Quadrupole Last‐Passage Algorithm for Charge Density on an L‐Shaped Conducting Surface

Abstract

The last-passage (LP) Monte Carlo algorithms for the charge density on an L-shaped conducting surface are further developed by deriving a quadrupole LP Green's function on the L-shaped flat surface. To demonstrate the algorithm, charge densities on an L-shaped conductor are computed in 3D space and it is found that these results agree very well with the ones from the Given-Hwang's original last-passage algorithm on a flat surface. Compared with the Given-Hwang's LP algorithm, the quadrupole LP one is very suitable for charge density near the L-shaped edge boundary.

Octupole Last‐Passage Algorithm for Charge Density on the Fichera‐Corner Conducting Surface

“…Since the analytic solution of the charge densities on the Fichera-corner conducting surface is unknown, we assume that the value obtained from a simulation with 10 12 Monte Carlo steps is the true value. The linear regression slope of the octupole LP algorithm (red line) is −0.50 (1) and that of the previous LP algorithm (blue line) −0.51 (3), which means both algorithms follow the typical Monte Carlo error convergence where the exact slope is well-known to be −1∕2. Also, in Figure 5, we note that the errors from the octupole LP algorithm are about ten times smaller than those from the previous LP algorithm used on the flat surface only.…”

“…In a series of papers, we have been developing last-passage algorithms for a charge density at a specific point on a conducting surface. [1][2][3][4][5] The first last-passage algorithm was developed for charge density at a point on a flat conducting surface. [5] Using the correspondence between an electrostatic potential problem and the Brownian diffusion probability one from the Feynman-Kac formula of probabilistic potential theory [1,6,7] (see Figure 1), one can write the electrostatic potential very near to the conducting surface as…”

“…Figure 7. Charge densities near the corner of the Fichera-corner conducting surface where r is the distance from the along the diagonal line.The slope of the linear regression is 2.00000(1). Each data point is obtained from 10 10 Monte Carlo steps with ten independent runs.…”

Sphere Last‐Passage Algorithm for Charge Density on a Smooth (Convex) Conducting Surface

“…In a series of papers, we have been developing the last-passage algorithms such as last-passage algorithm, [7] off-centered lastpassage algorithm on a flat conducting surface, [8] quadrupole last-passage algorithm on an L-shaped conducting surface, [4] and last-passage algorithm on a spherical conducting surface [9] and so on [10]. In addition, recently we have developed a last-passage algorithm for computing charge distribution over a finite region of a conducting object.…”

“…Figure 2. Schematic diagram for the relation of electrostatic potential 𝜙(r) and the diffusion probability p(r):The potential 𝜙(r) is the potential at r when the spherical conductor is at unit potential and the probability p(r) is the probability of the Brownian particle starting from the distance r and going to infinity without going back to the spherical conductor [4,5]. …”