Random motion with gamma steps in higher dimensions

Pogorui, A. A.; Rodríguez‐Dagnino, Ramón M.

doi:10.1016/j.spl.2013.03.011

Cited by 15 publications

(3 citation statements)

References 8 publications

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“…In addition to recent time-resolved Gamma-flight solutions[Le Caër 2011;Pogorui and Rodríguez-Dagnino 2011;Pogorui and Rodríguez-Dagnino 2013] we consider the steadystate collision densities of all orders, as well as the scalar flux, and diffusion approximations to both.7.1 Gamma Random Flights-Rod Model (d = 1) 7.1.1 Rod Model -Gamma k = 1/2…”

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confidence: 99%

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

d’Eon

2013

Transport Theory and Statistical Physics

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We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) for the stationary collision density and scalar flux about an isotropic point source in an infinite d-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentiallydistributed free paths in arbitrary dimensions as well as a number of non-classical transport theories (non-exponential random flights) that describe a broader class of transport processes within partially-correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean's moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean's approach produces compact, analytic approximations that are, overall, more accurate for high absorption and for small source-detector separations than either P 1 diffusion or rigorous asymptotic diffusion. These diffusion-based approximations are exact in the first two even spatial moments, which we derive explicitly for various non-classical transport types. We also discuss connections between the random-flight-theory derivation of the Green's function and the discrete spectrum of the transport operator and report some new observations regarding the discrete eigenvalues of the transport operator for general dimensions and free-path distributions.

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confidence: 99%

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

d’Eon

2013

Transport Theory and Statistical Physics

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“…Relationship to Stochastic Processes Literature The Monte Carlo and deterministic methods in this paper apply generally to any stochastic processes where a particle moves along straight paths at constant speed between collisions that change the particle's direction and possibly absorb the particle. This include some, but not all, of the stochastic processes that are studied under names such as Lévy walks [Uchaikin and Gusarov 1997;Zaburdaev et al 2015], random evolutions [Hersh 1974], velocity jump process [Othmer and Hillen 2000] and semi-Markov random flights [Grosjean 1951;Dutka 1985;Majumdar and Ziff 2008;Zoia et al 2011;Le Caër 2011;De Gregorio and Orsingher 2012;Pogorui and Rodríguez-Dagnino 2013;De Gregorio 2017].…”

Section: Related Workmentioning

confidence: 99%

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

d’Eon¹

2019

Journal of Computational and Theoretical Transport

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When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux / radiance because the macroscopic cross-section Σ t (s) depends on the distance s to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.

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“…Similarly, the case of exponentially distributed random intertimes with linearly increasing parameters has been treated in [15], thus modeling a damping behavior sometimes appearing in particle systems. We recall also the papers [41] and [42], where isotropic random motions in higher dimensions with, respectively, Erlang-and gamma-distributed steps are studied.…”

Section: Introductionmentioning

confidence: 99%

Some results on the telegraph process driven by gamma components

Martinucci

Meoli²,

Zacks³

2022

Adv. Appl. Probab.

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We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$ . The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion. Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).

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Random motion with gamma steps in higher dimensions

Cited by 15 publications

References 8 publications

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

Some results on the telegraph process driven by gamma components

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