Arlotti, L.; Banasiak, J.; Ciake, F. L. Ciake Conservative and non-conservative Boltzmann-type models of semiconductor theory. (English) Zbl 1139.47312 Math. Models Methods Appl. Sci. 16, No. 9, 1441-1468 (2006). Summary: We analyze the linear Boltzmann equation of semiconductor theory with unbounded collision term modelling both elastic and inelastic scattering of electrons on the crystalline lattice (corresponding to scattering on impurities and optical phonons), in both bounded and unbounded domains. We prove the existence of a substochastic semigroup solving this problem and, for a large class of scattering cross-sections, we also characterize the generator of this semigroup as the closure of the formal right-hand side operator showing thus that the semigroup is conservative (stochastic) in this case. On the other hand, we provide an example of a cross-section growing at an exponential rate for which the semigroup is not conservative. Cited in 6 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 45K05 Integro-partial differential equations 82C40 Kinetic theory of gases in time-dependent statistical mechanics 82D37 Statistical mechanics of semiconductors 47N20 Applications of operator theory to differential and integral equations 47N55 Applications of operator theory in statistical physics (MSC2000) 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow Keywords:linear Boltzmann equation; semiconductor theory; inelastic collisions; substochastic semigroups PDFBibTeX XMLCite \textit{L. Arlotti} et al., Math. Models Methods Appl. Sci. 16, No. 9, 1441--1468 (2006; Zbl 1139.47312) Full Text: DOI References: [1] DOI: 10.1016/0022-0396(87)90115-X · Zbl 0655.35068 [2] DOI: 10.1007/BF00048802 · Zbl 0734.45005 [3] DOI: 10.1016/j.jmaa.2004.01.028 · Zbl 1075.47023 [4] Banasiak J., Math. Mod. Meth. Appl. Sci. 10 pp 163– [5] Banasiak J., Taiwanese J. Math. 5 pp 169– [6] DOI: 10.1142/S0218202503002751 · Zbl 1063.47030 [7] DOI: 10.1142/S0218202504003325 · Zbl 1090.47026 [8] Banasiak J., Trans. Th. Statist. Phys. 30 pp 367– [9] DOI: 10.1081/TT-120014800 · Zbl 1029.76049 [10] DOI: 10.1016/S0022-247X(03)00154-9 · Zbl 1059.47046 [11] Banasiak J., Perturbations of Positive Semigroups with Applications (2006) · Zbl 1097.47038 [12] DOI: 10.1016/0022-247X(87)90252-6 · Zbl 0657.45007 [13] DOI: 10.1016/S0022-0396(03)00134-7 · Zbl 1133.82316 [14] DOI: 10.1007/978-3-0348-5478-8 [15] Hartman P., Ordinary Differential Equations (1964) · Zbl 0125.32102 [16] DOI: 10.1007/978-3-7091-6963-6 [17] DOI: 10.1080/00411459108203906 · Zbl 0800.76399 [18] DOI: 10.1137/S0036141095291397 · Zbl 0896.45006 [19] DOI: 10.1006/jmaa.2001.7444 · Zbl 0986.35112 [20] DOI: 10.1007/978-3-7091-6961-2 [21] DOI: 10.1142/S0218202595000309 · Zbl 0832.65142 [22] DOI: 10.1142/S0218202597000384 · Zbl 0884.45006 [23] DOI: 10.1103/PhysRevLett.58.892 [24] Norris J. R., Markov Chains (1998) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.