论文标题
小世界模型的平均场密度和堵塞的软球模型
Mean-field density of states of a small-world model and a jammed soft spheres model
论文作者
论文摘要
我们考虑了$ D $尺寸中的一类随机块矩阵模型，即$ d \ ge 1 $，这是由在等距点附近的软球振动密度（DOS）的研究中进行的。平均度的联系网络$ z = z_0 +ζ$由随机$ z_0 $ - 惯性图表示（仅$ d = 1 $中的圆形图，带$ z_0 = 2 $），而erdös-renyi图的平均水平$ $ζ$被叠加了。 在情况下，$ d = 1 $，对于$ζ$，带有参数$ z $的转移的kesten-mckay dos是DOS的平均场解决方案。 $ z_0 = 2 $模型中的数值模拟，即$ k = 1 $ newman-watts小世界型号，在$ z_0 = 3 $中，模型导致我们推测，$ζ\ to dos的累积功能均匀地收敛于转移的kesten-mckay-mckay dos $ $ $ $ [0] $ [0] \ sqrt {z_0-1} + 1 $。 对于$ 2 \ le d \ le 4 $，我们引入了一个截止参数$ k_d \ le 0.5 $建模球体排斥。情况$ k_d = 0 $是随机弹性网络情况，DOS接近带有参数的Marchenko-Pastur dos $ t = \ frac {z} {d} $。对于$ k_d $ the the dos the小$ω$的dos to to换移的kesten-mckay dos，带有参数$ t = \ frac {z} {d} $;在等静态情况下，DOS的$Ω= 0 $预期的高原。 $ d = 3 $的玻色子峰频率与$ k_3 $大的频率接近于分子动力学模拟中的$ z = 7 $和$ 8 $。
We consider a class of random block matrix models in $d$ dimensions, $d \ge 1$, motivated by the study of the vibrational density of states (DOS) of soft spheres near the isostatic point. The contact networks of average degree $Z = z_0 + ζ$ are represented by random $z_0$-regular graphs (only the circle graph in $d=1$ with $z_0=2$) to which Erdös-Renyi graphs having a small average degree $ζ$ are superimposed. In the case $d=1$, for $ζ$ small the shifted Kesten-McKay DOS with parameter $Z$ is a mean-field solution for the DOS. Numerical simulations in the $z_0=2$ model, which is the $k=1$ Newman-Watts small-world model, and in the $z_0=3$ model lead us to conjecture that for $ζ\to 0$ the cumulative function of the DOS converges uniformly to that of the shifted Kesten-McKay DOS, in an interval $[0, ω_0]$, with $ω_0 < \sqrt{z_0-1} + 1$. For $2 \le d \le 4$, we introduce a cutoff parameter $K_d \le 0.5$ modeling sphere repulsion. The case $K_d=0$ is the random elastic network case, with the DOS close to the Marchenko-Pastur DOS with parameter $t=\frac{Z}{d}$. For $K_d$ large the DOS is close for small $ω$ to the shifted Kesten-McKay DOS with parameter $t=\frac{Z}{d}$; in the isostatic case the DOS has around $ω=0$ the expected plateau. The boson peak frequency in $d=3$ with $K_3$ large is close to the one found in molecular dynamics simulations for $Z=7$ and $8$.
