Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models

2026-04-20 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

This research investigates phase transitions in repulsive-attractive mean-field free energies on the unit circle, focusing on conditions under which the critical coupling strength $K_{c}$ equals the linear stability threshold $K_{\#}$ of the uniform distribution, and when the phase transition is continuous. The authors prove that for a $\frac{1}{n+1}$-periodic interaction with specific Fourier coefficient decay, $K_{c}=K_{\#}$ and the transition is continuous, with the uniform distribution being the unique global minimizer at criticality. This proof leverages a sharp coercivity estimate derived from the constrained Lebedev\u2013Milin inequality. The findings are applied to three key models: the two-dimensional Doi\u2013Onsager model, where $K_{c}=K_{\#}=3\pi/4$ and the transition is continuous; the noisy transformer model, identifying a sharp threshold $\beta_{*}\approx 2.447$ below which the transition is continuous at $K_{c}(\beta)=K_{\#}(\beta)$, and discontinuous above; and the noisy Hegselmann\u2013Krause model, establishing a similar sharp dichotomy.

Key takeaway

For AI Scientists and Research Scientists modeling complex systems with mean-field interactions, understanding the conditions for continuous versus discontinuous phase transitions is crucial. Your choice of interaction potential and its Fourier characteristics directly impacts system behavior at criticality. Specifically, if your model's interaction potential satisfies the decay conditions outlined, you can expect a continuous phase transition where the critical coupling strength aligns with the linear stability threshold, simplifying equilibrium predictions. Be aware of specific parameter thresholds, like $\beta_{*}$ in transformer models, which dictate the nature of the transition.

Key insights

Phase transitions in multimodal mean-field models are continuous when critical coupling equals linear stability threshold.

Principles

• Critical coupling $K_c$ often equals linear stability threshold $K_{\#}$ for continuous transitions.
• Fourier coefficient decay conditions dictate phase transition continuity.
• Uniqueness of global minimizers at $K_c$ implies continuous phase transitions.

Method

The proof relies on a sharp coercivity estimate for free energy, derived from the constrained Lebedev\u2013Milin inequality, to establish conditions for continuous phase transitions.

In practice

• Doi\u2013Onsager model exhibits continuous phase transition at $K_c=3\pi/4$.
• Noisy transformer model has a critical $\beta_{*}\approx 2.447$ for transition continuity.
• Noisy Hegselmann\u2013Krause model shows similar critical parameter behavior.

Topics

• Phase Transitions
• Mean-Field Free Energy
• Doi–Onsager Model
• Noisy Transformer Model
• Hegselmann–Krause Model

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.