Statistical mechanics and machine learning of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -Rényi ensemble
Bibliographic record
Abstract
We study the statistical physics of the classical Ising model in the so-called <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>α</a:mi> </a:math> -Rényi ensemble, a finite-temperature thermal state approximation that minimizes a modified free energy based on the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>α</b:mi> </b:math> -Rényi entropy. We begin by characterizing its critical behavior in mean-field theory in different regimes of the Rényi index <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>α</c:mi> </c:math> . Next, we re-introduce correlations and consider the model in one and two dimensions, presenting analytical arguments for the former and devising a Monte Carlo approach to the study of the latter. Remarkably, we find that while mean-field predicts a continuous phase transition below a threshold index value of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>α</d:mi> <d:mo>∼</d:mo> <d:mn>1.303</d:mn> </d:mrow> </d:math> and a first-order transition above it, the Monte Carlo results in two dimensions point to a continuous transition at all <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>α</e:mi> </e:math> . We conclude by performing a variational minimization of the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>α</f:mi> </f:math> -Rényi free energy using a recurrent neural network (RNN) where we find that the RNN performs well in two dimensions when compared to the Monte Carlo simulations. Our work highlights the potential opportunities and limitations associated with the use of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>α</g:mi> </g:math> -Rényi ensemble formalism in probing the thermodynamic equilibrium properties of classical and quantum systems.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".