December 2018, 11(6): 1503-1526. doi: 10.3934/krm.2018059

Fractional diffusion limits of non-classical transport equations

1. 

Karlsruhe Institute of Technology, Steinbuch Center for Computing, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

2. 

Dept. of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada

* Corresponding author: Martin Frank

Received  July 2017 Revised  October 2017 Published  June 2018

Fund Project: The first author is supported by the German research foundation DFG under grant FR2841/6-1

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.

Citation: Martin Frank, Weiran Sun. Fractional diffusion limits of non-classical transport equations. Kinetic & Related Models, 2018, 11 (6) : 1503-1526. doi: 10.3934/krm.2018059
References:
[1]

N. B. AbdallahA. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900. doi: 10.3934/krm.2011.4.873.

[2]

E. Albano and H. Martin, Temperature-programmed reactions with anomalous diffusion, J. Phys. Chem., 92 (1988), 3594-3597. doi: 10.1021/j100323a054.

[3]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.

[4]

T. Camminady, M. Frank and E. Larsen, Nonclassical particle transport in heterogeneous materials, M & C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Korean Nuclear Society, 2017, On USB.

[5]

B. Carreras, V. Lynch and G. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096. doi: 10.1063/1.1416180.

[6]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous diffusion in plasma physics, Appl. Math. Lett., 25.

[7]

N. CrouseillesH. Hivert and M. Lemou, Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit, Comptes Rendus Mathematique, 353 (2015), 755-760. doi: 10.1016/j.crma.2015.05.003.

[8]

A. Davis and A. Marshak, Solar radiation transport in the cloudy atmosphere: A 3D perspective on observations and climate impacts, Rep. Prog. Phys., 73 (2010), 026801. doi: 10.1088/0034-4885/73/2/026801.

[9]

E. d'Eon, Rigorous asymptotic and moment-preserving diffusion approximations for generalized linear boltzmann transport in arbitrary dimension, Transp. Theory Stat. Phys., 42 (2013), 237-297. doi: 10.1080/00411450.2014.910231.

[10]

M. Frank and T. Goudon, On a generalized Boltzmann equation for non-classical particle transport, Kinet. Relat. Models, 3 (2010), 395-407. doi: 10.3934/krm.2010.3.395.

[11]

F. Golse, Recent results on the periodic Lorentz gas, in Nonlinear Partial Differential Equations, 39–99, Adv. Courses Math. CRM Barcelona, Birkh'auser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0191-1_2.

[12]

E. Larsen, A generalized Boltzmann equation for non-classical particle transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 112 (2011), 619-631. doi: 10.1016/j.jqsrt.2010.07.003.

[13]

E. Larsen and R. Vasques, A generalized linear boltzmann equation for non-classical particle transport, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 619-631. doi: 10.1016/j.jqsrt.2010.07.003.

[14]

J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math., 174 (2011), 225-298. doi: 10.4007/annals.2011.174.1.7.

[15]

J. Marklof and B. Toth, Superdiffusion in the periodic Lorentz gas, Commun. Math. Phys., 347 (2016), 933-981. doi: 10.1007/s00220-016-2578-y.

[16]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.

[17]

A. Mellet and S. Merino-Aceituno, Anomalous energy transport in FPU-beta chain, J. Stat. Phys., 160 (2015), 583-621. doi: 10.1007/s10955-015-1273-2.

[18]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[19]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[20]

K. Pfeilsticker, First geometrical path lengths probability density function derivation of the skylight from spectroscopically highly resolving oxygen A-band observations. 2. derivation of the levy-index for the skylight transmitted by mid-latitude clouds, J. Geophys. Res., 104 (1999), 4104-4116.

[21]

Y. Sagi, M. Brook, I. Almog and N. Davidson, Observation of anomalous diffusion and fractional self-similarity in one dimension, Phys. Rev. Lett., 108 (2012), 093002. doi: 10.1103/PhysRevLett.108.093002.

[22]

E. SchumacherE. Hanert and E. Deleersnijder, Front dynamics in fractional-order epidemic models, J. Theo. Biol., 279 (2011), 9-16. doi: 10.1016/j.jtbi.2011.03.012.

[23]

R. Vasques and E. Larsen, Anisotropic diffusion in model 2-d pebble-bed reactor cores, in Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications, American Nuclear Society, 2009, On CD-ROM.

[24]

G. ViswanathanV. AfanasyevS. BuldyrevE. MurphyP. Prince and H. Stanley, Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0.

[25]

K. VynckM. BurresiF. Riboli and D. Wiersma, Photon management in two-dimensional disordered media, Nature Materials, 11 (2012), 1017-1022. doi: 10.1038/nmat3442.

[26]

L. Wang and B. Yan, An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit, J. Comput. Phys., 312 (2016), 157-174. doi: 10.1016/j.jcp.2016.02.034.

show all references

References:
[1]

N. B. AbdallahA. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900. doi: 10.3934/krm.2011.4.873.

[2]

E. Albano and H. Martin, Temperature-programmed reactions with anomalous diffusion, J. Phys. Chem., 92 (1988), 3594-3597. doi: 10.1021/j100323a054.

[3]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.

[4]

T. Camminady, M. Frank and E. Larsen, Nonclassical particle transport in heterogeneous materials, M & C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Korean Nuclear Society, 2017, On USB.

[5]

B. Carreras, V. Lynch and G. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096. doi: 10.1063/1.1416180.

[6]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous diffusion in plasma physics, Appl. Math. Lett., 25.

[7]

N. CrouseillesH. Hivert and M. Lemou, Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit, Comptes Rendus Mathematique, 353 (2015), 755-760. doi: 10.1016/j.crma.2015.05.003.

[8]

A. Davis and A. Marshak, Solar radiation transport in the cloudy atmosphere: A 3D perspective on observations and climate impacts, Rep. Prog. Phys., 73 (2010), 026801. doi: 10.1088/0034-4885/73/2/026801.

[9]

E. d'Eon, Rigorous asymptotic and moment-preserving diffusion approximations for generalized linear boltzmann transport in arbitrary dimension, Transp. Theory Stat. Phys., 42 (2013), 237-297. doi: 10.1080/00411450.2014.910231.

[10]

M. Frank and T. Goudon, On a generalized Boltzmann equation for non-classical particle transport, Kinet. Relat. Models, 3 (2010), 395-407. doi: 10.3934/krm.2010.3.395.

[11]

F. Golse, Recent results on the periodic Lorentz gas, in Nonlinear Partial Differential Equations, 39–99, Adv. Courses Math. CRM Barcelona, Birkh'auser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0191-1_2.

[12]

E. Larsen, A generalized Boltzmann equation for non-classical particle transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 112 (2011), 619-631. doi: 10.1016/j.jqsrt.2010.07.003.

[13]

E. Larsen and R. Vasques, A generalized linear boltzmann equation for non-classical particle transport, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 619-631. doi: 10.1016/j.jqsrt.2010.07.003.

[14]

J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math., 174 (2011), 225-298. doi: 10.4007/annals.2011.174.1.7.

[15]

J. Marklof and B. Toth, Superdiffusion in the periodic Lorentz gas, Commun. Math. Phys., 347 (2016), 933-981. doi: 10.1007/s00220-016-2578-y.

[16]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.

[17]

A. Mellet and S. Merino-Aceituno, Anomalous energy transport in FPU-beta chain, J. Stat. Phys., 160 (2015), 583-621. doi: 10.1007/s10955-015-1273-2.

[18]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[19]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[20]

K. Pfeilsticker, First geometrical path lengths probability density function derivation of the skylight from spectroscopically highly resolving oxygen A-band observations. 2. derivation of the levy-index for the skylight transmitted by mid-latitude clouds, J. Geophys. Res., 104 (1999), 4104-4116.

[21]

Y. Sagi, M. Brook, I. Almog and N. Davidson, Observation of anomalous diffusion and fractional self-similarity in one dimension, Phys. Rev. Lett., 108 (2012), 093002. doi: 10.1103/PhysRevLett.108.093002.

[22]

E. SchumacherE. Hanert and E. Deleersnijder, Front dynamics in fractional-order epidemic models, J. Theo. Biol., 279 (2011), 9-16. doi: 10.1016/j.jtbi.2011.03.012.

[23]

R. Vasques and E. Larsen, Anisotropic diffusion in model 2-d pebble-bed reactor cores, in Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications, American Nuclear Society, 2009, On CD-ROM.

[24]

G. ViswanathanV. AfanasyevS. BuldyrevE. MurphyP. Prince and H. Stanley, Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0.

[25]

K. VynckM. BurresiF. Riboli and D. Wiersma, Photon management in two-dimensional disordered media, Nature Materials, 11 (2012), 1017-1022. doi: 10.1038/nmat3442.

[26]

L. Wang and B. Yan, An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit, J. Comput. Phys., 312 (2016), 157-174. doi: 10.1016/j.jcp.2016.02.034.

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