Perturbation of an Infinite Network of Identical Capacitors

Abstract

The capacitance between any two arbitrary lattice sites in an infinite square lattice is studied when one bond is removed (i.e. perturbed). A connection is made between the capacitance and the Lattice Green's Function of the perturbed network, where they are expressed in terms of those of the perfect network. The asymptotic behavior of the perturbed capacitance is investigated as the separation between the two sites goes to infinity. Finally, numerical results are obtained along different directions and a comparison is carried out with the perfect capacitances.

Two-Particle Lattice Green Functions on a Two-Dimensional Rectangular Lattice with Next-Nearest Neighbor Hopping

“…We begin with finding the solution of the simplest of Green functions of the type given by (13), namely G 0,0 . After standard u = tan(k x /2) substitution, integral over k x is replaced by integral over quadratic polynomial in u in the denominator.…”

“…where G m,n means dG m,n (E)/dE and is given by equation similar to (13), just with the denominator squared and minus in front. By integration by parts of one of the integrals in (13) we can get another type of relations involving derivatives [29]:…”

“…They have the general form of Watson integrals [2], which appeared in physics during the research of magnetism in the sixties (particularly Heisenberg model on three dimensional (3d) lattices [3]). Apart from that, LGF's have also found use, e.g., in lattice statistical problems, random walks and percolation theories [4], problems of transport and diffusion in solids [5], theories of impurities in solids [6], lattice dynamics [7], investigations of tight-binding models on the lattice [8], investigations of electron band structures in crystals [9], models of phase transition in particular in 2d and in calculations of resistivity and capacitance in infinite networks of resistors or capacitors [10][11][12][13].…”

Analysis of Fractional-Order 2xn RLC Networks by Transmission Matrices

“…Over the past few decades, researchers have published some papers on the integer circuits networks. However, the last efforts have been focused mainly on analyzing the resistors or capacitors of the fractional order single component circuit networks [6,7]. Comparatively little attention has been paid to the impedance of the fractional-order multiplecomponent circuit networks.…”

Remarks on the Lattice Green Function for the Anisotropic Face Centered Cubic Lattice

“…Nowadays, GF is one of the most important concepts in many branches of physics, as many quantities in solid state physics can be expressed in terms of the LGF. In the following there are some examples: statistical model of ferromagnetism such as the Ising model [5], the Heisenberg model [6], spherical model [7], random walk theory [8,9], diffusion [10], band structure [11], and in recent * corresponding author; e-mail: drjasad@yahoo.com years it becomes an important tool in analyzing infinite electric networks [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”