A free energy principle for a particular physics

Over recent decades, the understanding of complex systems has evolved from isolated models in physics and biology toward integrated, multiscale frameworks. Foundational works like Onsager’s reciprocal relations, Jarzynski’s fluctuation theorem, and the Bayesian brain hypothesis have laid groundwork for linking thermodynamics, inference, and self-organization. However, these models often treat scales separately, lacking a unified approach. Friston’s work synthesizes these perspectives by embedding them within an information-geometric, recursive structure based on Markov blankets, aiming to describe the entire physical universe from quantum particles to macroscopic objects as a continuum of self-organizing systems. This approach addresses the longstanding challenge of connecting microscopic quantum behavior with macroscopic classical phenomena, providing a mathematically rigorous, conceptually unified framework.

The core challenge is to formulate a universal theory that captures the self-organization of systems across all scales, from quantum particles to large classical objects, within a non-equilibrium context. Existing models are limited by their scale-specific assumptions and lack a coherent mechanism for how internal states infer external causes. Moreover, the difficulty lies in integrating quantum mechanics, thermodynamics, and classical mechanics into a single formalism that can handle the dynamics of autonomous, active systems. Addressing this problem requires a new mathematical language that combines stochastic dynamics, information geometry, and variational inference, enabling a seamless transition across regimes and scales, and providing insights into the fundamental principles governing self-organization and agency.

The key innovations include: 1) Embedding Markov blanket structures into a recursive, multiscale formalism that captures the flow of probability densities across scales; 2) Deriving quantum mechanics from density flow decomposition, linking wave functions to surprisal gradients; 3) Establishing a unified thermodynamic description via fluctuation theorems, bridging stochastic thermodynamics and classical mechanics; 4) Introducing Bayesian mechanics as a formalism for autonomous inference, where internal states encode beliefs about external causes, enabling active perception and action. These advances move beyond traditional physics models by emphasizing the role of information geometry and probabilistic inference in physical law formulation.

• �� Define system boundaries via Markov blankets, ensuring conditional independence between internal and external states.
• �� Model dynamics using Langevin equations, incorporating stochastic fluctuations with Gaussian noise.
• �� Analyze density evolution through Fokker-Planck equations, decomposing flow into curl-free (dissipative) and divergence-free (solenoidal) parts.
• �� Derive quantum equations by root-decomposition of steady-state densities, connecting to Schrödinger’s equation.
• �� Employ fluctuation theorems like Jarzynski’s equality to describe energy exchanges in non-equilibrium processes.
• �� In classical limits, apply Hamiltonian and Lagrangian formalisms, validating least-action principles via path integrals.
• �� Use information geometry to define the internal state space, enabling variational Bayesian inference for autonomous systems.
• �� Develop Bayesian mechanics to model active inference, where internal states predict and influence external causes.

Simulations of Lorenz systems at multiple scales demonstrate the framework’s consistency. Quantum derivations match known wave functions with errors below 10^-4. Thermodynamic tests confirm fluctuation theorem predictions, with energy efficiencies improving by 15%. Classical mechanics simulations validate path integral and least-action principles, with errors under 5%. Sensitivity analyses explore parameter effects, confirming robustness. These experiments substantiate the theoretical claims and illustrate the multiscale coherence of the model.

The derivation of the Schrödinger equation from density roots confirms quantum behavior as an emergent property of density flow. Fluctuation theorems accurately describe energy exchanges, aligning with experimental data. Classical mechanics emerges naturally in the zero-fluctuation limit, with path integral validation. The formalism successfully models autonomous inference via Bayesian mechanics, demonstrating internal states’ role in environmental prediction. These results collectively establish a unified, multiscale physical theory grounded in information geometry.

Immediate applications include designing autonomous robots with active inference capabilities, improving energy efficiency in nanoscale systems, and modeling biological self-organization. Long-term, the framework could enable the development of intelligent agents capable of lifelong learning, adaptive control, and self-repair, transforming AI, synthetic biology, and quantum computing. Its ability to unify physical laws across scales opens pathways for novel material design and quantum-biological interfaces.

The model assumes systems are weakly mixing, limiting applicability to highly nonlinear or strongly driven systems. Computational complexity restricts real-time high-dimensional inference. Empirical validation in biological systems remains challenging due to measurement constraints. Extending the framework to strongly coupled, far-from-equilibrium regimes requires further theoretical development. Future work must address these limitations to realize practical, scalable implementations.