构形空间中微观态密度的随机矩阵理论研究

Koretaka Yuge, Kazuhito Takeuchi, Tetuya Kishimoto
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引用次数: 3

摘要

在经典系统中,我们最近的理论研究为系统的空间约束如何与宏观性质联系提供了新的见解,从而导致无序状态下平衡宏观物理性质和结构的普遍表征。这些重要的特征依赖于这样一个事实,即微观态密度(DOMS)在构型空间中的统计相互依赖性在热力学极限下在广泛的空间约束下在数值上消失,而DOMS的这种行为到目前为止还没有得到很好的定量理解。本研究从理论上解决了这一问题,基于高斯正交集合的随机矩阵,在数学上保证了相应的统计独立性。利用广义Ising模型,我们证实了随系统规模的增大,随机矩阵的协方差矩阵的本征态密度(DOE)的低阶矩与随机矩阵的低阶矩具有渐近性。这一结果支持了我们发展的理论方法,其中无序状态下的平衡宏观性质可以分解为给定系统中足够多的组分的每个广义坐标的单个贡献,从而通过几个特殊的微观状态来表示平衡宏观性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Theoretical study on density of microscopic states in configuration space via Random Matrix
In classical systems, our recent theoretical study provides new insight into how spatial constraint on the system connects with macroscopic properties, which lead to universal representation of equilibrium macroscopic physical property and structure in disordered states. These important characteristics rely on the fact that statistical interdependence for density of microscopic states (DOMS) in configuration space appears numerically vanished at thermodynamic limit for a wide class of spatial constraints, while such behavior of the DOMS is not quantitatively well-understood so far. The present study theoretically address this problem based on the Random Matrix with Gaussian Orthogonal Ensemble, where corresponding statistical independence is mathematically guaranteed. Using the generalized Ising model, we confirm that lower-order moment of density of eigenstates (DOE) of covariance matrix of DOMS shows asymptotic behavior to those for Random Matrix with increase of system size. This result supports our developed theoretical approach, where equilibrium macroscopic property in disordered states can be decomposed into individual contribtion from each generalized coordinate with the sufficiently high number of constituents in the given system, leading to representing equilibrium macroscopic properties by a few special microscopic states.
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