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Atom as an Agent: Emergent Order by Collective Predictive Coding

Koji Hashimoto, Tadahiro Taniguchi

arXiv:2610.36673v1cond-mat.dis-nncond-mat.stat-mechcs.AIhep-th

TL;DR

The paper asks how macroscopic order parameters emerge from microscopic systems when their existence is not explained by the microscopic ingredients alone. It applies collective predictive coding by treating atoms as agents that infer and communicate through a shared variable. Across classical spin examples, the learned collective predictive order recovers known order parameters, and in Ising models its equilibrium distribution yields an emergent Landau-like potential.

  • Problem

    The reason macroscopic order emerges from microscopic systems remains unrevealed, despite emergence’s importance in physics.

    Source: §1 · p. 2

  • Method

    The Atom-as-an-Agent principle applies collective predictive coding by treating atoms as agents that infer a shared variable from observed spin configurations.

    Source: §1 · p. 2, §6 · p. 30

  • Results

    The learned collective predictive order recovers known order parameters across classical spin models, while Ising-model equilibrium distributions yield an emergent Landau potential.

    Source: §6 · p. 29

  • Takeaways & Limitations

    The examples demonstrate that macroscopic order parameters can emerge from spin observations without using the microscopic Hamiltonian’s structure.

    Source: §6 · pp. 30–31

    The demonstrated applications are classical systems; extending the approach to quantum systems is identified as important future work.

    Source: §6 · p. 31

Abstract

from arXiv · show

What is the intrinsic reason of emergence of macroscopic order parameter out of microscopic systems? In this article, we apply to physics the collective predictive coding (CPC), the theory of language/symbol emergence. As our language emerges out of our human communications, physical macroscopic order parameter emerges out of communications among atoms --- each atom is regarded as an agent who predicts each other's behavior through inference. We present three examples of classical spin systems: two-dimensional ferromagnetic and antiferromagnetic Ising models, a two-dimensional spin model with competing stripe orders, and the fully connected Sherrington--Kirkpatrick spin-glass model. In all of these examples, proper macroscopic order parameters including the replica overlap emerge automatically through communication informatics of CPC, only from observed spin configurations. We call the learned macroscopic coordinate the collective predictive order. Furthermore, for the ferromagnetic and antiferromagnetic Ising models, the equilibrium distribution of the collective predictive order yields an emergent potential with the symmetry and thermal evolution characteristic of a Landau potential. Notably, the emergence of the collective predictive order does not refer to any structure of microscopic Hamiltonians/Lagrangians. Thus our ``Atom-as-an-Agent" (AaaA) principle offers a completely data-driven method for the emergence of macroscopic order, that is, machine learning physics.

1. The problem of order parameter emergence

The paper frames the unexplained emergence of macroscopic order as a problem of how collective properties arise from microscopic systems. It applies collective predictive coding (CPC) to treat atoms as communicating agents, learning shared order parameters from observed spin configurations without Hamiltonian structure.

1. The problem of order parameter emergence

  • CPC is proposed as a mechanism for order-parameter emergence because shared physical notions, like language, develop through communication among agents.The paper identifies the unexplained existence of emergence as the motivating problem.Source: §1 · p. 2
  • The collective predictive order recovers known order parameters from spin observations alone across Ising, competing-stripe, and Sherrington–Kirkpatrick models, agreeing closely up to a nonlinear map.The recovered quantities include magnetization, staggered magnetization, the competing stripe orders, and replica overlap.Source: §1 · p. 3
  • The method is data-driven: it needs observed spin configurations, not the microscopic Hamiltonian’s structure or symmetries, to discover order parameters.This contrasts with textbook introductions of order parameters through Hamiltonian symmetries.Source: §1 · p. 3
  • For ferromagnetic and antiferromagnetic Ising models, the learned coordinate’s equilibrium distribution yields Landau-like potentials that change from two wells to one on heating.The resulting finite-size transition-temperature estimates are consistent with the known critical point.Source: §1 · p. 3
  • Local proposals, assessments by neighboring sites, and updates to local prediction models learn a shared ferromagnetic pattern from random initial weights.The authors call this communication procedure the Metropolis-Hastings naming game (MHNG).Source: §1 · pp. 3–4

2. Collective predictive coding

Collective predictive coding links agents’ separate observations through a shared latent variable, which is learned by minimizing an information-theoretic free energy. Applied to spin configurations, this procedure infers and learns a collective coordinate that can be read as a candidate order parameter.

2. Collective predictive coding

  • CPC models communication among agents through a shared variable that links their distinct observations into a collective description.The apple example illustrates how communication and perceptual learning establish a shared sign grounded in individual perspectives.Source: §2 · p. 4

2.1. Local observations and a shared variable

  • Each configuration supplies fixed local observations, and a shared latent variable w connects each site’s local probabilistic prediction model to the others.The observations are conditionally independent given w; correlations are represented through that common variable.Source: §2.1 · pp. 4–5

2.2.1. Variational free energy.

  • The variational free energy is minimized over configuration-specific distributions q_a(w) and shared model parameters α and θ, with the observed data held fixed.It is a dimensionless information quantity; the Hamiltonian and temperature enter only indirectly through the observed configurations.Source: §2.2.1 · pp. 5–6

2.2.2. Inference of latent states.

  • At fixed model parameters, minimizing over q_a yields the posterior distribution p(w | S(a)) for the observed configuration, providing inference of the latent state.Restricting q_a to a chosen family gives an approximate inference scheme.Source: §2.2.2 · p. 6

2.2.3. Learning the configuration distribution.

  • At fixed inferred distributions, learning updates shared parameters so the model’s configuration distribution approaches the empirical frequency distribution in KL divergence.Regularization balances this fit against constraints on the parameters.Source: §2.2.3 · p. 7

2.2.4. Numerical strategy: Alternating minimization and collective prediction.

  • The numerical procedure alternates inference and parameter-learning updates until convergence, but different initializations may reach different local minima.With approximate inference, the corresponding variational bound is optimized and the unrestricted equality may not be attained.Source: §2.2.4 · p. 7

2.3. Extracting the latent constituents

  • After learning, the conditional distribution of w given a new configuration provides a macroscopic coordinate called the collective predictive order.Its physical significance is assessed through predictive power, symmetries, and phase transitions; its form depends on the chosen model class.Source: §2.3 · p. 8

3. Emergent order parameter

Across Ising, competing-stripe, and SK spin systems, CPC learns collective coordinates from spin configurations without Hamiltonians or known order parameters; these coordinates recover familiar macroscopic order parameters, including replica overlap in the SK model, and support prediction of withheld spins.

3.1. A local probabilistic prediction model and the learned coordinate

  • For Ising observations, CPC uses site-specific biases and learned weights to predict spins from a latent coordinate; its conditional mean is tanh(bi + Wi · w).The conditional mean of the inferred latent coordinate is used as the collective predictive order; for K = 1, it depends on spins through a weighted sum, with uniform or alternating weights corresponding to magnetization or staggered magnetization.Source: §3.1 · pp. 8–9

3.2. Two criteria for a successful discovery

  • Discovery is assessed by agreement with known order parameters and by masked-spin prediction gain against an independent-site reference; positive gain indicates better CPC predictions.Training, validation, and test sets separate parameter learning, model selection, and final assessment; CPC receives no temperature labels, coupling constants, lattice coordinates, or known order parameters.Source: §3.2 · pp. 10–11

3.3. Ferromagnetic and antiferromagnetic order

  • Learned Ising weights align with uniform ferromagnetic or staggered antiferromagnetic patterns, and the coordinate tracks the corresponding magnetization through a nonlinear map.The normalized overlaps with the correct patterns are 0.99913 and 0.99916; overlaps with the incorrect patterns are below 2.5 × 10^-4.Source: §3.3 · pp. 11–13, Figure 2 · p. 12
  • For Ising models, prediction gain is about 0.651 at T = 1.5 and 0.28 at Tc, but only 1.5 × 10^-4 and 6.1 × 10^-4 at T = 3.5.The high-temperature coordinate still tracks magnetization fluctuations but provides essentially no predictive improvement over independent-site predictions.Source: §3.3 · p. 13, Figure 3 · p. 13

3.4. Two competing stripe orders

  • For stripe order, K = 2 spans both patterns (O = 0.9963), whereas K = 1 captures one direction (O = 0.4973); at T = 1.2, K = 2 also yields greater prediction gain.Prediction gains are 0.6557 for K = 2 and 0.4234 for K = 1 per hidden spin; the learned coordinates reconstruct (mx, my) with R2 = 0.9861. At T = 3.6, K = 2 has negligible or slightly negative gain (∆NLL ≃ −0.005). The dimension comparison is within the linear local-field model family.Source: §3.4 · pp. 14–15, Figure 4 · p. 15

3.5. Replica overlap in a spin glass

  • In the SK model, 98.55% of the squared norms of the non-common weight contrasts lies in the overlap channel, and the learned coordinate maps to replica overlap with test-set R2 = 0.99200.The test coupling realizations were unseen during CPC training and polynomial fitting.Source: §3.5 · p. 16
  • R2 = 0.99200 for the fitted odd-quintic map from replica overlap q12 to inferred collective predictive order bw on SK test configurations.The 960 configurations span eight temperatures and 12 independently generated coupling realizations; the map was fitted on separate validation realizations and held fixed.Source: Figure 5 · p. 17
  • At T = 0.55, CPC lowers prediction NLL by ∆NLL = 0.1289 per hidden site pair versus the independent-site reference, with positive gains across all 12 test coupling realizations.Shuffling the second replica’s sites removes the predictive improvement.Source: §3.5 · p. 17
  • Across all three examples, configuration snapshots recover familiar macroscopic order parameters, with physical relevance supported by prediction of observations withheld from inference.Results depend on the chosen local probabilistic prediction model, observation structure, and candidate latent dimensions.Source: §3.5 · p. 17

4. Emergent potential

The equilibrium distribution of the learned collective predictive order defines a constrained Landau potential for the ferromagnetic and antiferromagnetic Ising models. These data-derived potentials reproduce the models’ thermal evolution and yield finite-size transition estimates consistent with the known critical point.

4. Emergent potential

  • For each model and lattice size, CPC learns the map from configurations at several temperatures, then fixed-temperature test distributions determine the potential.The learned map is held fixed while test configurations are grouped by temperature; smoothing and a negative logarithm relative to the maximum density produce the numerical potential.Source: §4 · p. 18, §4.1 · pp. 18–19

4.1. Emergent Landau potential

  • The equilibrium distribution of the learned collective predictive order defines a constrained Landau potential, with frequent values forming wells and rare intermediate values forming barriers.The construction uses relative frequencies of inferred values from equilibrium configurations.Source: §4.1 · pp. 18–19
  • The potential is expressed in collective predictive order, not magnetization, and a nonlinear coordinate change can alter its shape and number of local minima.The potential is reconstructed from the learned coordinate’s equilibrium distribution.Source: §4.1 · p. 19

4.2. Emergent potential of Ising models and emergent symmetry

  • The reconstructed Ising potentials approximately preserve Z2 symmetry without the model being told the Hamiltonian symmetry.At low temperature they have two wells; on heating, the wells approach the origin and give way to one well near T = 2.55.Source: §4.2 · pp. 20–21

4.3. Landau coefficients and critical temperature estimate

  • The quadratic coefficient changes from negative to positive while the quartic coefficient remains positive, marking a transition in the fitted potential.The fits use |bw|/w0,L ≤ 0.9 and βΦ ≤ 6; temperatures above 2.55 are excluded because the observed coordinate range cannot reliably constrain the quartic term.Source: §4.3 · p. 21
  • The quadratic-coefficient extrapolations yield 95% intervals [2.203, 2.300] and [2.223, 2.298], both containing the exact Tc = 2.269185.The estimates use the temperature where the fitted quadratic coefficient vanishes and finite-size extrapolation.Source: §4.3 · pp. 21–22

5. Agentic communication between atoms

Section 5 moves CPC from centralized inference to communication among sites, then shows that learned local models recover a shared ferromagnetic coordinate that closely tracks magnetization.

5.1. A shared variable proposed and tested by individual spins

  • In four-neighbor MHNG, one site proposes a shared value from its own spin-conditioned distribution; each neighbor tests it, and any rejection leaves the state unchanged.All four listeners must accept for the shared state to change; acceptance uses a Metropolis–Hastings test based on how well the proposal explains each listener’s spin.Source: §5 · p. 23, Figure 8 · p. 24, §5.1 · p. 25

5.2. Numerical steps for learning

  • For L = 32 (N = 1024), training spans 13 temperatures from 1.5 to 3.5; each epoch records eight communication samples per configuration before sitewise parameter updates.The training set contains 3,120 configurations, with 780 for validation and 2,600 for testing.Source: §5.2 · pp. 25–26

5.3. The order parameter obtained through communication

  • Starting from random signs, local learning yields nearly uniform weights: overlap rises from 0.01033 to 0.999726, with the other eight runs ending between 0.999708 and 0.999748.The selected run’s final weights have mean 0.515033 and spatial standard deviation 0.012052.Source: §5.3 · p. 27, Figure 9 · p. 27
  • On 2,600 test configurations, the local-readout average correlates with magnetization at |corr(¯wloc, mF)| = 0.9999996, and their normalized temperature curves nearly overlap.The correlation is reported for the test configurations; the temperature curves are each normalized by their own value at T = 1.5.Source: §5.3 · p. 28, Figure 10 · p. 28

6. Conclusion and discussion

The paper concludes that CPC can recover diverse macroscopic order parameters from spin configurations and reconstruct Landau-like potentials for Ising models. Its communication experiment shows that local learning can also produce a shared variable that tracks ferromagnetic order.

Conclusion and discussion

  • CPC recovers magnetization, staggered magnetization, competing stripe orders, and replica overlap; for the Ising models, its emergent Landau potential evolves from two wells to one on heating.Extrapolating the temperature where the potential’s quadratic coefficient vanishes gives transition temperatures consistent with the known exact value.Source: §6 · p. 29
  • Four-neighbor communication learns nearly uniform weights and a shared variable that closely tracks magnetization, but it need not reproduce centralized inference’s shared-variable distribution.The four-neighbor stationary distribution generally differs from the centralized distribution for a fixed configuration and parameters.Source: §6 · p. 29
  • AaaA learns a shared variable that explains local observations, rather than using phase classification as its learning objective or extracting a coordinate solely as a configuration mapping.The paper contrasts this emphasis with phase-classification approaches and PCA or variational autoencoders.Source: §6 · pp. 30–31
  • The construction makes each microscopic site a participant: local prediction models connect observations through a shared variable that agents learn to use.The paper presents this as a concrete realization of the Atom-as-an-Agent principle.Source: §6 · pp. 29–31
  • The examples are classical; proposed extensions include quantum and topological systems, holographic spacetime, and communication architectures reflecting partial experimental access.A topological extension would require observations and local prediction models through which a shared variable can represent topological information.Source: §6 · pp. 31–32
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