Atom as an Agent: Emergent Order by Collective Predictive Coding

Koji Hashimoto; Tadahiro Taniguchi

cond-mat.dis-nnOct 27, 2026

Atom as an Agent: Emergent Order by Collective Predictive Coding

Authors: Koji Hashimoto, Tadahiro Taniguchi

Organizations: Physics Department, Kyoto University, Kyoto 606-8502, JAPAN · Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JAPAN

Abstract

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.

Figures & tables

Figure 1: E and M updates of the regularized free energy for the selected ferromagnetic (left) and antiferromagnetic (right) cases. The first ten epochs are shown. Blue circles mark individual Newton updates within an M-step; orange diamonds mark the subsequent E-step, with epoch zero denoting the initial E-step. Intermediate Newton updates are spaced evenly within each epoch for display; the horizontal axis is not elapsed time. The vertical axis is the difference from the final recorded free energy, divided by the number of training configurations and sites, on a logarithmic scale.

\mathcal{F}_{\mathrm{last}}

is the value at numerical termination, not an assumed exact minimum.

Figure 2: Spatial weights $W_{i}$ learned independently from ferromagnetic (left) and antiferromagnetic (right) configurations. The uniform and staggered patterns are results of learning, rather than constraints.

Figure 3: Temperature dependence and predictive performance of the learned Ising coordinate. (a) Solid curves: mean absolute collective predictive order divided by its value at

T=1.5

. Dashed curves: mean absolute magnetization for the ferromagnet (F) and staggered magnetization for the antiferromagnet (AF), without rescaling. The nonlinear maps in Eq. ( 23 ) relate the learned coordinate to these conventional order parameters. (b) Prediction gain

\Delta\mathrm{NLL}

, Eq. ( 20 ), for the hidden

75\%

of spins given the observed

25\%

. Positive values indicate a lower prediction NLL than the independent-site reference. Vertical dotted lines mark

T_{c}

Figure 4: Collective predictive order (

K=2

) inferred from low-temperature stripe configurations at

T=1.2

1.4

, and

1.6

. Each point represents one test configuration. The separate four sectors associated with

+m_{x}

-m_{x}

+m_{y}

, and

-m_{y}

are observed. Colors identify the conventional stripe sectors after learning; these labels were not supplied to CPC.

Figure 5: Relation between the replica overlap

q_{12}

and the inferred collective predictive order

\widehat{w}

in the fully connected SK model with

N=32

spins. Each point represents one replica-pair configuration from the test set. The 960 configurations comprise ten samples at each of eight temperatures,

0.55\leq T\leq 2.0

, for each of 12 independently generated realizations of the random couplings

\{J_{ij}\}

. The two replicas in each pair share the same couplings. The black curve is the odd-quintic map from

q_{12}

\widehat{w}

, fitted using the separate validation set of coupling realizations and then held fixed. It gives

R^{2}=0.99200

on the test configurations shown here.

Figure 6: Emergent Landau potentials of the collective predictive order at $L=32$ . Top: ferromagnet; bottom: antiferromagnet. From left to right, the temperatures are $T=2.22$ , $T=T_{c}=2/\log(1+\sqrt{2})\simeq 2.269185$ , $T=2.40$ , and $T=2.55$ , in units $|J|=k_{\rm B}=1$ . Gray solid curves show the potentials reconstructed using Eq. ( 36 ); colored dotted curves show the quartic fits of Eq. ( 40 ). The horizontal axis is $\widehat{w}/w_{0,L}$ , where $w_{0,L}$ is defined in Eq. ( 39 ), and each reconstructed potential has its minimum set to zero. Fit coefficients are listed in Table 1 .

Model	$T$	$a_{0}$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$
F	2.220	4.6	2.7	$-40$	$-15$	95
F	2.269	2.9	1.0	$-30$	$-12$	96
F	2.400	0.5	1.0	$-9$	$-17$	112
F	2.550	0.2	$-3.3$	35	10	59
AF	2.220	4.7	4.8	$-39$	$-26$	95
AF	2.269	2.9	2.4	$-28$	$-15$	81

Table 1: Representative quartic coefficients at

L=32

, in the coordinate and potential convention of Eqs. ( 39 ) and ( 40 ). F and AF denote the ferromagnet and antiferromagnet. Digits are rounded to reflect the sampling uncertainty.

	Ferromagnet		Antiferromagnet
$L$	$T_{*}(L)$	95% interval	$T_{*}(L)$	95% interval
12	2.512	[2.487, 2.595]	2.493	[2.467, 2.565]
16	2.513	[2.479, 2.592]	2.518	[2.485, 2.633]
24	2.485	[2.458, 2.560]	2.484	[2.451, 2.521]
32	2.413	[2.393, 2.435]	2.406	[2.386, 2.431]
48	2.356	[2.345, 2.372]	2.345	[2.338, 2.358]

Table 2: Temperatures

T_{*}(L)

at which the fitted quadratic Landau coefficient vanishes. Brackets give the 95% intervals from 300 resamplings of the test configurations.

Figure 7: Finite-size extrapolation of $T_{*}(L)$ , the temperature at which the linear fit to $a_{2}(T)$ vanishes. Points show the six system sizes; error bars are 95% intervals from resampling test configurations. Dashed lines fit all six sizes, while solid lines fit only $L=32,48,64$ using Eq. ( 42 ). Squares at $1/L=0$ mark the latter extrapolations. The horizontal dotted line is the exact $T_{c}=2.269185\ldots$ .

Figure 8: One communication step of the four-neighbor MHNG. The lower layer shows a fixed observed spin configuration; the upper layer represents values of the single variable shared by all sites. (a) Site $k$ (teal) proposes $v$ from its local conditional distribution, Eq. ( 45 ). (b) Its four nearest neighbors (orange) independently compare $v$ with the current value $w$ using Eq. ( 46 ); orange links depict these local evaluations. (c1) All four listeners accept (check marks), and the shared state changes from $w$ to $v$ . (c2) At least one listener rejects (cross), and the shared state remains $w$ , as indicated by the loop arrow. Panels (c1) and (c2) are alternative outcomes. Filled circles in the upper layer indicate the current shared value. Both outcomes count as one communication step, after which a new proposing site is chosen. The spin configuration stays fixed throughout.

Figure 9: Site weights $W_{i}$ learned with four-neighbor communication, at initialization and epochs 3, 10, and 60. All panels use the same color scale $[-0.6,0.6]$ and the same overall sign convention. Random initial signs evolve into a nearly uniform pattern.

Figure 10: Collective predictive order evaluated by local inference after learning through four-neighbor MHNG. (a) The spatial average $\bar{w}_{\mathrm{loc}}(S)$ of the local proposal means versus magnetization on 2600 test configurations. A common sign is chosen so that the mean learned weight is positive. (b) Temperature dependence of $\langle|\bar{w}_{\mathrm{loc}}|\rangle_{T}$ (open circles, solid line) and $\langle|m_{\mathrm{F}}|\rangle_{T}$ (squares, dashed line), each normalized by its own value at $T=1.5$ . Each temperature has 200 test configurations; the two curves almost overlap.

$T$	$\langle\|\bar{w}_{\mathrm{loc}}\|\rangle_{T}$	$\langle\|m_{\mathrm{F}}\|\rangle_{T}$
$1.5$	0.41602	0.98661
$T_{c}$	0.29536	0.70033
$3.5$	0.02678	0.06350

Table 3: Mean absolute local readout and magnetization on test configurations, with 200 configurations at each temperature.

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Igor Itkin

Independent Researcher, Tel Aviv, Israel

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Simulating societies of many large language model (LLM) agents is expensive, yet the questions asked of such simulations are usually macroscopic: phase behaviour, stylised facts, and scaling with the number of agents

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Abstract

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