Abstract
This paper provides a review of a new method of addressing problems in diffusion Monte Carlo: the Green’s function first-passage method (GFFP). In particular, we address three new strands of thought and their interaction with the GFFP method: the use of angle-averaging methods to reduce vector or tensor Laplace equations to scalar Laplace equations; the use of the simulation-tabulation (ST) method to dramatically expand the range of the GFFP method; and the development of last-passage diffusion methods; these drastically improve the efficiency of diffusion Monte Carlo methods. All of these claims are addressed in detail, with specific examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Haji-Sheikh and E. M. Sparrow. The solution of heat conduction problems by probability methods, Journal of Heat Transfer, 89, 121–131, 1967.
I. C. Kim and S. Torquato. Effective conductivity of suspensions of hard spheres by Brownian motion simulation, J. Appl. Phys., 69(4), 2280–2289, 1991.
I. C. Kim and S. Torquato. Effective conductivity of suspensions of overlapping spheres, J. Appl. Phys., 71(6), 2727–2735, 1992.
S. Torquato, I.-C. Kim, and D. Cule. Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first-passage-time equations, J. Appl. Phys., 85, 1560–1571, 1999.
C.-O. Hwang, J. A. Given, and M. Mascagni. On the rapid calculation of permeability for porous media using Brownian motion paths, Phys. Fluids A, 12(7), 1699–1709, 2000.
K. L. Chung. Green, Brown, and Probability. World Scientific, Singapore, 1995.
J. A. Given, J. B. Hubbard, and J. F. Douglas. A first-passage algorithmfor the hydrodynamic friction and diffusion-limited reaction rate of macromolecules, J. Chem. Phys., 106(9), 3721–3771, 1997.
M. E. MÜller. Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Stat., 27, 569–589, 1956.
T. E. Booth. Exact Monte Carlo solution of elliptic partial differential equations, J. Comput. Phys., 39, 396–404, 1981.
S. Torquato and I. C. Kim. Efficient simulation technique to compute effective properties of hetergeneous media, Appl. Phys. Lett., 55, 1847–1849, 1989.
C.-O. Hwang, J. A. Given, and M. Mascagni. Rapid diffusion Monte Carlo algorithms for fluid dynamic permeability, Mone Carlo Methods and Applications, 7(3–4), 213–222, 2001.
C.-O. Hwang, J. A. Given, and M. Mascagni. The simulation-tabulation method for classical diffusion Monte Carlo, J. Comput. Phys., 2000, (submitted).
M. Freidlin. Functional Integration and Partial Differential Equations, Princeton University Press, Princeton, New Jersey, 1985.
K. L. Chung and Z. Zhao. From Brownian Motion to SchrÖdinger’s Equation, Springer-Verlag, Berlin, 1995.
K. K. Sabelfeld. Integral and probabilistic representations for systems of elliptic equations, Mathematical and Computer Modelling, 23, 111–129, 1996.
R. Ettelaie. Solutions of the linearized Poisson-Boltzmann equation through the use of randomw alk simulation method, J. Chem. Phys., 103(9), 3657–3667, 1995.
C.-O. Hwang and M. Mascagni. Efficient modified “Walk On Spheres” algorithm for the linearized Poisson-Boltzmann equation, Appl. Phys. Lett., 78(6), 787–789, 2001.
C.-O. Hwang, M. Mascagni, and J. A. Given. A Feynman-Kac formula implementation for the linearized Poisson-Boltzmann equation, 2001, (in preparation).
J. B. Hubbard and J. F. Douglas. Hydrodynamic friction of arbitrarily shaped Brownian particles, Phys. Rev. E, 47, R2983–R2986, 1993.
J. F. Douglas, H.-X. Zhou, and J. B. Hubbard. Hydrodynamic friction and the capacitance of arbitrarily shaped objects, Phys. Rev. E, 49, 5319–5331, 1994.
H.-X. Zhou, A. Szabo, J. F. Douglas, and J. B. Hubbard. A Brownian dynamics algorithmfor caculating the hydrodynamic friction and the electrostatic capacitance of an arbitrarily shaped object, J. Chem. Phys., 100(5), 3821–3826, 1994.
N. S. Martys, S. Torquato, and D. P. Bentz. Universal scaling of fluid permeability for sphere packings, Phys. Rev. E, 50, 403–408, 1994.
H.-X. Zhou. Calculation of translational friction and intrinsic viscosity. I. General formulation for arbitrarily shaped particles, Biophys. J., 69, 2286–2297, 1995.
H.-X. Zhou. Calculation of translational friction and intrinsic viscosity. II. Application to globular proteins, Biophys. J., 69, 2298–2303, 1995.
H.-X. Zhou. Comparison of three Brownian-dynamics algorithms for calculating rate constants of diffusion-influenced reactions, J. Chem. Phys., 108(19), 8139–8145, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Given, J.A., Mascagni, M., Hwang, CO. (2001). Continuous Path Brownian Trajectories for Diffusion Monte Carlo via First- and Last-Passage Distributions. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds) Large-Scale Scientific Computing. LSSC 2001. Lecture Notes in Computer Science, vol 2179. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45346-6_4
Download citation
DOI: https://doi.org/10.1007/3-540-45346-6_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43043-8
Online ISBN: 978-3-540-45346-8
eBook Packages: Springer Book Archive