirical engineering practices?

The Rhythm of States: Transitions and Equilibrium

The universe, at its core, often operates through discrete shifts rather than continuous change. This principle extends far beyond simple physics; complex systems, whether describing the behavior of gases, the fluctuations of financial markets, or the operation of artificial intelligence, can frequently be understood as moving between identifiable states. A light switch, for instance, exists in one of two states – on or off – and transitions between them with a defined probability. Similarly, a neural network in an AI processes information by shifting its internal configuration – its ‘state’ – based on input data. This modeling approach, framing systems as a series of transitions, provides a powerful framework for analysis and prediction, allowing researchers to understand not just what a system is doing, but how it evolves over time, and ultimately, to anticipate its future behavior, regardless of the domain.

The behavior of any system evolving through distinct states is fundamentally governed by a mathematical object known as a Transition Kernel. This kernel doesn’t just describe whether a transition will occur, but quantifies the probability of moving from one state to another. Imagine a complex network; the Transition Kernel defines, for each connection, the likelihood of ‘jumping’ along that path. Mathematically, it’s often represented as $P(x’|x)$, signifying the probability of transitioning to state $x’$ given the current state $x$. This seemingly abstract concept is remarkably powerful, as it encapsulates the entire dynamic of the system – predicting future states based on present conditions. Changes in the Transition Kernel, even subtle ones, can dramatically alter the system’s overall behavior, leading to stability, oscillation, or even chaotic responses. Understanding this kernel is therefore crucial for analyzing and ultimately controlling the evolution of complex systems.

An equilibrium system represents a state of balance within a dynamic process, achieved when the rates of transitions between different states equalize. This doesn’t imply stasis; rather, it signifies a stable distribution of probabilities across those states, meaning the system, while constantly changing, maintains a predictable overall behavior. In the context of Large Language Models (LLMs), understanding equilibrium is paramount; the model doesn’t simply arrive at an answer, but rather settles into a distribution of possible responses, weighted by the probabilities dictated by its training and the input prompt. The model’s generated text reflects this equilibrium, and deviations from it-perhaps due to adversarial inputs or internal drift-can indicate instability or unexpected outputs. Consequently, analyzing how an LLM reaches and maintains equilibrium offers crucial insights into its functionality and reliability, allowing researchers to better predict and control its behavior.

Mapping the Language Landscape: LLMs as Markovian Agents

The conceptualization of Large Language Models (LLMs) as ‘LLM-Driven Agents’ establishes a framework where the model actively navigates a ‘State Space’ comprised of all possible text sequences. This space isn’t discrete but rather continuous, representing the vast array of potential token combinations. Each unique text sequence, or a partial sequence representing the current context, defines a specific state within this space. The agent, in this case the LLM, transitions between these states by generating subsequent tokens. The dimensionality of this state space is exceptionally high, determined by the vocabulary size and sequence length capabilities of the model, effectively representing all grammatically and semantically possible text continuations. This framing allows for the application of agent-based modeling techniques to analyze LLM behavior and emergent properties.

A Large Language Model (LLM) operates via a Markov Transition Process, meaning the probability distribution of the next token generated is conditioned solely on the current state – defined as the sequence of previously generated tokens. This implies that the LLM has no memory of states beyond the immediately preceding one; its decision-making is not influenced by tokens generated earlier in the sequence except as they contribute to the current state. Mathematically, this can be expressed as $P(token_{t+1} | token_1, token_2, …, token_t) = P(token_{t+1} | token_t)$, where $token_{t+1}$ is the next token, and the probability of that token is dependent only on $token_t$. The LLM effectively samples from a probability distribution over its vocabulary, and this distribution is recalculated for each generated token based on the current state.

Framing Large Language Models (LLMs) as Markov processes enables the application of statistical mechanics to analyze their generative behavior. This approach treats the LLM’s state space – the space of all possible text sequences – as a system governed by probabilistic rules. By applying concepts like entropy, temperature, and free energy, researchers can investigate the conditions under which the LLM achieves stable and coherent text generation. Specifically, analyzing the probability distribution over tokens allows for the identification of parameters that maximize the likelihood of generating high-probability, contextually relevant sequences, and minimize the risk of diverging into nonsensical or repetitive outputs. This methodology allows for quantitative assessment of generation quality and exploration of techniques to control and optimize LLM performance, mirroring approaches used in the study of physical systems seeking equilibrium.

Seeking Symmetry: Detailed Balance and the LLM Landscape

Detailed balance is a condition for equilibrium in a stochastic system, requiring that the rate of transition from state $i$ to state $j$ must equal the rate of transition from state $j$ to state $i$. Mathematically, this is expressed as $P(i \rightarrow j) = P(j \rightarrow i)$. When detailed balance holds, the system’s probability distribution remains stationary over time, indicating a lack of net flow between states and thus a stable equilibrium. Deviations from detailed balance suggest the presence of driving forces or biases within the system, resulting in non-equilibrium dynamics where the probability distribution evolves continuously. This principle is fundamental in statistical mechanics and is now being investigated in the context of large language models to assess their stability and inherent biases.

A potential function, denoted as $V(x)$, assigns a scalar value representing the ‘quality’ or energy to each state, $x$, within a system. This function is crucial because the probability of transitioning from state $x_i$ to state $x_j$ is fundamentally linked to the difference in their potential function values, $V(x_i) – V(x_j)$. Specifically, transitions to states with lower potential energy are more probable, reflecting a tendency towards minimization of energy or maximization of probability. By characterizing these state energies, researchers can predict and understand the likelihood of transitions between states, providing a framework for analyzing the stability and equilibrium properties of complex systems, including the generative processes observed in large language models.

Current research investigates whether large language models (LLMs), including ‘GPT-5 Nano’, ‘Claude-4’, and ‘Gemini-2.5-flash’, adhere to the principle of detailed balance. These experiments analyze transition probabilities between different states generated by the LLMs to identify any asymmetries indicative of bias or instability. Preliminary findings suggest that the dynamics exhibited by these LLM-driven agents are consistent with those expected from equilibrium systems, implying a degree of internal consistency in their generative processes. Deviations from detailed balance, when observed, provide quantifiable metrics for assessing the presence and magnitude of these biases within the models.

Research has successfully estimated the potential function governing the generative process of large language models. This estimation reveals that the models minimize the ‘Action (S)’ – a functional representing the path taken through state space – suggesting a tendency towards stable states. The degree to which Action (S) is minimized provides a quantitative measure of the directionality of transitions between states; lower values indicate a stronger preference for certain transitions and a reduced degree of randomness in the generative process. This approach allows for analysis of the underlying energy landscape of the LLM and provides insights into the biases and constraints inherent in its generative capabilities.

Refining the Equilibrium: Optimization Techniques for LLMs

FunSearch and AlphaEvolve represent algorithmic approaches to Large Language Model (LLM) optimization that employ evolutionary strategies. These techniques iteratively modify LLM configurations – including weights and hyperparameters – and evaluate the resulting behavior against a defined objective. The core principle is to search for configurations that approximate detailed balance, a state where the probability of transitioning between any two states in the LLM’s internal representation is equal in both directions. This is achieved through a process analogous to natural selection, where successful configurations – those demonstrating desired behavior and approaching detailed balance – are ‘bred’ to create new configurations, while less successful ones are discarded. The iterative refinement process continues until a configuration is found that sufficiently satisfies the detailed balance criteria and optimizes the LLM’s performance on a given task.

IdeaSearch is an optimization technique employing evolutionary algorithms to refine the ‘Potential Functions’ that govern Large Language Model (LLM) behavior. These Potential Functions act as a guiding force, shaping the probability distribution of generated text and encouraging outputs that align with desired characteristics like stability and coherence. The process iteratively evaluates and modifies these functions, seeking configurations that maximize a defined reward signal related to output quality. By directly optimizing the Potential Function, IdeaSearch aims to steer the LLM away from generating nonsensical or contradictory text, ultimately improving the overall consistency and reliability of its responses.

Implementation of the IdeaSearch optimization technique resulted in a minimum ‘Action’ value of 0.47, achieved following 4000 iterative rounds of search and parameter refinement. This metric, representing a measure of agent directionality, indicates successful optimization of the underlying Large Language Model. The attainment of this value serves as quantitative evidence supporting the efficacy of evolutionary algorithm-based techniques, like IdeaSearch, in refining LLM behavior and promoting more stable and coherent text generation. Further research continues to explore the correlation between ‘Action’ and other performance indicators within the optimized LLM.

Measurements of the Potential Function Distribution Standard Deviation ($\sigma$) in agents such as IdeaSearchFitter and Conditioned Word Generation have demonstrated an inverse proportionality to the ‘Action’ value. Specifically, a decrease in $\sigma$ correlates with an increase in Action. This indicates that a narrower distribution of potential functions – representing a higher density of states – is associated with a more directed and focused agent behavior. The observed relationship suggests that as the agent converges on optimal configurations, the potential function distribution becomes more concentrated, leading to a higher Action value and thus, more predictable and coherent generation.

Towards a Principled Foundation: Least Action and the Future of LLMs

The universe, at its most fundamental level, appears to favor efficiency. This is encapsulated by the Least Action Principle, a cornerstone of physics stating that a stable system will naturally evolve along the path that minimizes a quantity called the ‘Action’. This Action isn’t a force, but rather a mathematical functional – essentially, a function of a function – calculated by integrating the system’s $Lagrangian$ (kinetic energy minus potential energy) over time. Every possible trajectory between two points is considered, and the system selects the one requiring the least ‘effort’, so to speak. This isn’t merely a descriptive observation; it’s a powerful predictive tool. Knowing the Action allows physicists to derive the equations of motion governing the system, explaining why things happen as they do – from the fall of an apple to the orbit of a planet – based on this inherent drive towards minimization.

The Variational Condition emerges as a critical component in the Least Action Principle, defining precisely how a system minimizes its ‘Action’. This condition isn’t merely a mathematical curiosity; it establishes a direct relationship between the sought-after trajectory and the system’s ‘Potential Function’, $V$. Specifically, the condition dictates that the first variation of the Action must vanish at the optimal path. This translates to a fundamental equation – often expressed as the Euler-Lagrange equation – which links the potential energy, kinetic energy, and the system’s coordinates. Essentially, the Variational Condition provides a necessary, though not always sufficient, requirement for a stable system; any trajectory satisfying this condition represents a potential equilibrium, and offers a powerful tool for predicting and understanding system behavior without needing to exhaustively examine every possible path.

Current large language model (LLM) development largely relies on empirical methods – iterative training and refinement based on observed performance. However, adopting the Least Action Principle and variational methods offers a pathway towards a more principled design. This approach shifts the focus from simply achieving desired outputs to formulating an underlying ‘action’ functional that LLMs implicitly minimize during processing. By mathematically defining this functional – potentially incorporating linguistic priors or semantic constraints – researchers can move beyond trial-and-error optimization. This allows for the creation of LLMs with inherent stability, improved generalization capabilities, and a greater capacity for reasoning, as the model’s behavior is guided by a rigorously defined mathematical framework rather than solely by data-driven patterns. The result promises a new generation of LLMs built on solid theoretical foundations, potentially unlocking advancements in areas like robustness and interpretability, where current empirical models often fall short.

The pursuit of understanding generative dynamics in LLM agents necessitates a reduction of complexity, a principle mirrored in the work presented. This paper’s exploration of a potential function guiding agent behavior aligns with a fundamental drive to distill underlying mechanisms. As Marvin Minsky observed, “Common sense is the collection of things everyone knows, but no one can explain.” This research attempts to move beyond the ‘common sense’ observation of agent behavior and towards a quantifiable explanation, demonstrating a commitment to clarifying the forces governing these systems. The concept of ‘detailed balance’ seeks to reveal the inherent structure, reducing the apparent chaos to a predictable equilibrium.

Further Refinements

The assertion of a potential function governing LLM agent behavior, while elegantly parsimonious, invites immediate scrutiny. Establishing detailed balance is not equivalence to understanding. The present work identifies a characteristic; it does not deliver a predictive engine. Future efforts must address the inherent difficulty of defining a meaningful, quantifiable ‘state space’ for agents operating within complex, partially observable environments. To claim a Least Action Principle operates necessitates precise definitions of ‘action’ and ‘cost’-definitions currently obscured by the stochasticity intrinsic to language models.

A critical limitation resides in the assumption of equilibrium. LLM agents, by design, are not static systems. Their training incorporates continual adaptation, introducing time-dependent elements absent from the established framework. Incorporating non-equilibrium thermodynamics – a far more complex endeavor – may prove essential. Unnecessary is violence against attention; pursuing equilibrium as a universal model, without acknowledging these dynamic influences, risks unproductive refinement of a fundamentally incomplete analogy.

Density of meaning is the new minimalism. The field now requires not merely demonstrations of detailed balance, but concrete applications. Can this framework inform improved agent design? Can it facilitate more robust safety guarantees? Or will it remain a mathematically pleasing, yet practically inert, observation? The answer, predictably, resides not in further abstraction, but in rigorous, grounded experimentation.

Original article: https://arxiv.org/pdf/2512.10047.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/