# American Institute of Mathematical Sciences

March  2014, 7(1): 1-8. doi: 10.3934/krm.2014.7.1

## A kinetic equation for spin polarized Fermi systems

 1 Mathematical Sciences, S-41296 Gothenburg, Sweden

Received  December 2012 Revised  May 2013 Published  December 2013

This paper considers a kinetic Boltzmann equation, having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures. The distribution functions have values in the space of positive hermitean $2\times2$ complex matrices. Global existence of weak solutions is proved in $L^1\cap L^{\infty}$ for the initial value problem of this Boltzmann equation in a periodic box.
Citation: Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
##### References:
 [1] J. Dolbeault, Kinetic models and quantum effects,, Arch. Rat. Mech. Anal., 127 (1994), 101.  doi: 10.1007/BF00377657.  Google Scholar [2] R. El Hajj, Étude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique,, Ph.D Thèse, (2008).   Google Scholar [3] R. El Hajj, Diffusion Models for Spin Transport Derived from the Spinor Boltzmann Equation,, to appear in Comm. in Math. Sci., ().   Google Scholar [4] J. Jeon and W. Mullin, Kinetic equation for dilute, spin-polarized quantum systems,, J. Phys. France, 49 (1988), 1691.  doi: 10.1051/jphys:0198800490100169100.  Google Scholar [5] D. S. Jin and C. A. Regal, Fermi Gas Experiments,, Proc. Int. School of Physics Enrico Fermi, (2008).   Google Scholar [6] P. L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications III,, J. Math. Kyoto Univ., 34 (1994), 539.   Google Scholar [7] J. Lukkarinen, P. Mei and H. Spohn, Global well-posedness of the spatially homogeneous Hubbard-Boltzmann equation,, , ().   Google Scholar [8] P. C. Nasher, G. Taslevin, M. Leduc, S. B. Crampton and F. Laloë, Spin rotation effects and spin waves in gaseous $^3He\uparrow$,, J. Phys. Lett., 45 (1984), 441.   Google Scholar [9] V. P. Silin, Introduction to the Kinetic Theory of Gases,, (in Russian), (1971).   Google Scholar

show all references

##### References:
 [1] J. Dolbeault, Kinetic models and quantum effects,, Arch. Rat. Mech. Anal., 127 (1994), 101.  doi: 10.1007/BF00377657.  Google Scholar [2] R. El Hajj, Étude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique,, Ph.D Thèse, (2008).   Google Scholar [3] R. El Hajj, Diffusion Models for Spin Transport Derived from the Spinor Boltzmann Equation,, to appear in Comm. in Math. Sci., ().   Google Scholar [4] J. Jeon and W. Mullin, Kinetic equation for dilute, spin-polarized quantum systems,, J. Phys. France, 49 (1988), 1691.  doi: 10.1051/jphys:0198800490100169100.  Google Scholar [5] D. S. Jin and C. A. Regal, Fermi Gas Experiments,, Proc. Int. School of Physics Enrico Fermi, (2008).   Google Scholar [6] P. L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications III,, J. Math. Kyoto Univ., 34 (1994), 539.   Google Scholar [7] J. Lukkarinen, P. Mei and H. Spohn, Global well-posedness of the spatially homogeneous Hubbard-Boltzmann equation,, , ().   Google Scholar [8] P. C. Nasher, G. Taslevin, M. Leduc, S. B. Crampton and F. Laloë, Spin rotation effects and spin waves in gaseous $^3He\uparrow$,, J. Phys. Lett., 45 (1984), 441.   Google Scholar [9] V. P. Silin, Introduction to the Kinetic Theory of Gases,, (in Russian), (1971).   Google Scholar
 [1] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [2] Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493 [3] Jean-Pierre Françoise, Hongjun Ji. The stability analysis of brain lactate kinetics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2135-2143. doi: 10.3934/dcdss.2020182 [4] François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417 [5] Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87 [6] Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006 [7] Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure & Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77 [8] Karl Peter Hadeler. Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1541-1560. doi: 10.3934/mbe.2013.10.1541 [9] Yan Li. Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5461-5480. doi: 10.3934/dcdsb.2019066 [10] Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot. Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1225-1242. doi: 10.3934/mbe.2018056 [11] Robert P. Gilbert, Philippe Guyenne, Ying Liu. Modeling of the kinetics of vitamin D$_3$ in osteoblastic cells. Mathematical Biosciences & Engineering, 2013, 10 (2) : 319-344. doi: 10.3934/mbe.2013.10.319 [12] Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719 [13] Sarthok Sircar, Anthony Roberts. Ion mediated crosslink driven mucous swelling kinetics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1937-1951. doi: 10.3934/dcdsb.2016030 [14] Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami Radunskaya, Ian Tucker. Predicting the drug release kinetics of matrix tablets. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 261-277. doi: 10.3934/dcdsb.2009.12.261 [15] Carlos J. García-Cervera, Xiao-Ping Wang. A note on 'Spin-polarized transport: Existence of weak solutions'. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2761-2763. doi: 10.3934/dcdsb.2015.20.2761 [16] M. I. Alomar, David Sánchez. Thermopower of a graphene monolayer with inhomogeneous spin-orbit interaction. Conference Publications, 2015, 2015 (special) : 1-9. doi: 10.3934/proc.2015.0001 [17] Lei Yang, Xiao-Ping Wang. Dynamics of domain wall in thin film driven by spin current. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1251-1263. doi: 10.3934/dcdsb.2010.14.1251 [18] Roy Malka, Vered Rom-Kedar. Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 475-502. doi: 10.3934/mbe.2011.8.475 [19] Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159 [20] Evans K. Afenya, Calixto P. Calderón. Growth kinetics of cancer cells prior to detection and treatment: An alternative view. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 25-28. doi: 10.3934/dcdsb.2004.4.25

2018 Impact Factor: 1.38