Solution of the non-isotropic random flight problem in the k-dimensional space

Grosjean, C. C.

doi:10.1016/s0031-8914(53)80004-2

Cited by 20 publications

(21 citation statements)

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“…The literature on stochastic processes includes a number of fully general studies in R d (see [34,4,35,36,37,38] and references therein) that reveal the influence of dimension. Such studies often permit d ≥ 1 to be a free real parameter [39] by using the general surface area Ω d of the unit sphere S d−1 ≡ {x ∈ R d , ||x|| = 1} in d dimensions,…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…For infinite homogeneous media, Green's functions for monoenergetic linear transport with isotropic [28,41,42,38,43,44] and anisotropic [4,45,46] scattering are known in domains apart from 3D. The unidirectional point source was also considered in Flatland [47].…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…It is, however, occasionally useful to consider Euclidean domains apart from R 3 . For infinite homogeneous media, monoenergetic problems in R d have been mostly solved [4]. In this paper, we solve the classic problem of diffuse reflection from an isotropically-scattering half space in R d for general d ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

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Radiative Transfer in Half Spaces of Arbitrary Dimension

d’Eon¹,

McCormíck

2019

Journal of Computational and Theoretical Transport

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We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain R d for arbitrary d ≥ 1. A continuous family of pseudo-problems and related H functions arises and includes the classical 3D solutions, as well as 2D "Flatland" and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension d are highlighted and we find that a discrete diffusion mode is not universal for d > 3 in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.

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Section: Motivation and Related Workmentioning

confidence: 99%

Section: Motivation and Related Workmentioning

confidence: 99%

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Radiative Transfer in Half Spaces of Arbitrary Dimension

d’Eon¹,

McCormíck

2019

Journal of Computational and Theoretical Transport

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“…Few references exist on this topic (see for instance [1,10]), while, as previously observed, the literature is essentially devoted to the analysis of random walks with uniform distribution. For the sake of completeness, we point out that other multidimensional random models with finite velocity have been proposed in literature: see, for example, [6,12,13,18,21].…”

Section: Introductionmentioning

confidence: 99%

On Random Flights with Non-uniformly Distributed Directions

Gregorio

2012

J Stat Phys

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This paper deals with a new class of random flights in a"e (d) , d >= 2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter nu >= 0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. The analysis of the position (X) under bar (d) t > 0, obtained as projection onto the lower space R-m , m < d, of the original random motion in R-d . In its general framework, the analysis of (X) under bar (d) t > 0, is very complicated; nevertheless for some values of nu, we provide some explicit results about the process. Indeed, for nu=1 we get the characteristic function of the random flight moving in a"e (d) . By inverting the obtained characteristic function, we derive the analytic form (up to some constants) of the probability distribution of (X) under bar (d), t > 0. It is worth to mention that the stochastic processes considered in this paper belong to the class of the non-isotropic random walks, which has several applications in the mechanical statistics

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“…This was found to lead to simple explicit results in certain special cases, such as the case of segment vectors with independent, identically normally distributed components. Grosjean (1953) allowed for random variation in segment length, and for a general orientation distribution, though he retained the assumptions of independence between segments, and between length and orientation of the same segment.…”

Section: Introduction and Historical Outlinementioning

confidence: 99%