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Conservative and non-conservative Boltzmann-type models of semiconductor theory. (English) Zbl 1139.47312

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.

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
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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)
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