Space Homogeneous Solutions of the Linear Semiconductor Boltzmann Equation

Majorana, Armando; Milazzo, C.

doi:10.1006/jmaa.2001.7444

Cited by 20 publications

(16 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, we merely restrict ourselves here to a few remarks on the critical case in the whole space. This equation is formally similar to the linear Boltzmann equation of gas; see for instance [3,7,[16][17][18]. Actually, the difference between them resides in the peculiarity of scattering events.…”

Section: Remarks On the Linear Semiconductor Equationmentioning

confidence: 92%

See 1 more Smart Citation

New generation theorems in transport theory

Mokhtar-Kharroubi

2011

Afr. Mat.

View full text Add to dashboard Cite

New generation theorems for singular kinetic equations in L 1 -spaces are given. We show that space homogeneous linear kinetic equations in L 1 spaces and full transport equations with advection termv. ∂ ∂ x in L 1 spaces on arbitrary spatial domains share several generation properties. In particular, in the subcritical case, they share the so-called "closure property" of the generator. We also show, for subcritical equations, that a principle of detailed balance insures this "closure property". We show, in the subcritical case, that the full transport operator in bounded geometries is always a generator for sublinear collision frequencies. We provide also generation theorems based on (weak) compactness methods and, in the case of space homogeneous kinetic equations, study the essential spectrum of the corresponding semigroup. The diffusion models (used in nuclear reactor theory) where the advection operator v. ∂ ∂ x is replaced by a Laplacian operator in space variable are also dealt with.

show abstract

Section: Remarks On the Linear Semiconductor Equationmentioning

confidence: 92%

“…We do not know whether there is something physical corresponding to (17). We note however that for the hard spheres scattering model (see for instance [8, p. 290]), the collision frequency is subaffine only, i.e.…”

Section: Remark 22 (I)mentioning

confidence: 99%

New generation theorems in transport theory

Mokhtar-Kharroubi

2011

Afr. Mat.

View full text Add to dashboard Cite

show abstract

“…For more details about analysis on fragmentation differential equations, we refer the reader to [6,16,20,24,34,41,50,54] and the references therein. The classic model for fragmentation process is given by the integrodifferential equation…”

Section: Fragmentation Differential Equationmentioning

confidence: 99%

Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

Goufo¹,

Mugisha²

2015

Open Mathematics

View full text Add to dashboard Cite

Classical models of clusters' fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order (with 0 < Ä 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters' fission process. We make use of the theory of strongly continuous solution operators for fractional models (analogues of C 0 -semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993 to analyze and show existence results for clusters' splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He's homotopy perturbation (He, 1999) and Kato's type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive. Model's motivation and introductionThe concepts of fractional derivatives and fractional integral started in 1695 when L'Hospital questioned about the meaning of the operator d n y=dx n if n D 1=2I that is "what if n is fractional?". Leibniz then replied as "d 1=2 x=dx 1=2 will be equal to x p dx W x". Despite its three centuries of age, fractional calculus remains lightly unpopular amongst science and engineering community. However, there is a growing interest in extending the normal calculus with interger orders to noninteger orders (real or complex order) [13,36,39,44] because its applications have attracted a great range of attention in the past few years. Indeed, it has turned out recently that many phenomena in different fields, including sciences, engineering and technology can be described very successfully by the models using fractional order differential equations. As example, the concept of fractional Laplacian operator in the theory of Lévy flights [12] is a typical application of fractional derivatives, which leads to the theory of sub-and superdiffusion, well applicable in reaction-diffusion systems. In the field of mathematical epidemiology, especially the

show abstract

“…In radiative transfer, the phase function integrated over outgoing directions is dominated by the extinction coefficient [7,29]. In cell growth modeling [28], electron transport in weakly ionized gases [15], rarefied gas dynamics [6], and modeling of electron-phonon interaction in semiconductors [18,19], the integrated (nonnegative) collision kernel is exactly equal to the collision frequency. In fact, Eq.…”

Section: Introductionmentioning

confidence: 99%

“…The situation as described in the preceding paragraph occurs in various applications, such as electron transport in weakly ionized gases [15], cell growth modeling [20], and modeling of electron-phonon interactions in semiconductors [19]. In these papers, sufficient conditions for having the isometry condition (3.15) are derived and the possibility of not having (3.15) is discussed, though without providing explicit models in which the isometry relation (3.15) is not satisfied.…”

mentioning

confidence: 99%