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June  2020, 13(3): 507-529. doi: 10.3934/krm.2020017

Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

2. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

* Corresponding author: K. Hopf

Current address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Received  May 2019 Revised  December 2019 Published  March 2020

Kaniadakis and Quarati (1994) proposed a Fokker–Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily long time, and thus enabling a numerical study of the condensation process in the Kaniadakis–Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis–Quarati model in $ 3 $D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

Citation: José A. Carrillo, Katharina Hopf, Marie-Therese Wolfram. Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons. Kinetic & Related Models, 2020, 13 (3) : 507-529. doi: 10.3934/krm.2020017
References:
[1]

L. AlmeidaF. BubbaB. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, Netw. Heterog. Media, 14 (2019), 23-41.  doi: 10.3934/nhm.2019002.  Google Scholar

[2]

R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipative schemes for nonlinear nonlocal equations with a gradient flow structure, arXiv e-prints, 2018. arXiv: 1811.11502. Google Scholar

[3]

J. Bandyopadhyay and J. J. L. Velázquez, Blow-up rate estimates for the solutions of the bosonic Boltzmann–Nordheim equation, J. Math. Phys., 56 (2015), 063302, 27pp. doi: 10.1063/1.4921917.  Google Scholar

[4]

W. Bao, Mathematical models and numerical methods for Bose–Einstein condensation, In Proceedings of the International Congress of Mathematicians–-Seoul 2014. Vol. IV, 971–996. Kyung Moon Sa, Seoul, 2014.  Google Scholar

[5]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[6]

W. Bao, L. Pareschi and P. A. Markowich, Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation, In Modeling and Computational Methods for Kinetic Equations, Model. Simul. Sci. Eng. Technol., 287–320. Birkhäuser Boston, Boston, MA, 2004.  Google Scholar

[7]

N. Ben AbdallahI. M. Gamba and G. Toscani, On the minimization problem of sub-linear convex functionals, Kinet. Relat. Models, 4 (2011), 857-871.  doi: 10.3934/krm.2011.4.857.  Google Scholar

[8]

A. BlanchetV. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical patlak–keller–segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

[9]

J. A. CañizoJ. A. CarrilloP. Laurençot and J. Rosado, The Fokker–Planck equation for bosons in 2D: Well-posedness and asymptotic behavior, Nonlinear Anal., 137 (2016), 291-305.  doi: 10.1016/j.na.2015.07.030.  Google Scholar

[10]

V. Calvez and T. O. Gallouët, Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., 36 (2016), 1175-1208.  doi: 10.3934/dcds.2016.36.1175.  Google Scholar

[11]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

[12]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Art. 53, 53 pp. doi: 10.1007/s00526-019-1486-3.  Google Scholar

[13]

J. A. CarrilloM. Di Francesco and G. Toscani, Condensation phenomena in nonlinear drift equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 145-171.   Google Scholar

[14]

J. A. Carrillo, B. Düring, D. Matthes and D. S. McCormick, A lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, J. Sci. Comput., 75 {2018), 1463–1499. doi: 10.1007/s10915-017-0594-5.  Google Scholar

[15]

J. A. Carrillo, K. Hopf and J. L. Rodrigo, On the singularity formation and relaxation to equilibrium in 1D Fokker–Planck model with superlinear drift, Adv. Math., 360 (2020), 106883, 66pp. doi: 10.1016/j.aim.2019.106883.  Google Scholar

[16]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[17]

J. A. CarrilloP. Laurenñot and J. Rosado, Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics, J. Differential Equations, 247 (2009), 2209-2234.  doi: 10.1016/j.jde.2009.07.018.  Google Scholar

[18]

J. A. CarrilloS. LisiniG. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258 (2010), 1273-1309.  doi: 10.1016/j.jfa.2009.10.016.  Google Scholar

[19]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Scientific Computing, 31 (2009), 4305-4329.  doi: 10.1137/080739574.  Google Scholar

[20]

J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202.  doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

[21]

J. A. CarrilloJ. Rosado and F. Salvarani, 1d nonlinear Fokker–Planck equations for fermions and bosons, Appl. Math. Lett., 21 (2008), 148-154.  doi: 10.1016/j.aml.2006.06.023.  Google Scholar

[22]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717.  doi: 10.1090/mcom3033.  Google Scholar

[23]

F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709.  doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[24]

J. DolbeaultB. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.  Google Scholar

[25]

M. Escobedo and S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl., 80 (2001), 471-515.  doi: 10.1016/S0021-7824(00)01201-0.  Google Scholar

[26]

M. EscobedoS. Mischler and J. Velázquez, Asymptotic description of Dirac mass formation in kinetic equations for quantum particles, J. Differential Equations, 202 (2004), 208-230.  doi: 10.1016/j.jde.2004.03.031.  Google Scholar

[27]

M. Escobedo and J. J. L. Velázquez, On the blow up and condensation of supercritical solutions of the nordheim equation for bosons, Comm. Math. Phys., 330 (2014), 331-365.  doi: 10.1007/s00220-014-2034-9.  Google Scholar

[28]

M. Escobedo and J. J. L. Velázquez, Finite time blow-up and condensation for the bosonic Nordheim equation, Invent. Math., 200 (2015), 761-847.  doi: 10.1007/s00222-014-0539-7.  Google Scholar

[29]

M. Escobedo and J. J. L. Velázquez, On the theory of weak turbulence for the nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 238 (2015), v+107pp. doi: 10.1090/memo/1124.  Google Scholar

[30]

L. C. EvansO. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM J. Math. Anal., 37 (2005), 737-751.  doi: 10.1137/04061386X.  Google Scholar

[31]

F. FilbetJ. Hu and S. Jin, A numerical scheme for the quantum {B}oltzmann equation with stiff collision terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 443-463.  doi: 10.1051/m2an/2011051.  Google Scholar

[32]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227.  doi: 10.1137/050628015.  Google Scholar

[33]

K. Hopf, On the Singularity Formation and Long-time Asymptotics in a Class of Nonlinear Fokker–Planck Equations, Thesis (Ph.D.)–University of Warwick, 2019. Google Scholar

[34]

J. HuQ. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, J. Sci. Comput., 62 (2015), 555-574.  doi: 10.1007/s10915-014-9869-2.  Google Scholar

[35]

K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., New York-London, 1963.  Google Scholar

[36]

G. Kaniadakis and P. Quarati, Classical model of bosons and fermions, Phys. Rev. E, 49 (1994), 5103-5110.   Google Scholar

[37]

R. LacazeP. LallemandY. Pomeau and S. Rica, Dynamical formation of a Bose–Einstein condensate, Phys. D, 152/153 (2001), 779-786.  doi: 10.1016/S0167-2789(01)00211-1.  Google Scholar

[38]

X. Lu, The Boltzmann equation for Bose–Einstein particles: Condensation in finite time, J. Stat. Phys., 150 (2013), 1138-1176.  doi: 10.1007/s10955-013-0725-9.  Google Scholar

[39]

X. Lu, Long time convergence of the Bose–Einstein condensation, J. Stat. Phys., 162 (2016), 652-670.  doi: 10.1007/s10955-015-1427-2.  Google Scholar

[40]

X. Lu, Long time strong convergence to Bose–Einstein distribution for low temperature, inet. Relat. Models, 11 (2018), 715-734.  doi: 10.3934/krm.2018029.  Google Scholar

[41]

P. A. Markowich and L. Pareschi, Fast conservative and entropic numerical methods for the boson Boltzmann equation, Numer. Math., 99 (2005), 509-532.  doi: 10.1007/s00211-004-0570-5.  Google Scholar

[42]

D. Matthes and H. Osberger, Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[43]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker–Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[44]

D. V. Semikoz and I. I. Tkachev, Kinetics of bose condensation, Phys. Rev. Lett., 74 (1995), 3093-3097.  doi: 10.1103/PhysRevLett.74.3093.  Google Scholar

[45]

D. V. Semikoz and I. I. Tkachev, Condensation of bosons in the kinetic regime, Phys. Rev. D, 55 (1997), 489-502.  doi: 10.1103/PhysRevD.55.489.  Google Scholar

[46]

A. Soffer and M.-B. Tran, On the dynamics of finite temperature trapped Bose gases, Adv. Math., 325 (2018), 533-607.  doi: 10.1016/j.aim.2017.12.007.  Google Scholar

[47]

J. Sopik, C. Sire and P.-H. Chavanis, Dynamics of the Bose–Einstein condensation: Analogy with the collapse dynamics of a classical self-gravitating Brownian gas, Phys. Rev. E (3), 74 (2006), 011112, 15pp. doi: 10.1103/PhysRevE.74.011112.  Google Scholar

[48]

H. Spohn, Kinetics of the Bose–Einstein condensation,, Phys. D, 239 (2010), 627-634.  doi: 10.1016/j.physd.2010.01.018.  Google Scholar

[49]

G. Toscani, Finite time blow up in kaniadakis–quarati model of bose–einstein particles, Comm. Partial Differential Equations, 37 (2012), 77-87.  doi: 10.1080/03605302.2011.592236.  Google Scholar

show all references

References:
[1]

L. AlmeidaF. BubbaB. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, Netw. Heterog. Media, 14 (2019), 23-41.  doi: 10.3934/nhm.2019002.  Google Scholar

[2]

R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipative schemes for nonlinear nonlocal equations with a gradient flow structure, arXiv e-prints, 2018. arXiv: 1811.11502. Google Scholar

[3]

J. Bandyopadhyay and J. J. L. Velázquez, Blow-up rate estimates for the solutions of the bosonic Boltzmann–Nordheim equation, J. Math. Phys., 56 (2015), 063302, 27pp. doi: 10.1063/1.4921917.  Google Scholar

[4]

W. Bao, Mathematical models and numerical methods for Bose–Einstein condensation, In Proceedings of the International Congress of Mathematicians–-Seoul 2014. Vol. IV, 971–996. Kyung Moon Sa, Seoul, 2014.  Google Scholar

[5]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[6]

W. Bao, L. Pareschi and P. A. Markowich, Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation, In Modeling and Computational Methods for Kinetic Equations, Model. Simul. Sci. Eng. Technol., 287–320. Birkhäuser Boston, Boston, MA, 2004.  Google Scholar

[7]

N. Ben AbdallahI. M. Gamba and G. Toscani, On the minimization problem of sub-linear convex functionals, Kinet. Relat. Models, 4 (2011), 857-871.  doi: 10.3934/krm.2011.4.857.  Google Scholar

[8]

A. BlanchetV. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical patlak–keller–segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

[9]

J. A. CañizoJ. A. CarrilloP. Laurençot and J. Rosado, The Fokker–Planck equation for bosons in 2D: Well-posedness and asymptotic behavior, Nonlinear Anal., 137 (2016), 291-305.  doi: 10.1016/j.na.2015.07.030.  Google Scholar

[10]

V. Calvez and T. O. Gallouët, Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., 36 (2016), 1175-1208.  doi: 10.3934/dcds.2016.36.1175.  Google Scholar

[11]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

[12]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Art. 53, 53 pp. doi: 10.1007/s00526-019-1486-3.  Google Scholar

[13]

J. A. CarrilloM. Di Francesco and G. Toscani, Condensation phenomena in nonlinear drift equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 145-171.   Google Scholar

[14]

J. A. Carrillo, B. Düring, D. Matthes and D. S. McCormick, A lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, J. Sci. Comput., 75 {2018), 1463–1499. doi: 10.1007/s10915-017-0594-5.  Google Scholar

[15]

J. A. Carrillo, K. Hopf and J. L. Rodrigo, On the singularity formation and relaxation to equilibrium in 1D Fokker–Planck model with superlinear drift, Adv. Math., 360 (2020), 106883, 66pp. doi: 10.1016/j.aim.2019.106883.  Google Scholar

[16]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[17]

J. A. CarrilloP. Laurenñot and J. Rosado, Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics, J. Differential Equations, 247 (2009), 2209-2234.  doi: 10.1016/j.jde.2009.07.018.  Google Scholar

[18]

J. A. CarrilloS. LisiniG. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258 (2010), 1273-1309.  doi: 10.1016/j.jfa.2009.10.016.  Google Scholar

[19]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Scientific Computing, 31 (2009), 4305-4329.  doi: 10.1137/080739574.  Google Scholar

[20]

J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202.  doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

[21]

J. A. CarrilloJ. Rosado and F. Salvarani, 1d nonlinear Fokker–Planck equations for fermions and bosons, Appl. Math. Lett., 21 (2008), 148-154.  doi: 10.1016/j.aml.2006.06.023.  Google Scholar

[22]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717.  doi: 10.1090/mcom3033.  Google Scholar

[23]

F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709.  doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[24]

J. DolbeaultB. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.  Google Scholar

[25]

M. Escobedo and S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl., 80 (2001), 471-515.  doi: 10.1016/S0021-7824(00)01201-0.  Google Scholar

[26]

M. EscobedoS. Mischler and J. Velázquez, Asymptotic description of Dirac mass formation in kinetic equations for quantum particles, J. Differential Equations, 202 (2004), 208-230.  doi: 10.1016/j.jde.2004.03.031.  Google Scholar

[27]

M. Escobedo and J. J. L. Velázquez, On the blow up and condensation of supercritical solutions of the nordheim equation for bosons, Comm. Math. Phys., 330 (2014), 331-365.  doi: 10.1007/s00220-014-2034-9.  Google Scholar

[28]

M. Escobedo and J. J. L. Velázquez, Finite time blow-up and condensation for the bosonic Nordheim equation, Invent. Math., 200 (2015), 761-847.  doi: 10.1007/s00222-014-0539-7.  Google Scholar

[29]

M. Escobedo and J. J. L. Velázquez, On the theory of weak turbulence for the nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 238 (2015), v+107pp. doi: 10.1090/memo/1124.  Google Scholar

[30]

L. C. EvansO. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM J. Math. Anal., 37 (2005), 737-751.  doi: 10.1137/04061386X.  Google Scholar

[31]

F. FilbetJ. Hu and S. Jin, A numerical scheme for the quantum {B}oltzmann equation with stiff collision terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 443-463.  doi: 10.1051/m2an/2011051.  Google Scholar

[32]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227.  doi: 10.1137/050628015.  Google Scholar

[33]

K. Hopf, On the Singularity Formation and Long-time Asymptotics in a Class of Nonlinear Fokker–Planck Equations, Thesis (Ph.D.)–University of Warwick, 2019. Google Scholar

[34]

J. HuQ. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, J. Sci. Comput., 62 (2015), 555-574.  doi: 10.1007/s10915-014-9869-2.  Google Scholar

[35]

K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., New York-London, 1963.  Google Scholar

[36]

G. Kaniadakis and P. Quarati, Classical model of bosons and fermions, Phys. Rev. E, 49 (1994), 5103-5110.   Google Scholar

[37]

R. LacazeP. LallemandY. Pomeau and S. Rica, Dynamical formation of a Bose–Einstein condensate, Phys. D, 152/153 (2001), 779-786.  doi: 10.1016/S0167-2789(01)00211-1.  Google Scholar

[38]

X. Lu, The Boltzmann equation for Bose–Einstein particles: Condensation in finite time, J. Stat. Phys., 150 (2013), 1138-1176.  doi: 10.1007/s10955-013-0725-9.  Google Scholar

[39]

X. Lu, Long time convergence of the Bose–Einstein condensation, J. Stat. Phys., 162 (2016), 652-670.  doi: 10.1007/s10955-015-1427-2.  Google Scholar

[40]

X. Lu, Long time strong convergence to Bose–Einstein distribution for low temperature, inet. Relat. Models, 11 (2018), 715-734.  doi: 10.3934/krm.2018029.  Google Scholar

[41]

P. A. Markowich and L. Pareschi, Fast conservative and entropic numerical methods for the boson Boltzmann equation, Numer. Math., 99 (2005), 509-532.  doi: 10.1007/s00211-004-0570-5.  Google Scholar

[42]

D. Matthes and H. Osberger, Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[43]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker–Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[44]

D. V. Semikoz and I. I. Tkachev, Kinetics of bose condensation, Phys. Rev. Lett., 74 (1995), 3093-3097.  doi: 10.1103/PhysRevLett.74.3093.  Google Scholar

[45]

D. V. Semikoz and I. I. Tkachev, Condensation of bosons in the kinetic regime, Phys. Rev. D, 55 (1997), 489-502.  doi: 10.1103/PhysRevD.55.489.  Google Scholar

[46]

A. Soffer and M.-B. Tran, On the dynamics of finite temperature trapped Bose gases, Adv. Math., 325 (2018), 533-607.  doi: 10.1016/j.aim.2017.12.007.  Google Scholar

[47]

J. Sopik, C. Sire and P.-H. Chavanis, Dynamics of the Bose–Einstein condensation: Analogy with the collapse dynamics of a classical self-gravitating Brownian gas, Phys. Rev. E (3), 74 (2006), 011112, 15pp. doi: 10.1103/PhysRevE.74.011112.  Google Scholar

[48]

H. Spohn, Kinetics of the Bose–Einstein condensation,, Phys. D, 239 (2010), 627-634.  doi: 10.1016/j.physd.2010.01.018.  Google Scholar

[49]

G. Toscani, Finite time blow up in kaniadakis–quarati model of bose–einstein particles, Comm. Partial Differential Equations, 37 (2012), 77-87.  doi: 10.1080/03605302.2011.592236.  Google Scholar

Figure 1.  Long-time behaviour for symmetric mass-supercritical initial datum (P1) (d = 1, γ = 2:9).
Figure 2.  Long-time behaviour for asymmetric mass-supercritical initial datum (P2) (d = 1, γ = 2:9).
Figure 3.  The mass-subcritical cases (P3) and (P4), d = 1, γ = 2:9, A = 1:5:
Figure 4.  Long-time behaviour in mass-subcritical case (P5) (γ = 1, d = 3).
Figure 5.  Long-time behaviour for mass-supercritical initial value (P6) (d = 3, γ = 1, ε = 10−12, δ = 0).
Figure 6.  Transient condensate in the mass-subcritical case (P7) (d = 3, γ = 1, ε = δ = 10−10).
Figure 7.  Spatial blow-up profile in (P7).
Table 1.  Convergence to reference solution at time T = 0:025.
timesteps meshsize Lx2 error rate
1000 50 7.3825e-3 -
1000 100 2.1290e-3 1.7939
1000 200 5.6056e-4 1.9253
1000 400 1.4222e-4 1.9788
1000 800 3.5598e-5 1.9982
1000 1600 8.8061e-6 2.0152
1000 3200 2.0991e-6 2.0687
timesteps meshsize Lx2 error rate
1000 50 7.3825e-3 -
1000 100 2.1290e-3 1.7939
1000 200 5.6056e-4 1.9253
1000 400 1.4222e-4 1.9788
1000 800 3.5598e-5 1.9982
1000 1600 8.8061e-6 2.0152
1000 3200 2.0991e-6 2.0687
Table 2.  Convergence to reference solution (on space-time grid).
timesteps meshsize Lt, x2 error rate
10 50 6.1372e-3 -
20 100 3.1393e-3 0.9671
40 200 1.5817e-3 0.9890
80 400 7.8542e-4 1.0099
160 800 3.8200e-4 1.0399
320 1600 1.7877e-4 1.0955
640 3200 7.6728e-5 1.2203
timesteps meshsize Lt, x2 error rate
10 50 6.1372e-3 -
20 100 3.1393e-3 0.9671
40 200 1.5817e-3 0.9890
80 400 7.8542e-4 1.0099
160 800 3.8200e-4 1.0399
320 1600 1.7877e-4 1.0955
640 3200 7.6728e-5 1.2203
Table 3.  Convergence to reference solutions using CN and (P3).
timesteps meshsize $ L^2_{t, x} $ error rate
10 50 5.2392e-3 -
20 100 1.1085 e-3 2.2408
40 200 2.4257 e-4 2.1921
80 400 5.6873e-05 2.0926
160 800 1.3983e-05 2.0241
timesteps meshsize $ L^2_{t, x} $ error rate
10 50 5.2392e-3 -
20 100 1.1085 e-3 2.2408
40 200 2.4257 e-4 2.1921
80 400 5.6873e-05 2.0926
160 800 1.3983e-05 2.0241
Table 4.  Convergence to exact solution at the final time $ T = 0.04 $.
number of mesh size $ L^2 $ error rate
time points (at time $ T $)
4000 25 6.2783e-3 -
4000 50 2.2323e-3 1.4919
4000 100 7.9661e-4 1.4866
4000 200 2.6080e-4 1.6109
4000 400 7.7921e-5 1.7428
4000 800 1.9283e-5 2.0147
number of mesh size $ L^2 $ error rate
time points (at time $ T $)
4000 25 6.2783e-3 -
4000 50 2.2323e-3 1.4919
4000 100 7.9661e-4 1.4866
4000 200 2.6080e-4 1.6109
4000 400 7.7921e-5 1.7428
4000 800 1.9283e-5 2.0147
Table 5.  Convergence to reference solution (space-time grid).
number of mesh size full $ L^2 $ error rate
time points
4 25 8.3850e-4 -
8 50 4.1295e-4 1.0218
16 100 2.0813e-4 0.9885
32 200 1.0427e-4 0.9971
64 400 5.1996e-5 1.0039
128 800 2.5774e-5 1.0125
number of mesh size full $ L^2 $ error rate
time points
4 25 8.3850e-4 -
8 50 4.1295e-4 1.0218
16 100 2.0813e-4 0.9885
32 200 1.0427e-4 0.9971
64 400 5.1996e-5 1.0039
128 800 2.5774e-5 1.0125
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