Wittmer J P, Semenov A N, Baschnagel J
Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France.
Phys Rev E. 2023 Jul;108(1-2):015002. doi: 10.1103/PhysRevE.108.015002.
Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.
各向同性系统中二阶张量场分量的相关函数可以简化为一个由几个不变相关函数（ICF）表征的各向同性四阶张量场。需要强调的是，该场的分量通常取决于场矢量变量的坐标，因此也取决于坐标系的取向。这些角度依赖性与普通各向异性系统的不同。作为获得ICF过程的一个简单例子，我们讨论了各向同性玻璃中时间平均应力的相关性，其中在倒易空间中只有一个ICF在大采样时间和小波矢时成为一个有限常数e。结果表明，e由垂直于波矢的冻结应力分量的典型尺寸决定，即它是由每个独立构型的应力对称性破缺引起的。利用所给出的关于各向同性张量场的一般数学形式，这一发现反过来解释了在实空间中观察到的长程应力相关性。在额外但相当一般的假设下，e被证明由一个热力学量，即平衡杨氏模量E给出。因此，对于某些各向同性非晶物体，我们将有限杨氏模量或剪切模量的存在与倒易空间中应力分量的对称性破缺联系起来。
