dorsal/arxiv
View SchemaDiagrammar For Random Flight Motion
| Authors | S. Tim Hatamian |
|---|---|
| Categories | |
| ArXiv ID | physics/0309048 |
| URL | https://arxiv.org/abs/physics/0309048 |
Abstract
We present a perturbation theory by extending a prescription due to Feynman for computing the probability density function for the random flight motion. The method can be applied to a wide variety of otherwise difficult circumstances. The series for the exact moments, if not the distribution itself, for many important cases can be summed for arbitrary times. As expected, the behavior at early time regime, for the sample processes considered, deviate significantly from diffusion theory; a fact with important consequences in various applications such as financial physics. A half dozen sample problems are solved starting with one posed by Feynman who originally solved it only to first order. Another illustrative case is found to be a physically more plausible substitute for both the Uhlenbeck-Ornstein and the Wiener processes. The remaining cases we've solved are useful for applications in regime-switching and isotropic scattering of light in turbid media. It is demonstrated that under isotropic random flight with invariant-speed, the description of motion in higher dimensions is recursively related to either the one or two dimensional movement. Also for this motion, we show that the solution heretofore presumed correct in one dimension, applies only to the case we've named the "Shooting Gallery" problem; a special case of the full problem. A new class of functions dubbed "Damped-exponential-integrals" are also identified.
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"abstract": "We present a perturbation theory by extending a prescription due to Feynman\nfor computing the probability density function for the random flight motion.\nThe method can be applied to a wide variety of otherwise difficult\ncircumstances. The series for the exact moments, if not the distribution\nitself, for many important cases can be summed for arbitrary times. As\nexpected, the behavior at early time regime, for the sample processes\nconsidered, deviate significantly from diffusion theory; a fact with important\nconsequences in various applications such as financial physics. A half dozen\nsample problems are solved starting with one posed by Feynman who originally\nsolved it only to first order. Another illustrative case is found to be a\nphysically more plausible substitute for both the Uhlenbeck-Ornstein and the\nWiener processes. The remaining cases we\u0027ve solved are useful for applications\nin regime-switching and isotropic scattering of light in turbid media. It is\ndemonstrated that under isotropic random flight with invariant-speed, the\ndescription of motion in higher dimensions is recursively related to either the\none or two dimensional movement. Also for this motion, we show that the\nsolution heretofore presumed correct in one dimension, applies only to the case\nwe\u0027ve named the \"Shooting Gallery\" problem; a special case of the full problem.\nA new class of functions dubbed \"Damped-exponential-integrals\" are also\nidentified.",
"arxiv_id": "physics/0309048",
"authors": [
"S. Tim Hatamian"
],
"categories": [
"physics.class-ph",
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"title": "Diagrammar For Random Flight Motion",
"url": "https://arxiv.org/abs/physics/0309048"
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