Singularities in Hessian element distributions of amorphous media

Vishnu V. Krishnan, S. Karmakar, K. Ramola
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Abstract

We show that the distribution of elements $H$ in the Hessian matrices associated with amorphous materials exhibit cusp singularities $P(H) \sim {\lvert H \rvert}^{\gamma}$ with an exponent $\gamma < 0$, as $\lvert H \rvert \to 0$. We exploit the rotational invariance of the underlying disorder in amorphous structures to derive these exponents exactly for systems interacting via radially symmetric potentials. We show that $\gamma$ depends only on the degree of smoothness $n$ of the potential of interaction between the constituent particles at the cut-off distance, independent of the details of interaction in both two and three dimensions. We verify our predictions with numerical simulations of models of structural glass formers. Finally, we show that such cusp singularities affect the stability of amorphous solids, through the distributions of the minimum eigenvalue of the Hessian matrix.
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非晶介质黑森元分布的奇异性
我们表明,在与非晶材料相关的Hessian矩阵中,元素$H$的分布表现出顶点奇点$P(H) \sim {\lvert H \rvert}^{\gamma}$,其指数为$\gamma < 0$,为$\lvert H \rvert \to 0$。我们利用非晶结构中潜在无序的转动不变性,精确地推导出通过径向对称势相互作用的系统的这些指数。我们表明$\gamma$仅取决于组成粒子之间在截止距离处的相互作用势的平滑程度$n$,而与二维和三维相互作用的细节无关。我们用结构玻璃成形器模型的数值模拟验证了我们的预测。最后,我们通过Hessian矩阵最小特征值的分布证明了这种尖点奇点影响非晶固体的稳定性。
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