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First page of More particles or the pair? An exact test of how to improve mode-coupling theory

More particles or the pair? An exact test of how to improve mode-coupling theory

Hu Cang

cond-mat.soft Nov 9, 2026 · v1 physics.optics
formalization physics
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TL;DR
Finite-order bounds and a factorial estimate for the generalized mode-coupling hierarchy are proved as machine-checked theorems in Lean 4.
Abstract
How a dense liquid arrests into a glass remains a central problem in physics. Mode-coupling theory (MCT), the leading microscopic theory of the glass transition, fails to reproduce mean-field caging. Two routes are proposed to repair it: tracking more-particle correlations, or enforcing repulsion within each colliding pair. Testing both on an exact benchmark liquid, we show that MCT lets touching spheres pass through each other at all orders: adding particles dilutes memory and worsens the transition, whereas restoring pair impenetrability recovers mean-field arrest and the cage size.
Problem

Mode-coupling theory (MCT) of the glass transition fails to reproduce mean-field caging. Two repairs have been proposed: tracking more-particle correlations, or enforcing repulsion within each colliding pair. Neither had been tested on an exact benchmark.

Approach

Both routes are tested on an exact benchmark liquid with known mean-field behavior. A generalized MCT hierarchy with n-particle memory kernels is analyzed. Finite-order bounds and a factorial estimate on the hierarchy are proved as machine-checked theorems in Lean 4. A pair-impenetrability closure is then constructed and compared against the exact solution.

Results

At every order, MCT lets touching spheres pass through each other. Adding more particles dilutes memory and worsens the predicted transition. Restoring pair impenetrability recovers mean-field arrest and the correct cage size.