Percolation of k×k square tiles deposited under equilibrium conditions on square lattices.

IF 2.4 3区 物理与天体物理 Q1 Mathematics
N M De La Cruz Feliz, F O Sanchez-Varretti, N De La Cruz Félix, A J Ramirez-Pastor
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Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study the percolation behavior of square tiles of side k (k-tiles) on two-dimensional square lattices. The k-tiles, containing k×k identical units (each one occupying a lattice site), were reversibly adsorbed on the lattice. The process was monitored by following the probability R_{L,k}(θ) that a lattice composed of L×L sites percolates at a concentration θ of sites occupied by particles of side k. The classical percolation problem is recovered for k=1 giving the well-known site percolation threshold θ_{c,k=1}=0.592746⋯. A slight decrease is observed at the percolation threshold when k goes from 1 to 2, with θ_{c,k=2}=0.58483(9). For k≥2, the percolation threshold θ_{c,k} monotonically increases with k and asymptotically converges toward a definite value for large k-tiles θ_{c,k→∞}≈0.963. Accordingly, the model presents a percolation transition for the whole range of k. This behavior is completely different from that observed for the percolation problem of k×k square tiles irreversibly deposited on square lattices, where the percolation threshold is an increasing function of k in the range of 1≤k≤3. For k≥4, the percolation phase transition disappears. This stark contrast between the behaviors of reversible and irreversible adsorption models is a significant outcome of this research. Our findings could guide future studies exploring possible formation mechanisms in conductivity experiments of composite materials. Finally, the accurate determination of the critical exponents α, β, and γ, along with the measurement of the fractal dimension of the percolating cluster and the shortest-path exponent, indicates that although the deposition mechanism drastically affects the behavior of the percolation threshold with k, it does not alter the nature of the percolation transition occurring in the system. Accordingly, the universality class of the reversible adsorption model was found to be the same as for the random percolation model.

平衡条件下沉积在方格上的k×k方瓦的渗透。
通过数值模拟和有限尺度尺度分析,研究了k边方砖在二维方砖上的渗透行为。含有k×k相同单元(每个单元占据一个晶格位点)的k-tiles被可逆地吸附在晶格上。通过遵循R_{L,k}(θ)的概率来监测该过程,即由L×L位点组成的晶格在侧k粒子占据的位点的浓度θ处渗透。经典的渗透问题在k=1时恢复,给出了众所周知的位点渗透阈值θ_{c,k=1}=0.592746⋯。当k从1到2时,渗流阈值略有下降,θ_{c,k=2}=0.58483(9)。当k≥2时,渗透阈值θ_{c,k}随k单调增大,对于大k瓦片θ_{c,k→∞}≈0.963渐近收敛。因此,该模型在整个k范围内呈现出一个渗透过渡。这种行为与k×k不可逆沉积在方形格子上的方形瓦片的渗透问题完全不同,在1≤k≤3的范围内,渗透阈值是k的递增函数。当k≥4时,渗流相变消失。可逆和不可逆吸附模型之间的鲜明对比是本研究的重要成果。我们的发现可以指导未来的研究探索复合材料电导率实验中可能的形成机制。最后,对临界指数α、β和γ的精确测定,以及对渗透簇的分形维数和最短路径指数的测量表明,尽管沉积机制极大地影响了k的渗透阈值行为,但它并没有改变体系中发生的渗透转变的性质。因此,发现可逆吸附模型的通用性与随机渗透模型相同。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Physical review. E
Physical review. E 物理-物理:流体与等离子体
CiteScore
4.60
自引率
16.70%
发文量
0
审稿时长
3.3 months
期刊介绍: Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.
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