Kinetics of First Passage in a Cone
E. Ben-Naim and P.L. Krapivsky
We study statistics of first passage inside a cone in 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.
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