Recent research shows that Monte Carlo diffusion methods are often the most efficient algorithms for solving certain elliptic boundary value problems. In this paper, we extend this research by providing two efficient algorithms based on the concept of "last-passage diffusion." These algorithms are qualitatively compared with each other (and with the best first-passage diffusion algorithm) in solving the classical problem of computing the charge distribution on a conducting disk held at unit voltage. All three algorithms show detailed agreement with the known analytic solution to this problem.

It is well known that there is no analytical expression for the electrical capacitance of a cube, even though it has been claimed that one can compute this capacitance numerically to high precision. However, there have been some disparities between reported numerical results of the capacitance of the unit cube. In this article, the ''walk on planes'' ͑WOP͒ algorithm ͓M. L. Mansfield, J. F. Douglas, and E. J. Garboczi, Phys. Rev. E 64, 061401 ͑2001͔͒ is used to compute the capacitance of the unit cube. With WOP, we remove the error from the ⑀-absorption layer commonly used in ''walk on spheres'' computations so that there is no inherent error introduced in these WOP computations except the intrinsic Monte Carlo sampling error of size O(N 1/2). This WOP technique comes from the isomorphism, provided by probabilistic potential theory, between the electrostatic Dirichlet problem on a conducting surface, and the corresponding Brownian motion first-passage expectation. The numerical result we obtain with WOP, 0.660 678 2Ϯ1ϫ10 Ϫ7 , supports the deterministic calculations by Read ͓F.

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We describe two efficient methods of estimating the fluid permeability of standard models of porous media by using the statistics of continuous Brownian motion paths that initiate outside a sample and terminate on contacting the porous sample. The first method associates the “penetration depth” with a specific property of the Brownian paths, then uses the standard relation between penetration depth and permeability to calculate the latter. The second method uses Brownian paths to calculate an effective capacitance for the sample, then relates the capacitance, via angle-averaging theorems, to the translational hydrodynamic friction of the sample. Finally, a result of Felderhof is used to relate the latter quantity to the permeability of the sample. We find that the penetration depth method is highly accurate in predicting permeability of porous material.

We develop and test the last-passage diffusion algorithm, a charge-based Monte Carlo algorithm, for the mutual capacitance of a system of conductors. The first-passage algorithm is highly efficient because it is charge based and incorporates importance sampling; it averages over the properties of Brownian paths that initiate outside the conductor and terminate on its surface. However, this algorithm does not seem to generalize to mutual capacitance problems. The last-passage algorithm, in a sense, is the time reversal of the first-passage algorithm; it involves averages over particles that initiate on an absorbing surface, leave that surface, and diffuse away to infinity. To validate this algorithm, we calculate the mutual capacitance matrix of the circular-disk parallel-plate capacitor and compare with the known numerical results. Good agreement is obtained.