AI 中文总结
研究硬球在各维度下熔化、玻璃化和堵塞问题，核心方法是基于三个几何要素推导出无参数主方程，主要贡献是能精准定位相关物理状态，且理论预测的林德曼常数等得到证实。
AI 中文摘要
林德曼准则指出，一旦原子振动达到粒子间间距的十分之一左右，固体就会失去稳定性，这一经验法则已存在一个多世纪。数值由模式耦合和复本理论再现，但从未作为简单可验证论据的结果被分离出来。本文表明，对于d维硬球，该准则源于三个精确的几何要素。通过状态方程确定配位数的接触定理、确定非接触邻居如何投影到逃逸方向的各向同性恒等式以及推导的首次通过论证。由此产生的无参数主方程定位了动力学玻璃化转变、随机密堆积、考兹曼点、玻璃密堆积和平衡晶体熔化，理论预测的林德曼常数和二维中的相关体积分数等已得到独立模拟和实验的证实。
英文摘要
The Lindemann criterion that a solid loses stability once atomic vibrations reach roughly a tenth of the interparticle spacing, has remained an empirical rule for over a century. The numerical value was reproduced by mode-coupling and replica theories but never isolated as the consequence of a simple, verifiable argument. Here we show that for hard spheres in $d$ dimensions the rule follows from three exact geometric ingredients. The contact theorem fixing the coordination number from the equation of state, an isotropy identity fixing how non touching neighbors project onto an escape direction, and a first-passage argument which is derived, in which the elementary hop spans one interparticle spacing rather than one particle diameter. The resulting parameter-free master equation locates the kinetic glass transition, random close packing, the Kauzmann point, glass close packing, and equilibrium crystal melting in $d=3$--$12$, each to within a few per cent of reported independent simulation and replica-theory values, and places all five on a single barrier surface. The theory makes two predictions that are verifiable, the Lindemann constant, $\c_L(3)=0.13$ per neighbor spacing in $3$ dimensions derived from the theory, which must fall systematically with increasing dimensions. The other being in two dimensions, the current theory predicts the arrest in the volume fraction $η_g=0.781$, the jamming at $η=0.832$, and both steps of the two-stage melting scenario, all of which are already corroborated by independent simulations and experiments.
