玻璃化转变统一模型：Lennard-Jones势的不变点研究

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本文探讨了玻璃化转变的统一模型理论，特别是在固-液玻璃化转变过程中，局域集群增长相变中的自相似Lennard-Jones (L-J) 势函数的固定点性质。作者吴嘉麟，来自东华大学材料科学与工程学院，通过分析自相似势函数涨落的“增强-限制”递推方程，证明了这种潜在的固定点存在。以下是主要发现： 1. **固定点的存在性**：在玻璃化转变的局部集群增长相变中，自相似L-J势函数存在一个固定的点。这个固定点对于理解玻璃态物质的结构和动力学特性至关重要。 2. **自我相似硬球秩序**：研究结果显示，在玻璃化转变过程中，仅有8种不同的自相似硬球阶次存在。这揭示了系统在转变过程中的简化结构模式。 3. **通用行为**：两个正交的减缩几何相位势（Reduced Geometric Phase Potentials, RGPP）被观察到，分别是快速的RGPP（3/8）和慢速的RGPP（5/8），它们分别对应于不同的集群生长相变阶段，展示了系统的非平凡动态特征。 4. **稳定性条件**：势函数涨落的强化和限制的稳定性条件即为林德曼比（Lindemann ratio），本文精确地计算出其值为0.1047，这是确定玻璃状态稳定性的关键参数。 5. **界面激发能量**：界面激发的能量保持不变，具体为ε0的1/8。这表明了界面现象在玻璃化过程中起着稳定或驱动作用。 6. **界面激发的起源与转移**：界面激发的出现和转移是由于自相似性的平衡机制，这暗示了潜在的物理过程可能涉及势能分布和集群形成之间的动态相互作用。这篇首发论文深入剖析了玻璃化转变中自相似Lennard-Jones势函数的固定点及其与集群生长、几何相位和稳定性等因素的关联，为我们理解复杂凝聚态系统的行为提供了新的视角。通过这些发现，我们可以更好地设计和预测玻璃材料的性能，并对相关领域的理论模型进行进一步发展。

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Unified model theory of the glass transition II．Fixed point

of self-similar Lennard Jones potentials in the glass

transition

Wu Jialin

College of Material Science and Engineering, Donghua University, Shanghai, China (200051)

E-mail：jlwu@dhu.edu.cn

Abstract

The existence of fixed point for self-similar Lennard-Jones (L-J) potentials, in local cluster growth

phase transition in the glass transition, has been proved using the recursive equation of reinforce-

restraint of self-similar potential fluctuation. A series of results have been directly deduced. (a) There

are only 8 orders of self-similar hard-spheres in the glass transition. (b) A universal behavior: two

orthogonal reduced geometric phase potentials (RGPP), the fast RGPP 3/8 and the slow RGPP 5/8,

accompanied with 8 orders of cluster growth phase transitions. (c) The stability condition for

fluctuation reinforce-restraint is just the Lindemann ratio that is exactly deduced as 0.1047in this paper.

(d) The interface excited energy is an invariable of 1/8

. The origin and transfer of interface excitation

comes of the balance between self-similar L-J potential fluctuation and geometric phase potential

fluctuation, which also gives rise to a new attractive potential of –17/16

, lower than potential well

energy –

, in the self-similar mean field of mean fields of different sizes. (e) The delocalization energy

of two-body is exactly the transfer energy of excited interface, and the delocalization path is along 8

orders of geodesics in topological analyses.

Keywords: Glass transition, Fixed point, Geometric phase, Potential fluctuation, Lindemann ratio

In the previous paper [1], a theoretical framework has been proposed that the glass transition

(GT) is only determined by the intrinsic 8 orders of instant 2-D mosaic geometric structures,

without any presupposition and relevant parameter. According to the theory of thermally excited

ripplon [2] and the results of [3], the interface excitation between two reference particles in GT

has been illustrated. An interface excitation is a vector with 8 orders of relaxation times, 8 orders

of additional restoring torques, quantized interface excited energy (QIEE) of 1/8

(

is potential

well energy) and extra volume. Each order of harmonic restoring torque gives rise to an additional

position-asymmetry of Lindemann displacement [4, 5]; thus, in order to eliminate the additional

position-asymmetry, the accompanied 8 orders of transient 2-D clusters with the 4 neighboring

interface relaxations of the reference particle have been formed. Here, a 2-D cluster is strictly

defined by an instant interface excited loop on projection plane. In order to step by step eliminate

the anharmonic interface tensions, the 8 orders of dynamical 3-D hard-spheres should be adopted.

It is interesting that the physical picture provides a unified mechanism to interpret hard-sphere [6],

compacting cluster [7], non-ergodic [8], free volume [9], cage [10], percolation [11], instantaneous

normal mode [12], geometrical frustration [7] and jamming behaviors [13]. It is clarified [1] that

the clusters in mosaic structures turn out to be the special hard-spheres with the quantized

interface interaction potential of 1/8

and the appearance of 8 orders of 3-D hard-spheres

respectively correspond to the 8 orders of local cluster growth phase transitions (LCGPT) in GT.

However, the central assumption of the intrinsic 8 orders of instant 2-D mosaic geometric

structures is mainly based on the existence of the 8 orders of domain wall vibration frequencies,

proposed by Wolynes and co-workers [3]. In this paper, the independent testification for fixed

point of self-similar Lennard-Jones (L-J) potentials will directly educe that there are only 8 orders

of self-similar hard-spheres in GT and explain the origin of QIEE. The existence of fixed point of

L-J potentials will be a very important theoretical result to out and away simplify discussions on

pairwise interactions, especially in computer simulations. L-J potentials are widely used to

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玻璃化转变统一模型：Lennard-Jones势的不变点研究

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