Universal Thermodynamic Interatomic Potentials for Crystalline Materials

Free energies govern solid-state phase stability, yet computational materials discovery still relies largely on ground-state energies because free energy calculations require ensemble averages. We introduce the thermodynamic interatomic potential (TIP), which extends an interatomic potential from its static energy to a thermodynamically consistent Gibbs free energy model, with thermodynamic responses following from temperature and pressure by automatic differentiation. We implement TIP[UMA] using the universal potential UMA, train it on free energies from quasi-harmonic to molecular dynamics fidelity, and calibrate it to higher-resolution calculations or experiment. From a single evaluation, it returns the equation of state of a crystal and locates phase transitions among competing branches, including dynamically stabilized phases. Fine-tuning extends the model to alloy solubility limits and miscibility gaps. TIP makes the free energy as accessible as the potential energy, opening finite-temperature phase stability to high-throughput discovery.

Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<β\leqβ_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<β\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<β\leqβ_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

Free energy landscape of Dense Associative Memory

Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.

BMFA: Boundary-Minority Free-Energy Adaptive Screening

Vision Transformers process spatially redundant tokens efficiently only when coarse token summaries preserve the evidence required by exponential attention aggregation. We identify a boundary-minority underestimation failure in which a spatially small, high-response region contributes dominant Gibbs mass while remaining nearly invisible to a block mean. We formalize the failure through the discrepancy between normalized log-mean-exp free energy and mean summarization, prove that minority Gibbs mass can remain non-vanishing as its spatial support and mean contribution vanish, and characterize the limitations of finite-order moment corrections. Building on the resulting analysis, we introduce Boundary-Minority Free-Energy Adaptive Screening (BMFA), which constructs a hierarchical piecewise-constant approximation and recursively refines blocks according to a computable lower-bound increment of local free energy. Controlled synthetic tests, COCO and LVIS diagnostic probes, closed-loop DeiT-Tiny evaluations, and ImageNet-1K experiments establish a consistent evidence chain. BMFA reduces the mean synthetic underestimate from 2.582 to 0.261 at a 5.794% leaf ratio, lowers the COCO image-edge mean gap from 2.254 to 0.526, and preserves 71.520% ImageNet Top-1 accuracy at a 55.861% leaf ratio. The current prototype evaluates selection quality after full QK computation; the reported leaf ratio therefore characterizes representation granularity rather than verified sparse-kernel speedup.

Gradient descent at the Edge of Stability: free energy model and kinetic description of the two-layer network

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the loss and the sharpness. We propose a continuous-time effective model that tracks the evolution of the average trajectory coupled with the time-averaged covariance of its fast oscillations. Our analysis reveals that the natural quantity to monitor in such unstable regimes is an effective free energy, which combines the original risk functional with a curvature-related "entropic" term. Our model allows us to track the envelope of the oscillations even in situations where its dynamics evolve on similar timescales as the averaged weights. Otherwise stated, we can track the spikes that occur during the training of some neural network architectures. For wide two-layer neural networks optimized under stable non-vanishing oscillations, we derive a mean-field limit that results in a novel kinetic equation describing the joint distribution of weights and their fluctuations. We show that this equation can be interpreted as a Wasserstein-2 gradient flow of a macroscopic free energy. Finally, we provide numerical evidence on matrix factorization and deep learning tasks (CIFAR-10) to demonstrate the model's accuracy in capturing the envelope of the oscillations and the predictive power of the effective free energy.

Phase transitions for the noisy transformer model in arbitrary dimension

We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $β_*^{(d)}>0$ such that \begin{equation*} \frac{I_{d/2+1}(β_*^{(d)})}{I_{d/2}(β_*^{(d)})}=\frac1d, \end{equation*} where $I_ν$ is the modified Bessel function of the first kind. For $0<β\le β_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation*} K_\#^{(d)}(β)=\frac{β^{d/2}}{2^{d/2}Γ(d/2)I_{d/2}(β)}, \end{equation*} and the phase transition is continuous. For $β>β_*^{(d)}$, the uniform density is not globally minimizing at $K_\#^{(d)}(β)$, so the critical coupling satisfies $K_c<K_\#^{(d)}(β)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner--Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk--Hecke/Bessel coefficient computation and a degree-two quartic obstruction.