Izmailian N Sh, Huang Ming-Chang
Department of Physics, Chung-Yuan Christian University, Chungli 320, Taiwan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Jul;82(1 Pt 1):011125. doi: 10.1103/PhysRevE.82.011125. Epub 2010 Jul 19.
We analyze the exact formulas for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximally separated nodes on an M×N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/s (R(M×N))(r,s) = c(ρ)ln S + c(0)(ρ,ξ) + ∑(p=1)(∞) (c(2p)(ρ,ξ))/S(p) with S = MN, ρ = r/s and ξ = M/N. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ξeff = square root(ρ)ξ for free and periodic boundary conditions and ξeff = square root(ρ)ξ/2 for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ξeff→1/ξeff.
我们分析了吴[《物理学报A》37, 6653 (2004)]所得到的在自由、周期和圆柱边界条件下矩形电阻网络中任意两个节点间电阻的精确公式。基于此表达式，我们接着应用伊瓦什凯维奇、伊兹迈利安和胡[《物理学报A》35, 5543 (2002)]的算法，推导出在具有电阻r和s的二维空间方向的M×N矩形电阻网络中两个最大分离节点间电阻的精确渐近展开式。我们的结果是1/s (R(M×N))(r,s) = c(ρ)ln S + c(0)(ρ,ξ) + ∑(p = 1)(∞) (c(2p)(ρ,ξ))/S(p)，其中S = MN，ρ = r/s且ξ = M/N。该展开式中的所有系数均通过解析函数表示。对于自由和周期边界条件，我们引入了有效纵横比ξeff = square root(ρ)ξ，对于圆柱边界条件，ξeff = square root(ρ)ξ/²，并表明所有有限尺寸修正项在变换ξeff→1/ξeff下是不变的。
