Brownian Motion in Parabolic Space
- 1 Department of Applied Mathematics, Faculty of Engineering, Oita University, Oita, Japan
Abstract
A new mathematical system applicable to whatever Brownian problems where the Fickian diffusion equation (F-equation) is applicable was established. The F-equation, which is a parabolic type partial differential equation in the evolution equation, has ever been used for linear diffusion problems in the time-space (t, x, y, z). In the parabolic space (xt –0.5 , yt –0.5 , zt –0.5 ), the present study reveals that the F-equation becomes an ellipse type Poisson equation and furthermore the elegant analytical solutions are possible. Applying the new system to one-dimension nonlinear interdiffusion problems, the solutions were previously obtained as the analytical expressions. The obtained solutions were also elegant in accordance with the experimental results. In the present study, nonlinear diffusion problems are discussed in the two and three dimensional cases. The Brownian problem is widely relevant not only to material science but also to other various science fields. Hereafter, the new mathematical system will be thus extremely useful for the analysis of the Brownian problem in various science fields.
- A. Einstein, “Die von der Molekularkinetischen Theorie der Warme Geforderte Bewegung von in Ruhenden Flussiigkeiten Suspendierten Teilchen,” Annalen der Physik, Vol. 18, No. 8, 1905, pp. 549-560. doi:10.1002/andp.19053220806
- J. Perrin, “Mouvement Brownien et Réalité Moléculaire,” Annales de chimie et de Physique, Vol. 18, No. 8, 1909, pp. 5-114.
- I. Minoura, E. Katayama, K. Sekimoto and E. Muto, “One-Dimensional Brownian Motion of Charged Nano- particles along Microtubules: A Model System for Weak Binding Interactions,” Biophysical Journal, Vol. 98, No. 8, 2010, pp. 1589-1597. doi:10.1016/j.bpj.2009.12.4323
- C. C. Chen, J. Daponte and M. Fox, “Fractal Feature Analysis and Classification in Medical Imaging,” IEEE Transactions on Medical Imaging, Vol. 8, No. 1, 1989, pp. 133-142. doi:10.1109/42.24861
- K. Arakawa and E. Krotkov, “Modeling of Natural Ter- rain Based on Fractal Geometry,” Systems and Compu- tors in Japan, Vol. 25, No. 11, 1994, pp. 99-113. doi:10.1002/scj.4690251110
- L. M. Wein, “Brownian Networks with Discretionary Rout- ing,” Operations Research, Vol. 39, No. 2, 1990, pp. 322- 340. doi:10.1287/opre.39.2.322
- A. Fick, “On Liquid Diffusion,” Philosophical Magazine, Vol. 4, No. 10, 1855, pp. 30-39.
- L. Boltzmann, “Zur Integration der Diffusionsgleichung bei Variabeln Diffusionscoefficienten,” Annual Review Physical Chemistry, Vol. 53, No. 2, 1894, pp. 959-964.
- C. Matano, “On the Relation between Diffusion-Coeffi- cients and Concentrations of Solid Metals,” Japanese Journal of Physics, Vol. 8, No. 8, 1933, pp. 109-113.
- T. Okino, “New Mathematical Solution for Analyzing Interdiffusion Problems,” Materials Transactions, Vol. 52, No. 12, 2011, pp. 2220-2227. doi:10.2320/matertrans.M2011137