F. Golse and F. Poupaud, Limite fluide des e'quations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 135-160 (1992).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which are used for nonlinear parabolic equations [27, 18] There are some deep connections between entropies for kinetic equations and for nonlinear diffusions, which are out of the scope of this paper. However, to illustrate this point, we will derive a diffusive limit, at a formal level (see [55, 74, 52, 13, 33, 91, 83] for rigorous results) Further references corresponding to more specific aspects will be mentioned in the rest of the paper. We will not provide all details for each proof and will systematically refer to papers in which details or similar ideas can be found. Some of the results presented here ....

....function is the one which was exhibited in Examples 2 and 3 of Section 1 (also see [18, 75, 76, 47] with the above notations, for any (x; v) 2 IR f 1 (x; v) fl OE 0 OE Gamma with fl(u) ff e where ff 0 is a parameter related to Planck s constant. We refer to [55] for a mathematically rigourous justification of the diffusive limit. The function ae 1 (x) Gamma(OE 0 OE Gamma ) is the unique equilibrium density of the nonlinear diffusion equation ae t = r Delta (r(ae) aerOE) where (u) Gamma (s) ds and Gamma(u) jS 0 (2s) fl (s ....

F. Golse, F. Poupaud, Limite fluide des 'equations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac [Fluid limit of Boltzmann semiconductor equations for a Fermi-Dirac statistic], Asymptotic Anal. 6 no. 2 (1992), 135--160.

....type functionals. Propagation of moments, in the Vlasov Poisson context, is studied in [25] Finally some basic application aspects can be found in [6] for the diffusive asymptotic limit of the neutron transport equation, in [7] for the study of the radiative transfer model problem, and in [17] where a compactness argument is used to study a semiconductor model. Compared to the analytical studies the numerical analysis of the VPFP system, both in theory and implementations, is much less developed. In this setting the Monte Carlo simulations are explained in transport diffusion context ....

F. Golse and F. Poupaud, Limite fluide des 'equations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, J. Asympt. Anal., 6 (1992), 135--160.

....of the drift velocity parameter U and ffi = j 2 with j small. In fact, taking standard expansions in j yields, at the fluid level, the drift diffusion approximation with a small scaled Debye length which is an adequate approximation for relative low field regimes. See Poupaud [P1] Golse Poupaud [GP]. In addition, Poupaud [P1] P2] considered a high field scaling comparable to ours in the case where the electric field E is given, so that the system is reduced to the linearized Boltzmann equation. P1] contains a derivation of fluid equations by Hilbert expansion procedure and [P2] contains a ....

F. Golse and F. Poupaud, Limite fluide des e'quations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 135-160 (1992).

....small, and when simultaneously, the ratio of the mean time between collisions to the characteristic time scale is of order ff 2 . This limit has been extensively studied in the literature, in various contexts (see e.g. 23] 7] 4] for neutron transport, 3] for radiative transfer and in [24] [22] for semiconductors) In the present work, we investigate the diffusion limit for the collisionless Boltzmann equation describing trapped particles in a surface potential subject to elastic collisions with the surface. We shall focus on a formal result, leaving the rigorous proof of convergence to ....

F. Golse and F. Poupaud, Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 (1992), pp. 135--160.

....During the last recent years, many studies have been performed in this direction. The derivation of the Drift Diffusion model by a diffusive limit of the linear Boltzmann equation of semiconductors has been justified mathematically in [13] The case of Fermi Dirac statistics has been dealt with in [10]. More recently, the spherical harmonics expansion (SHE) model and the Energy Transport (ET)model have been obtained as diffusion limits of the Boltzmann equation [5, 4, 8, 17, 7] The SHE model is obtained by assuming that the dominant collision process is elastic collisions whereas the ET model ....

F. Golse, F. Poupaud, Limite fluide des 'equations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, J. on Asympt. Analysis 6, pp. 135--160, 1992.

.... for an introduction to the subject) Its application to bulk semiconductors is reviewed from a physics view point in [28] 12] The modern mathematical view of the theory has been set up in [2] in the context of neutron transport and its application to semiconductors has been developed in [27] [22]. In these works, the resulting macroscopic model is the Drift Diffusion model which is the basic tool in semiconductor modeling [25] 30] and which deals with the electron number density in position space. By analyzing the various collision scales, it has recently been possible to derive a ....

....that f ff formally converges to a function of (x; k) t) only. The second and third ones correspond to the derivations of the continuity and current equations (5.1) 5. 2) To achieve these goals, two methods can be developped: the Hilbert expansion method [2] 13] and the moment method [22]. We shall choose the latter because it involves more straightforward computations. In the present case, the establishment of the continuity equation is more difficult than in the usual case [22] due to the time discreteness of the scattering model (3.15) 3.16) The derivation of the current ....

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F. Golse and F. Poupaud, Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 (1992), pp. 135--160.

....framework of [6] Let us mention that in [5] is performed the derivation of the ET model under a different assumption on the dominant collisions, which leads to the same model with different expressions of the diffusion coefficients. The approach used here has been developed by Golse and Poupaud [22] for the DD model and is based on an entropy estimate and a mean compactness lemma [4] Here it is also necessary to study the link between the conservative and entropic variables, which was immediate in [22] The entropy estimate stated in the present paper is similar to the one established by ....

....the diffusion coefficients. The approach used here has been developed by Golse and Poupaud [22] for the DD model and is based on an entropy estimate and a mean compactness lemma [4] Here it is also necessary to study the link between the conservative and entropic variables, which was immediate in [22]. The entropy estimate stated in the present paper is similar to the one established by Desvillettes in [17] However, in the framework of [17] the theory of rarefied gazes) the energy is a parabolic function of the kinetic variable, which is not true in the present study. Due to this non ....

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F. Golse, F. Poupaud, Limite fluide des 'equations de Boltzmann des semi-- conducteurs pour une statistique de Fermi--Dirac, Asympt. Anal., 6, (1992), pp. 135-- 160.

....(the so called Larmor radius) This limitation results in the finiteness of the diffusivity without time rescaling. Mathematically, the present problem belongs to the class of diffusion approximation problems for kinetic equations (see e.g. 7] 4] in the context of neutron transport, 24] [20] for semiconductors and [15] for plasmas) Two methods are usually developped: the Hilbert expansion method [4] and the moment method [20] The latter, although providing only weak convergence results without explicit rates, is more flexible as it requires only mild regularity assumptions on the ....

.... the present problem belongs to the class of diffusion approximation problems for kinetic equations (see e.g. 7] 4] in the context of neutron transport, 24] 20] for semiconductors and [15] for plasmas) Two methods are usually developped: the Hilbert expansion method [4] and the moment method [20]. The latter, although providing only weak convergence results without explicit rates, is more flexible as it requires only mild regularity assumptions on the solution. It proceeds in three steps: i) prove that the time asymptotic profile of the distribution function is of the form F ( t) ....

F. Golse and F. Poupaud, Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 (1992), pp. 135--160.

....correctly represents the physics. By the Hilbert expansion method (assuming smallness of the scaled mean free path) macroscopic models can be derived from this kinetic equation. In particular, the low field DD model with field independent transport parameters has been justified in this way [5]. The inclusion of high field effects makes the problem significantly more difficult. Only with unrealistic assumptions on the scattering mechanisms, a DD model with field dependent mobility has been derived in [11] There the DD model is the result of a two step procedure. First, by a limit ....

F. Golse, F. Poupaud, Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptot. Anal. 6 (1992), pp. 135--160.

....function, and the free parameters are determined from the conservation laws corresponding to the collision invariants. Here this procedure is applied to the SHE model. A direct derivation from the Boltzmann equation (resulting in different transport coefficients) is also possible (see [3] [9]) The advantage of passing through the SHE model will become apparent below: When the relaxation time model for the elastic collision operator is used, then explicit formulas for the transport coefficients will be obtained. The elements of the null set of the electron electron collision operator ....

....equation in (4.1) by the relation T = T L . Then the model reduces to the charge conservation equation n( T L ) t Gamma r x Delta D 11 ( T L ) T L r x ( Gamma V ) # = 0 (4.4) for the determination of . Models of this form have been derived from the Boltzmann equation in [9] and [13] using simplified collision models driving the distribution function towards a Fermi Dirac distribution with T = T L . In [13] an explicit formula for the diffusivity has been derived. It differs from (4.2) which can be easily explained by the differences in the collision model. We ....

F. Golse, F. Poupaud, Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptot. Anal. 6 (1992), pp. 135--160.

.... 1 Introduction The drift diffusion (DD) model of semiconductors is, by far, the most simple and widely used macroscopic semiconductor model [23] 14] Its derivation from the more fundamental kinetic model, the semiconductor Boltzmann equation, has been made mathematically rigorous in [12], 19] However, the search for more accurate descriptions of carrier transport in semiconductors has been motivated by the recent developments of the microelectronics technology. Several models have been proposed to improve the DD model, including the hydrodynamic (HD) model [1] the ....

....6 Application to the energy transport and driftdiffusion models. The energy transport (ET) and drift diffusion (DD) models are usually directly derived from the semi conductor Boltzmann equation (2. 1) by a diffusion approximation using appropriate assumptions on the dominant scattering (see [4] [12]) However, using the knowledge that the equilibrium distribution of electrons is a Fermi Dirac distribution, they can also be derived from the SHE model by the moment method, using a Fermi Dirac Ansatz to compute the fluxes. We shall use this method to derive the ET and DD models for an ....

F. GOLSE, F. POUPAUD: "Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac ", Asymptotic Analysis 6 (1992), pp. 135-160.

....at the interface are considered. Using asymptotic analysis similiar to the usual boundary layer considerations as, e.g. in Bardos et al. 1] and Bensoussan et al. 2] for neutron transport, Cercignani [4] and Sone and coworkers [19] in the gas dynamics case and Poupaud [17] Golse Poupaud [8] and Yamnahakki [20] for semiconductor equations, we develop accurate coupling conditions in this case by an analysis of the interface layer between the two domains. This leads to kinetic linear half space problem. The essential point is then to find a fast approximate solution procedure of the ....

F. Golse, F. Poupaud, Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique Fermi-Dirac, J. Asympt. Anal. 6, 135, 1992

....than one in v occurs in the right hand side of the transport equation. 3 Nonlinear coefficient Let us now consider a nonlinear coefficient a 2 L 1 loc (R M ; R N ) and the associated transport operator t a(v) Delta r x . It appears either in relativistic or quantum kinetic equations [9], with M = N ; or in the kinetic formulation of scalar conservation laws [12] with M = 1. In this situation, a non degeneracy assumption on a is necessary to get the regularity of averages in v. Namely, for any direction oe 2 S N Gamma1 , a(v) Delta oe has to be non constant, otherwise the ....

F. Golse, F. Poupaud, Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asympt. Anal., 6, (1992), 135-- 160.

.... and in particular the popular drift diffusion equations (DD) and the hydrodynamic Euler Poisson system (HD) see [An] or again the reference book [MRS] and the literature quoted therein) The (rigorous) derivation of the DD model from the Boltzmann Poisson equations can be found in [P] and [GP]. The dominant collision mechanism in the derivation is the electron phonon scattering. This model works very well under the assumptions of low carrier densities and small electric fields. By contrast, hydrodynamic models are usually considered to describe high field phenomena or submicronic ....

F. Poupaud and F. Golse, Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6, pp.135-160, 1992.

....this allows to obtain the local compactness of such averages, and hence to pass to the limit in a sequence of solutions to kinetic equations, each time that averages in velocity are present. It is the case notably in Boltzmann s equation, in Vlasov Maxwell system [7] in quantum kinetic equations [16] or in the kinetic formulation of some conservation laws [27] Although these results are local, we can always reduce to global estimates by considering cutoff functions in x. Of course, we have to make an assumption on the function a(v) because obviously we have no further regularity if for ....

F. Golse, F. Poupaud, Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asympt. Anal., 6, (1992), 135-- 160.

....discuss the corresponding results for the low field scaling x x=fi, E fiE , t t=fi 2 (instead of (6.1) In this case, S(F ) is the dominant term in (4. 7) The limiting distribution is a Fermi Dirac distribution with a chemical potential determined by the low field drift diffusion model [8]. The formulas for the transport coefficients are, however, different from those in [8] since there the drift diffusion model has been derived directly from a kinetic equation with a different scattering operator and without the detour via the SHE equation. In the limit fi 0, 6.2) reduces to ....

....t=fi 2 (instead of (6.1) In this case, S(F ) is the dominant term in (4. 7) The limiting distribution is a Fermi Dirac distribution with a chemical potential determined by the low field drift diffusion model [8] The formulas for the transport coefficients are, however, different from those in [8], since there the drift diffusion model has been derived directly from a kinetic equation with a different scattering operator and without the detour via the SHE equation. In the limit fi 0, 6.2) reduces to Gamma E tr DE F = Phi 0 F F (1 Gamma F ) ....

F. Golse, F. Poupaud, Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptot. Anal. 6 (1992), pp. 135--160.

....at the interface are considered. Using asymptotic analysis similiar to the usual boundary layer considerations as, e.g. in Bardos et al. 1] and Bensoussan et al. 2] for neutron transport, Cercignani [4] and Sone and coworkers [20] in the gas dynamics case and Poupaud [17] Golse Poupaud [7] and Yamnahakki [21] for semiconductor equations, we develop accurate coupling conditions in this case by an analysis of the interface layer between the two domains. This leads to kinetic linear half space problem. The essential point is then to find a fast approximate solution procedure of the ....

F. Golse and F. Poupaud, Limite fluide des equations de Boltzmann des semi-conducteurs pour une statistique Fermi-Dirac, J. Asympt. Anal., 6 (1992), p. 135. DOMAIN DECOMPOSITION FOR SEMICONDUCTOR EQUATIONS 19

.... some links between models in different frameworks: Vlasov, drift diffusion, Euler, This kind of results are also related to diffusion approximation techniques which have been used widely in various contexts: transport equation of neutronics [2] radiative transfer [3] semiconductors physics [10, 12]. The techniques used in this paper are mainly based on the control of the kinetic and potential energy, the entropy of the system and also of some moments associated with the density. This implies, via the Dunford Pettis Theorem, the weak L 1 (IR 2N ) compactness of the sequence fae ffl g ....

Golse F., Poupaud F., Limite fluide des 'equations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, J. on Asympt. Analysis , 6 (1992), pp. 135-160.

....of the smooth function . This can be done with the original methods of proof in the references quoted above. In order to deal with equations of the type (1.1. 10) specifically, to be able to treat fractional derivatives in v in the right hand side) we use the method of [DP, L] and [G, Po] and adapt the computations to our case. In the sequel, we say that f(z) g(z) when f; g are two real valued functions of z) if there exists some constant C 0 (independant of z) such that f(z) C g(z) Let us first establish the following technical result. Lemma 2 For all x and y 2 R, ....

....term in the right hand side of (4.5) is O(y Gamma1=2 ) a similar computation shows that the first term is of exactly the same order. We proceed next to stating the main result in this section; it is an amplification of the Velocity Averaging results of [DP, L] and of the Appendix of [G, Po] We first need the following Notation 1 1. For ff 2]0; 2[ the following seminorm will be used in the sequel: khk 2;ff = Z T 1 Z 1=2 Gamma1=2 jh(w ) Gamma h(w)j 2 j j Gamma1 Gammaff d dw 1=2 ; 4:6) khk1;2;ff = sup w2T 1 Z 1=2 Gamma1=2 jh(w ) Gamma h(w)j 2 ....

Golse, F.; Poupaud, F., Limite fluide des 'equations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asymptotic Anal., 6, (1992), no. 2, 135--160.

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F. Golse and F. Poupaud, Limite fluide des e'quations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6 135-160 (1992).

No context found.

F. Golse and F. Poupaud, "Limite fluide des 'equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac," Asymptotic Analysis, Vol. 6, p. 135 (1992).

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