In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. 7 and 8, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. 9, the comparison of results is provided with reference to other existing research.

中文翻译：

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一组简谐振子的稳定性分析

在本文中，我们研究了与位移平方成正比的谐振子势配置的稳定性。我们推导出由于参数变化而导致的配分函数波动的表达式，即。振荡器的质量、温度和频率。在这里，我们引入了配分函数的 Hessian 矩阵作为从参数空间到实数集的模型嵌入函数。在这个框架中，我们对谐振子波动的参数空间中的区域进行分类，在这些区域中它们产生稳定的统计配置。稳定性机制源于涨落理论的概念。在秒。7和8，我们提供了系统产生稳定或不稳定统计基础或经历几何相变的局部和全局相关性和稳定区域的性质。最后，在秒。9、结果对比参考其他已有研究。