Life as we know it

What to take away

1. 1. Any ergodic random dynamical system that possesses a Markov blanket will—almost surely—appear to minimize variational free energy and engage in active inference, making biological self-organization a near-inevitable consequence of generic coupled dynamics rather than a special emergent phenomenon.
2. 2. The heuristic proof proceeds in three steps: Helmholtz decomposition of the flow, solution of the Fokker–Planck equation yielding p(x|m)=exp(−G(x)), and the ergodic theorem, which together show that internal and active state flows perform a circuitous gradient ascent on the log marginal ergodic density (equation 2.5–2.6).
3. 3. Variational free energy F(s,a,l) is an upper bound on surprise (−ln p(s,a,l|m)) by the Kullback–Leibler divergence term (equation 2.8), so action that minimizes free energy also bounds the entropy of sensory and Markov blanket states, providing the thermodynamic basis for homeostasis.
4. 4. A primordial soup of 128 heterogeneous subsystems—each with Lorenz electrochemical dynamics and rate parameters k(i) drawn from a distribution giving most systems rates near 1—was integrated for 2048 s at 1/512 s steps using forward Euler, with the principal Markov blanket recovered by spectral graph theory using the k=8 largest eigenvector components.
5. 5. Internal-state eigenvariates from singular value decomposition (with ±16 s temporal embedding, retaining 32 principal components) predicted the motion of the most distant external subsystem via canonical variates analysis, with the true analysis yielding five subsystems exceeding the maximum null statistic versus p=0.00052 against time-reversed controls (82 external elements tested).
6. 6. Simulated lesions—rendering active, sensory, or internal subsystems functionally closed while preserving Newtonian coupling—each produced progressive structural disintegration (oscillator death) within 512 s, confirming that autopoietic integrity depends on the full circular causality of the Markov blanket.
7. 7. Functionally closed subsystems (one-third of the ensemble, unable to influence neighbors electrochemically) were invariably expelled to the periphery across all simulations, and no simulation ever produced a functionally closed internal state, demonstrating a selection pressure intrinsic to the dynamics.
8. 8. The paper raises the open hypothesis that minimum-entropy Markov blankets—whose constituency changes more slowly than the states they separate—may be the defining quantitative criterion distinguishing biological from non-biological self-organizing systems, with the marginal ergodic entropy of the blanket serving as the relevant measure.
9. 9. The methodology is replicable: all simulation code is distributed as part of the SPM academic freeware, using stochastic differential equations (3.1–3.3) with unit-normal random fluctuations, a quadratic confining potential, and an inverse-square repulsive plus electrochemically gated attractive force between subsystems.
10. 10. Evolution and adaptation are conjectured to be continuous with the same free energy minimization: somatic learning corresponds to optimizing generative model parameters (analogous to synaptic strengths) to minimize free energy, while evolutionary timescales may represent the slow unfolding of a trajectory toward the universe's global random attractor—making adaptation as mechanistically inevitable as self-organization.

Peer brief — for seminar discussion

Friston (2013, J. R. Soc. Interface, doi:10.1098/rsif.2013.0475) advances a mathematical argument that biological self-organization—ergodicity, Markov blanket formation, active inference, and autopoiesis—is not a special property of living matter but the generic behavior of any ergodic coupled dynamical system with short-range interactions. The argument is formalized through the free energy principle, with variational free energy F(s,a,l) introduced as the central quantity: applying the Helmholtz decomposition to the system's flow and solving the Fokker–Planck equation yields p(x|m)=exp(−G(x)) as the equilibrium density, after which the ergodic theorem implies that internal and active state flows perform a gradient ascent on the log marginal ergodic density (equations 2.5–2.6). Because free energy bounds surprise by a KL divergence term that is non-negative by Gibbs inequality, minimizing free energy simultaneously bounds the entropy of Markov blanket states and renders internal states equivalent to a posterior over hidden external states—exact Bayesian inference in the limit where variational and posterior densities coincide. The load-bearing empirical demonstration is a synthetic primordial soup of 128 subsystems, each carrying Lorenz electrochemical dynamics (with heterogeneous rate parameters k(i) and one-third rendered functionally closed), integrated for 2048 s at 1/512 s time steps using forward Euler in SPM freeware. Spectral graph theory and the Perron–Frobenius theorem applied to the adjacency matrix (built from spatial proximity over the final 256 s) identified the principal Markov blanket: the k=8 subsystems with the largest eigenvector components were designated internal states. Canonical variates analysis of 32 SVD eigenvariates (±16 s temporal embedding) of internal functional states predicted external subsystem motion with p=0.00052 against a time-reversed null across 82 external elements, with 5 subsystems exceeding the maximum null statistic. Selective lesioning—rendering active, sensory, or internal subsystems functionally closed—produced structural disintegration within 512 s (oscillator death), while functionally closed subsystems were universally expelled to the ensemble periphery. The implications the paper draws are broad: if ergodicity and short-range coupling suffice to generate a Markov blanket, and if any Markov blanket generically produces active inference, then perception, homeostasis, and even evolution are not biological curiosities but corollaries of descent onto a global random attractor. Evolution is explicitly framed as free energy minimization at a slower timescale, analogous to Bayesian model selection. An alternative formal treatment might have used information-geometric methods (e.g., Fisher information metrics on the space of generative models) rather than the Helmholtz/Fokker–Planck route, which would have made the connection to natural gradient learning more direct. The key hypothesis the paper floats is that minimum-entropy Markov blankets—those whose constituency changes more slowly than the states they separate—characterize biological systems specifically, with marginal ergodic entropy of the blanket as the quantitative discriminant. A critical reader will push back on the following: the core claim that any ergodic system with a Markov blanket 'appears to' minimize free energy is essentially analytic—it follows by construction from the definitions of ergodicity and the Fokker–Planck solution, so the argument risks circularity. The Lorenz attractor was chosen as an 'arbitrary' electrochemical substrate, but the Lorenz system's specific chaotic and synchronization properties (nonlinear coupling modifying the Rayleigh parameter 32+q̄1(j)) may load the simulation in favor of the result; it is not demonstrated that an ensemble with qualitatively different attractor geometry would produce the same blanket morphology and inferential coupling. More broadly, showing that a 128-node toy system with tuned interaction rules forms something resembling a membrane-enclosed cluster does not establish that ergodic Markov blankets are probable—rather than merely possible—in prebiotic chemistry, leaving the 'almost inevitable' claim undersubstantiated.

Methods (5)

• Euler integration
  Numerical method used to integrate stochastic differential equations of the primordial soup.
• Fokker-Planck equation
  Equation describing the evolution of probability density over states; used to find ergodic density.
• Singular Value Decomposition
  Used to summarize principal patterns of internal functional states.
• Synthetic primordial soup simulation
  An ensemble of coupled dynamical subsystems with Newtonian and electrochemical states used to demonstrate emergence of life-like properties.
• Temporal embedding
  Lagged time series used to capture dynamical dependencies.

Frameworks (4)

• Free Energy Principle
  A foundational variational principle from statistical physics that formalizes how self-organizing systems maintain structural integrity and adapt to their environment by minimizing free energy—a mathematical bound on surprise or prediction error. Originally developed by Karl Friston, the framework unifies action, perception, and learning as processes of active inference, where systems both update internal models of the world and act upon it to reduce the divergence between predictions and observations.
• Homeokinetics
  Soodak and Iberall's physical science for complex systems.
• Maximum Entropy Principle
  Jaynes' principle that systems maximize entropy under constraints.
• Synergetics
  Haken's theory of self-organizing non-equilibrium phase transitions.

Findings (9)

• The principal Markov blanket identified by spectral graph theory formed a clustered structure with internal subsystems enshrouded by sensory and active subsystems.

  Visual and quantitative observation of Markov blanket emergence.

• Functionally closed subsystems were rusticated to the periphery of the ensemble; no simulation produced a functionally closed internal state.

  Emergent spatial segregation of closed subsystems.

• Internal states significantly predicted motion of external subsystems; best prediction for the farthest subsystem (magenta circle, Fig 4d).

  Result of canonical variates analysis showing statistical dependency between internal states and external motion.

• Internal subsystem dynamics significantly predict external subsystem motion via canonical variates analysis (χ²-distributed, p=0.00052).

  Empirical validation from primordial soup that internal states encode information about hidden environmental states.

• Lesions to active, sensory, or internal states caused rapid dispersion and structural disintegration of the Markov blanket.

  Simulation result demonstrating autopoietic maintenance and oscillator death after lesions.

• Lesioning active, sensory, or internal states causes rapid structural disintegration and loss of spatial organization.

  Demonstrates autopoietic maintenance: Markov blanket integrity is necessary for preserving internal state configuration.

• Five out of 82 external subsystems had χ² values above the maximum null distribution, p=0.00052.

  Statistical significance of the prediction after time-flip control.

• Functionally closed subsystems are systematically expelled to the periphery of the ensemble.

  Simulated result showing that subsystems unable to influence others cannot invade internal organization, supporting Markov blanket partition.

• Slow subsystems were distributed among internal and external states, not segregated.

  Observation about heterogeneous rate constants in the simulation.

Claims (11)

• Any ergodic random dynamical system that possesses a Markov blanket will appear to actively maintain its structural and dynamical integrity.

  The lemma that leads to the main claim.

• Life requires only a boundary (Markov blanket), ergodicity, active inference capacity, and autopoiesis—not reproduction or DNA.

  Reframes definition of life focusing on dynamical self-organization rather than genetic reproduction.

• Action places an upper bound on the entropy of biological states, thereby conserving structural and dynamical integrity.

  Autopoietic aspect of active inference.

• The flow of internal and active states can be described as a gradient descent on variational free energy.

  A formal result from the proof.

• Biological systems are ergodic, possess a Markov blanket, exhibit active inference, and are autopoietic.

  Summary of the four criteria for biological self-organization.

• Any system that exists will appear to minimize free energy and therefore engage in active inference.

  The reworked argument that free energy minimization is a corollary of existence, not a prerequisite.

• A Markov blanket is (almost) inevitable in coupled dynamical systems with short-range interactions.

  Argument that physical laws inevitably produce Markov blankets.

• Biological self-organization is not as remarkable as one might think—and is (almost) inevitable.

  Central claim of the paper that life-like behavior emerges necessarily from coupled dynamical systems with Markov blankets.

• Internal states appear to encode Bayesian beliefs about hidden external states.

  The inferential interpretation of internal dynamics.

• Evolution and adaptation may be as inevitable as simple self-organization.

  Speculation that descent onto a global random attractor implies evolutionary free energy minimization.

Hypotheses (3)

• If internal states encode a probability density over external states, then it should be possible to predict external states from internal states.

  The testable hypothesis driving the active inference analysis in the simulation.

• Any system minimizing free energy will appear to engage in implicit Bayesian inference of hidden external causes.

  Predicts that internal states encode posterior beliefs about external world through gradient descent on free energy.

• If systems are ergodic and possess a Markov blanket, they will show lifelike behaviour.

  The main hypothesis the paper attempts to verify heuristically and with simulations.

Questions (5)

• If the minimization of free energy is just a corollary of descent onto a global random attractor, does this mean that adaptation and evolution are just ways of describing the same thing?

  Question about the relationship between adaptation, evolution, and free energy minimization.

• Does this mean there is lifelike behaviour everywhere or is there something special about the Markov blankets of systems we consider to be alive?

  Philosophical question arising from the ubiquity of Markov blankets.

• Is there a unique Markov blanket for any given system, or do nested multi-scale Markov blankets all exhibit life-like behavior?

  Poses challenge to definition: if every Markov blanket induces active inference, is there lifelike behavior everywhere?

• What would happen if systems believed their attracting sets had low entropy?

  Question about the consequences of intrinsic beliefs about entropy.

• Is there a unique Markov blanket for any given system?

  Question about the multiplicity of Markov blankets across scales.

Related work— refs + corpus + external arXiv

Cited / in-corpus / arXiv badges show which signals surfaced each row. Multi-source rows weighted higher.

• Bayesian Mechanics for Stationary Processes

  Karl Friston, Conor Heins and Grigorios A. Pavliotis Lancelot Da Costa

  2021
  ≈ 85%
• Active inference on discrete state-spaces: a synthesis

  in corpus

  2020
  ≈ 85%
• Sentient Self-Organization: Minimal dynamics and circular causality

  Biswa Sengupta and Karl Friston

  2017
  ≈ 85%
• Knitting a Markov blanket is hard when you are out-of-equilibrium: two examples in canonical nonequilibrium models

  \'Angel Poc-L\'opez, Conor Heins, Christopher L. Buckley Miguel Aguilera

  2022
  ≈ 85%
• Markov Blanket Density and Free Energy Minimization

  Luca M. Possati

  2025
  ≈ 85%
• Self-orthogonalizing attractor neural networks emerging from the free energy principle

  Tamas Spisak and Karl Friston

  2025
  ≈ 84%
• A Free Energy Principle for Biological Systems

  cited

  2012
  ≈ 84%
• Cognition coming about: self-organisation and free-energy

  Maxwell Ramstead, Axel Constant, Karl Friston Ines Hipolito

  2020
  ≈ 83%
• Free energy and inference in living systems

  Chang Sub Kim

  2022
  ≈ 83%
• Self-Evidencing Through Hierarchical Gradient Decomposition: A Dissipative System That Maintains Non-Equilibrium Steady-State by Minimizing Variational Free Energy

  Michael James McCulloch

  2025
  ≈ 83%
• Markov Blankets in the Brain

  Maxwell Ramstead, Laura Convertino, Anjali Bhat, Karl Friston, Thomas Parr Ines Hipolito

  2020
  ≈ 83%
• Internalized Morphogenesis: A Self-Organizing Model for Growth, Replication, and Regeneration via Local Token Exchange in Modular Systems

  Takeshi Ishida

  2026
  ≈ 83%
• How causal analysis can reveal autonomy in models of biological systems

  Hyunju Kim, Sara I. Walker, Giulio Tononi and Larissa Albantakis William Marshall

  2018
  ≈ 82%
• Active Inference: A Process Theory

  in corpus

  2017
  ≈ 82%
• Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. II: construction of SDS with nonlinear force and multiplicative noise

  cited

  2008
  ≈ 82%
• The Physical Basis of Prediction: World Model Formation in Neural Organoids via an LLM-Generated Curriculum

  Brennen Hill

  2025
  ≈ 82%
• Integrated Information and Autonomy in the Thermodynamic Limit

  Ezequiel Di Paolo Miguel Aguilera

  2019
  ≈ 82%
• Recognition Dynamics in the Brain under the Free Energy Principle

  Chang Sub Kim

  2019
  ≈ 82%
• Active Inference and Intentional Behaviour

  Tommaso Salvatori, Takuya Isomura, Alexander Tschantz, Alex Kiefer, Tim Verbelen, Magnus Koudahl, Aswin Paul, Thomas Parr, Adeel Razi, Brett Kagan, Christopher L. Buckley, and Maxwell J. D. Ramstead Karl J. Friston

  2023
  ≈ 82%
• Topological constraints on self-organisation in locally interacting systems

  in corpus

  2025
  ≈ 82%
• Active Inference with a Self-Prior in the Mirror-Mark Task

  in corpus

  2026
  ≈ 81%
• Self-Improvising Memory: A Perspective on Memories as Agential, Dynamically Reinterpreting Cognitive Glue

  in corpus

  2024
  ≈ 81%
• Topological constraints on self-organization in locally interacting systems

  in corpus

  2026
  ≈ 81%
• Potential in stochastic differential equations: novel construction

  cited

  2004
  ≈ 81%
• The computational boundary of a 'self': developmental bioelectricity drives multicellularity and scale-free cognition

  in corpus

  2019
  ≈ 81%
• There is no self-evidence: A physics of emptiness realisation

  in corpus

  2026
  ≈ 80%
• A Free energy principle for the brain (lecture summary)

  in corpus

  2008
  ≈ 80%
• Active Inference, Curiosity and Insight

  in corpus

  2017
  ≈ 80%
• Exploration Through Introspection: A Self-Aware Reward Model

  in corpus

  2026
  ≈ 80%
• Design for an Individual: Connectionist Approaches to the Evolutionary Transitions in Individuality

  in corpus

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