Abstract
In this chapter we develop a general mathematical theory for the moment equations for charged particles obtained by applying Levermore’s method (maximum entropy principle). We will show that the main drawbacks of this method disappear when it is applied to the semiclassical Boltzmann equation for semiconductors. In this case, the phase space is given by a space variable x, as usual, and a variable k which accounts for the crystal wave number, so that ħk has the dimension of a momentum. The variable varies over a bounded subset ß of Rn, called a Brillouin region. In this model, the velocity is a given function of k, which depends on the crystal energySince ß(k) the particles described by this model carry a charge, the Boltzmann equation is coupled to a Poisson equation for the electric potential which drives the particles. For this model, we prove a local existence result, and a global existence result of smooth solutions around equilibria.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alì, G.: Global existence of smooth solutions of the N-dimensional Euler-Poisson model. SIAM J. Math. Anal., 35, 389–422 (2003)
Alì, G., Bini, D., Rionero, S.: Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors. SIAM J. Math. Anal., 32, 572–587 (2000)
Alì, G., Jiingel, A.: Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas. J. Diff. Eqs., 190, 663–685 (2003)
Anile, A.M., Muscato, O.: Improved hydrodynamical model for carrier transport in semiconductors. Phys. Rev. B, 51,16728–16740 (1995)
Anile, A.M., Romano, V.: Non parabolic band transport in semiconductors: closure of the moment equations. Cont. Mech. Thermodyn., 11, 307–325 (1999)
Anile, A.M., Romano, V.: Hydrodynamical modeling of charge transport in semi-conductors. Meccanica, 35,219–296 (2000)
Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy, connexity and subcharacteristic conditions. Arch. Rational Mech. Anal., 137, 305–320 (1997)
Cercignani, C.: The Boltzmann Equation And Its Applications. Springer, Berlin (1988)
Chen, G.-Q., Jerome, J., Zhang, B.: Existence and the singular relaxation limit for the inviscid hydrodynamic energy model. In: Jerome, J. (ed.) Modelling and Computation for Application in Mathematics, Science and Engineering. Clarendon Press, Oxford (1998)
Fisher, A., Marsden, D.P.: The Einstein evolution equations as a first order quasilinear symmetric hyperbolic system. Commun. Math. Phys., 28, 1–38 (1972)
Hsiao, L., Markowich, P., Wang, S.: The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors. J. Differential Equations, 192, 111–133 (2003)
Jeffrey, A.: Quasi-Linear Hyperbolic Systems and Waves. Pitman, S. Francisco (1976)
Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (1993)
Kato, T.: The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Rational Mech. Anal., 58, 181–205 (1975)
Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Reg. Conf. Lecture No.ll, Philadelphia (1973)
Levermore, CD.: Moment closure hierarchies for the Boltzmann-Poisson equation. VLSI Design, 8, 97–101 (1995)
Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83,331–407(1996)
Luo, T., Natalini, R., Xin, Z.P.: Large-time behaviour of the solutions to a hydrodynamic model for semiconductors. SIAM J. Appl. Math., 59, 810–830 (1998)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Majorana, A.: Space homogeneous solutions of the Boltzmann equation describing electron-phonon interactions in semiconductors. Transp. Theory Stat. Phys., 20, 261–279(1991)
Majorana, A.: Conservation laws from the Boltzmann equation describing electron-phonon interactions in semiconductors. Transp. Theory Stat. Phys., 22, 849–859 (1993)
Majorana, A.: Equilibrium solutions of the non-linear Boltzmann equation for an electron gas in a semiconductor. II Nuovo Cimento, 108B, 871–877 (1993)
Markowich, P., Ringhofer, CA., Schmeiser, C: Semiconductor Equations. Springer, Wien (1990)
Mascali, G., Romano, V.: Hydrodynamical model of charge transport in GaAs based on the maximum entropy principle. Cont. Mech. Thermodyn., 14,405–423 (2002)
Miiller, I., Ruggeri,T.: Rational Extended Thermodynamics. Springer, Berlin (1998)
Romano, V: Maximum entropy principle for electron transport in semiconductors. In: Ciancio, V, Donato, A., Oliveri, F, Rionero, S. (eds) Proceedings “WASCOM 99 ” 10-th Conference on Waves and Stability in Continuous Media (7–12th June, 1999, Italy). World Scientific, Singapore (2001)
Romano, V: Non parabolic band transport in semiconductors: closure of the production terms in the moment equations. Cont. Mech. Thermodyn., 12,31–51 (2000)
Taylor, M.E.: Partial Differential Equations, I. Basic Theory. Springer, New York (1996)
Wu, N.: The Maximum Entropy Method. Springer, Berlin (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Giuseppe, A., Anile, A.M. (2004). Moment equations for charged particles: global existence results. In: Degond, P., Pareschi, L., Russo, G. (eds) Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8200-2_2
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8200-2_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6487-3
Online ISBN: 978-0-8176-8200-2
eBook Packages: Springer Book Archive