Kinetics of first passage in a cone
Abstract
We study statistics of first passage inside a cone in an arbitrary spatial dimension. The probability that a diffusing particle avoids the cone boundary decays algebraically with time. The decay exponent depends on two variables: the opening angle of the cone and the spatial dimension. In four dimensions, we find an explicit expression for the exponent, and in general, we obtain it as a root of a transcendental equation involving associated Legendre functions. At large dimensions, the decay exponent depends on a single scaling variable, while roots of the parabolic cylinder function specify the scaling function. Consequently, the exponent is of order one only if the cone surface is very close to a plane. We also perform asymptotic analysis for extremely thin and extremely wide cones.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- December 2010
- DOI:
- 10.1088/1751-8113/43/49/495007
- arXiv:
- arXiv:1009.0238
- Bibcode:
- 2010JPhA...43W5007B
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Mathematical Physics;
- Mathematics - Probability
- E-Print:
- 9 pages, 5 figures, 2 tables