Chakraborty Abhijit, Manna S S
Satyendra Nath Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata-700098, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032103. doi: 10.1103/PhysRevE.89.032103. Epub 2014 Mar 7.
A region of two-dimensional space has been filled randomly with a large number of growing circular disks allowing only a "slight" overlapping among them just before their growth stops. More specifically, each disk grows from a nucleation center that is selected at a random location within the uncovered region. The growth rate δ is a continuously tunable parameter of the problem which assumes a specific value while a particular pattern of disks is generated. When a growing disk overlaps for the first time with at least one other disk, its growth is stopped and is said to be frozen. In this paper we study the percolation properties of the set of frozen disks. Using numerical simulations we present evidence for the following: (i) The order parameter appears to jump discontinuously at a certain critical value of the area coverage; (ii) the width of the window of the area coverage needed to observe a macroscopic jump in the order parameter tends to vanish as δ→0; and on the contrary (iii) the cluster size distribution has a power-law-decaying functional form. While the first two results are the signatures of a discontinuous transition, the third result is indicative of a continuous transition. Therefore we refer to this transition as a sharp but continuous transition similar to what has been observed in the recently introduced Achlioptas process of explosive percolation. It is also observed that in the limit of δ→0, the critical area coverage at the transition point tends to unity, implying that the limiting pattern is space filling. In this limit, the fractal dimension of the pore space at the percolation point has been estimated to be 1.42(10) and the contact network of the disk assembly is found to be a scale-free network.
二维空间的一个区域已被大量不断增长的圆盘随机填充，在它们停止生长之前，彼此之间仅允许“轻微”重叠。更具体地说，每个圆盘都从一个成核中心开始生长，该成核中心在未覆盖区域内的随机位置选取。生长速率δ是该问题的一个可连续调节的参数，在生成特定圆盘图案时它取特定值。当一个正在生长的圆盘首次与至少一个其他圆盘重叠时，其生长停止，并称其被冻结。在本文中，我们研究了冻结圆盘集合的渗流特性。通过数值模拟，我们给出了以下证据：（i）序参量在面积覆盖率的某个临界值处似乎出现不连续跳跃；（ii）观察到序参量出现宏观跳跃所需的面积覆盖率窗口宽度随着δ→0趋于消失；相反，（iii）簇尺寸分布具有幂律衰减的函数形式。虽然前两个结果是不连续转变的特征，但第三个结果表明是连续转变。因此，我们将这种转变称为类似于在最近引入的爆炸渗流的阿克利奥普塔斯过程中所观察到的尖锐但连续的转变。还观察到，在δ→0的极限情况下，转变点处的临界面积覆盖率趋于1，这意味着极限图案是空间填充的。在此极限下，渗流点处孔隙空间的分形维数估计为1.42(10)，并且圆盘组件的接触网络被发现是一个无标度网络。
