Brownian Motion on a Sphere: Distribution of Solid Angles
We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our re...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Text |
| Language: | English |
| Published: |
2000
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| Subjects: | |
| Online Access: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.310.7844 http://arxiv.org/pdf/cond-mat/0005345v1.pdf |
| Summary: | We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on curved surfaces like micelles and biological membranes. Our theoretical analysis agrees well with the results of computer experiments. 1 Let a diffusing particle start from the north pole of a sphere at time τ = 0. We join the final position of the particle at time β to its initial one (the north pole) by the shorter geodesic. This rule is well defined, unless the final position is exactly at the south pole, a zero probability event. The path followed by the diffusing particle (closed by the geodesic rule) encloses a solid angle Ω. The question we address is: At time β, what is the distribution P β (Ω) of solid angles? An experimental motivation for this question comes from recent time-resolved fluorescence studies[1, 2, 3, 4] on the Brownian motion of rod-like molecules on curved surfaces such as micelles and lipid vesicles. The experimentally measured fluorescence anisotropy |
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