Conditions for runaway phenomena in the kinetic theory of particle swarms

Frosali, Giovanni; Mee, C. V. M. van der; Paveri‐Fontana, S. L.

doi:10.1063/1.528339

Cited by 35 publications

(26 citation statements)

References 14 publications

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“…where t¿0, plus the rule that {S(t)f} t¿0 can be obtained by iterating (13). The question is if there exists more than one substochastic semigroup {S(t)} t¿0 on X (with corresponding resolvent {T } Re ¿0 ) which satisÿes (13) [or equivalently: for which the resolvent satisÿes (12)].…”

Section: Proofmentioning

confidence: 99%

“…The results are based on the spectral analysis of B( − A) −1 and it will in fact turn out that the three situations are characterized by whether 1 is in the resolvent set, the continuous spectrum or the residual spectrum of B( − A) −1 , respectively. In Section 4 we revise the existing treatment of the runaway problem in the kinetic theory of particle swarms [10,11] and characterize the generator. In the last section we apply the general theory to the fragmentation problem describing polymer degradation [12], perform a spectral analysis of the problem and completely characterize the generated semigroup.…”

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confidence: 99%

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A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

Frosali

Mee

Mugelli

2004

Math Methods in App Sciences

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We consider a class of abstract evolution problems characterized by the sum of two unbounded linear operators $A$ and $B$, where $A$ is assumed to generate a positive semigroup of contractions on an $L^1$-space and $B$ is positive. We study the relations between the semigroup generator $G$ and the operator $A + B$. $A$ characterization theorem for $G =A+B$ is stated. The results are based on the spectral analysis of $B(\lambda-A)^{-1}$. The main point is to study the conditions under which the value 1 belongs to the resolvent set, the continuous spectrum, or the residual spectrum of \ud $B(\lambda - A)^{-1}$

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Section: Proofmentioning

confidence: 99%

mentioning

confidence: 99%

A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

Frosali

Mee

Mugelli

2004

Math Methods in App Sciences

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“…The operator S + (A + C)/ generates an uniformly bounded semigroup in X k , Z (t), cf. [40]. Thus, the mild solution of (9.6) is The estimate of S(φ 1 +ψ 1 ) (s/ ) X k is a bit tedious, thus we simply sketch it.…”

Section: Estimate Of the Errormentioning

confidence: 99%

Diffusive corrections to asymptotics of a strong-field quantum transport equation

Manzini

Frosali

2010

Physica D: Nonlinear Phenomena

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a b s t r a c tThe asymptotic analysis of a linear high-field Wigner-BGK equation is developed by a modified ChapmanEnskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number , evolution equations are derived for the terms of zeroth and first order in . In particular, a quantum driftdiffusion equation for the position density of electrons, with an -order correction on the field terms, is obtained. Well-posedness and regularity of the approximate problems are established, and a rigorous proof that the difference between exact and asymptotic solutions is of order 2 , uniformly in time and for arbitrary initial data is given.

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“…Diffusions are field independent, and consequently they cannot approximate the ballistic electron distribution func-tions that have been observed by Barenger and Wilkins [4,5]. After the work initiated by [14] and [22] on strong forcing scaling for external fields, where the dominant term in the scaled Eq.…”

Section: Preliminaries Of the High Field Modelmentioning

confidence: 99%

Applicability of the High Field Model:An Analytical Study Via Asymptotic ParametersDefining Domain Decomposition

et al. 1998

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In this paper, we present a mesoscopic-macroscopic model of self-consistent charge transport. It is based upon an asymptotic expansion of solutions of the Boltzmann Transport Equation (BTE). We identify three dimensionless parameters from the BTE.These parameters are, respectively, the quotient of reference scales for drift and thermal velocities, the scaled mean free path, and the scaled Debye length. Such parameters induce domain dependent macroscopic approximations. Particular focus is placed upon the so-called high field model, defined by the regime where drift velocity dominates thermal velocity. This model incorporates kinetic transition layers, linking mesoscopic to macroscopic states. Reference scalings are defined by the background doping levels and distinct, experimentally measured mobility expressions, as well as locally determined ranges for the electric fields. The mobilities reflect a coarse substitute for reference scales of scattering mechanisms. See [9] for elaboration.The high field approximation is a formally derived modification of the augmented drift-diffusion model originally introduced by Thornber some fifteen years ago [25]. We are able to compare our approach with the earlier kinetic approach of Baranger and Wilkins [5] and the macroscopic approach of Kan, Ravaioli and Kerkhoven [20].

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Conditions for runaway phenomena in the kinetic theory of particle swarms

Cited by 35 publications

References 14 publications

A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

Diffusive corrections to asymptotics of a strong-field quantum transport equation

Applicability of the High Field Model:An Analytical Study Via Asymptotic ParametersDefining Domain Decomposition

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