Abstract
The survival probability of a particle diffusing in the two-dimensional domainx>0 near a “windy cliff” atx=0 is investigated. The particle dies upon reaching the edge of the cliff. In addition to diffusion, the particle is influenced by a steady “wind shear” with velocityv(x, y)=v sign(y)x, i.e., no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time ast −1/4, compared tot −1/2 in the absence of wind. Scaling descriptions are developed to elucidate this behavior, as well as the survival probability within a semi-infinite strip of finite width |y|<w with particle absorption atx=0. The behavior in the strip geometry can be described in terms of Taylor diffusion, an approach which accounts for the crossover to the t−1/4 decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.
Similar content being viewed by others
References
G. H. Weiss and R. J. Rubin,Adv. Chem. Phys. 52:363 (1983), and references therein.
J. Lee and J. Koplik,J. Stat. Phys. 79:895 (1995).
W. Feller,An introduction to Probability Theory and Its Applications (Wiley, New York, 1968), Vol. I, Chapter 3.
E. Ben-Naim, S. Redner, and D. ben-Avraham,Phys. Rev. A 45:7207 (1992).
G. Zumofen and Y. Klafter,Phys. Rev. E 51 (1995).
L. Breiman, inProceedings Fifth Berkeley Symposium on Mathematics and Probability, Volume II (1967); K. Uchiyama,Z. Wahrsch. verw. Gebiete 54:75 (1980); P. Salminen,Adv. Appl. Prob. 20, 411 (1988); L. Turban,J. Phys. A 25:L127 (1992); F. Iglói,Phys. Rev. A 45:7024 (1992).
P. L. Krapivsky and S. Redner,Am. J. Phys. (1966), to appear.
G. I. Taylor,Proc. R. Soc. Lond. A 219:186 (1953) R. Aris,Proc. R. Soc. Lond. A 235:67 (1956); see also J. Koplik, S. Redner, and D. Wilkinson,Phys. Rev. A 37:2619 (1988).
E. Ben-Naim, Private communication.
G. Matheron and G. de Marsily,Water Resources Res. 16:901 (1980).
J.-P. Bouchaud, A. Georges, J. Koplik, A. Provata, and S. Redner,Phys. Rev. Lett. 64:2503 (1990); S. Redner,Physica A 168:551 (1990).
P. S. Hagan, C. R. Doering, and C. D. Levermore,J. Stat. Phys. 54:1321 (1989); see also C. R. Doering, P. S. Hagan, and C. D. Levermore,Phys. Rev. Lett. 59:2129 (1987); P. S. Hagan, C. R. Doering, and C. D. Levermore,SIAM J. Appl. Math. 49:1480 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Redner, S., Krapivsky, P.L. Diffusive escape in a nonlinear shear flow: Life and death at the edge of a windy cliff. J Stat Phys 82, 999–1014 (1996). https://doi.org/10.1007/BF02179799
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179799