Abstract
We suggest in this paper a Random Walk on Spheres (RWS) method for solving transient drift-diffusion-reaction problems which is an extension of our algorithm we developed recently [26] for solving steady-state drift-diffusion problems. Both two-dimensional and three-dimensional problems are solved. Survival probability, first passage time and the exit position for a sphere (disc) of the drift-diffusion-reaction process are explicitly derived from a generalized spherical integral relation we prove both for two-dimensional and three-dimensional problems. The distribution of the exit position on the sphere has the form of the von Mises–Fisher distribution which can be simulated efficiently. Rigorous expressions are derived in the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion-reaction problems with arbitrary varying drift velocity vector. The method can efficiently be applied to calculate the fluxes of the solution to any part of the boundary. This can be done by applying a reciprocity theorem which we prove here for the drift-diffusion-reaction problems with general boundary conditions. Applications of this approach to methods of cathodoluminescence (CL) and electron beam induced current (EBIC) imaging of defects and dislocations in semiconductors are presented.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 14-11-00083
Funding statement: Support of the Russian Science Foundation under Grant 14-11-00083 is kindly acknowledged.
References
[1] H. Beirão da Veiga, On the semiconductor drift diffusion equations, Differential Integral Equations 9 (1996), no. 4, 729–744. 10.57262/die/1367969884Search in Google Scholar
[2] D. J. Best and N. I. Fisher, Efficient simulation of the von Mises distribution, J. Roy. Statist. Soc. Ser. C 28 (1979), 152–157. 10.2307/2346732Search in Google Scholar
[3] G. W. Brown, Monte Carlo methods, Modern Mathematics for the Engineer, McGraw–Hill, New York (1956), 279–303. Search in Google Scholar
[4] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., The Clarendon Press, New York, 1988. Search in Google Scholar
[5] M. Deaconu and A. Lejay, A random walk on rectangles algorithm, Methodol. Comput. Appl. Probab. 8 (2006), no. 1, 135–151. 10.1007/s11009-006-7292-3Search in Google Scholar
[6] L. Devroye, The series method for random variate generation and its application to the Kolmogorov–Smirnov distribution, Amer. J. Math. Management Sci. 1 (1981), no. 4, 359–379. 10.1080/01966324.1981.10737080Search in Google Scholar
[7] L. Devroye, Simulating Bessel random variables, Statist. Probab. Lett. 57 (2002), no. 3, 249–257. 10.1016/S0167-7152(02)00055-XSearch in Google Scholar
[8] B. S. Elepov, A. A. Kronberg, G. A. Mihaĭlov and K. K. Sabelfeld, Solution of Boundary Value Problems by the Monte Carlo Method (in Russian), Izdatel’stvo “Nauka” Sibirskoe Otdelenie, Novosibirsk, 1980. Search in Google Scholar
[9]
B. S. Elepov and G. A. Mihaĭlov,
The solution of the Dirichlet problem for the equation
[10] S. M. Ermakov, V. V. Nekrutkin and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics, Math. Appl. (Soviet Series) 34, Kluwer Academic, Dordrecht, 1989. 10.1007/978-94-009-2243-3Search in Google Scholar
[11] S. M. Ermakov and A. S. Sipin, The “walk in hemispheres” process and its applications to solving boundary value problems, Vestnik St. Petersburg Univ. Math. 42 (2009), no. 3, 155–163. 10.3103/S1063454109030029Search in Google Scholar
[12] M. Evans, N. Hastings and B. Peacock, Statistical Distributions, 3rd ed., Wiley Ser. Probab. Stat., Wiley-Interscience, New York, 2000. Search in Google Scholar
[13] R. Fisher, Dispersion on a sphere, Proc. Roy. Soc. London. Ser. A. 217 (1953), 295–305. 10.1098/rspa.1953.0064Search in Google Scholar
[14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood, 1964. Search in Google Scholar
[15] J. A. Given, J. B. Hubbard and J. F. A. Douglas, First-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of macromolecules, J. Chem. Phys. 106 (1997), no. 9, 3761–3771. 10.1063/1.473428Search in Google Scholar
[16] N. Golyandina, Convergence rate for spherical processes with shifted centres, Monte Carlo Methods Appl. 10 (2004), no. 3–4, 287–296. 10.1515/mcma.2004.10.3-4.287Search in Google Scholar
[17] A. Haji-Sheikh and E. M. Sparrow, The floating random walk and its application to Monte Carlo solutions of heat equations, SIAM J. Appl. Math. 14 (1966), 370–389. 10.1137/0114031Search in Google Scholar
[18] T. Lagache and D. Holcman, Extended narrow escape with many windows for analyzing viral entry into the cell nucleus, J. Stat. Phys. 166 (2017), no. 2, 244–266. 10.1007/s10955-016-1691-9Search in Google Scholar
[19] W. Liu, J. F. Carlin, N. Grandjean, B. Deveaud and G. Jacopin, Exciton dynamics at a single dislocation in GaN probed by picosecond time-resolved cathodoluminescence, Appl. Phys. Lett. 109 (2016), no. 4, Article ID 042101. 10.1063/1.4959832Search in Google Scholar
[20] M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist. 27 (1956), 569–589. 10.1214/aoms/1177728169Search in Google Scholar
[21] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002. 10.1201/9781420035322Search in Google Scholar
[22] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series. Elementary Functions (in Russian), “Nauka”, Moscow, 1981. Search in Google Scholar
[23] W. Rosenheinrich, Tables of some indefinite integrals of Bessel functions, preprint, http://www.eah-jena.de/~rsh/Forschung/Stoer/besint.pdf. Search in Google Scholar
[24] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Ser. Comput. Phys., Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Search in Google Scholar
[25] K. K. Sabelfeld, Random walk on semi-cylinders for diffusion problems with mixed Dirichlet–Robin boundary conditions, Monte Carlo Methods Appl. 22 (2016), no. 2, 117–131. 10.1515/mcma-2016-0108Search in Google Scholar
[26] K. K. Sabelfeld, Random walk on spheres method for solving drift-diffusion problems, Monte Carlo Methods Appl. 22 (2016), no. 4, 265–275. 10.1515/mcma-2016-0118Search in Google Scholar
[27] K. K. Sabelfeld, A mesh free floating random walk method for solving diffusion imaging problems, Statist. Probab. Lett. 121 (2017), 6–11. 10.1016/j.spl.2016.10.006Search in Google Scholar
[28] K. K. Sabelfeld, V. M. Kaganer, C. Pfüller and O. Brandt, Dislocation contrast in cathodoluminescence and electron-beam induced current maps on GaN{0001}, Phys. Rev. Appl., to appear. Search in Google Scholar
[29] K. K. Sabelfeld, A. Kireeva, V. M. Kaganer, Carsten Pfüller and Oliver Brandt, Drift and diffusion of excitons at threading dislocations in GaN{0001}, Phys. Rev. Appl., to appear. Search in Google Scholar
[30] K. K. Sabelfeld and I. A. Shalimova, Spherical and Plane Integral Operators for PDEs. Construction, Analysis, and Applications, De Gruyter, Berlin, 2013. 10.1515/9783110315332Search in Google Scholar
[31] K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems. Numerics for High-Dimensional PDEs and Applications, De Gruyter, Berlin, 2016. 10.1515/9783110479454Search in Google Scholar
[32] N. A. Simonov, Walk-on-spheres algorithm for solving third boundary value problem, Appl. Math. Lett. 64 (2017), 156–161. 10.1016/j.aml.2016.09.008Search in Google Scholar
[33] S. Steišūnas, On the sojourn time of the Brownian process in a multidimensional sphere, Nonlinear Anal. Model. Control 14 (2009), no. 3, 389–396. 10.15388/NA.2009.14.3.14502Search in Google Scholar
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