• Previous Article
    Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions
  • KRM Home
  • This Issue
  • Next Article
    A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[48]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[48]

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

[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.

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
[1]

Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks and Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002

[2]

Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793

[3]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[4]

Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic and Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1

[5]

Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1181-1197. doi: 10.3934/dcdss.2011.4.1181

[6]

Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic and Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029

[7]

Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185

[8]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[9]

Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic and Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169

[10]

Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125

[11]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[12]

Simone Calogero, Stephen Pankavich. On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field. Kinetic and Related Models, 2018, 11 (5) : 1063-1083. doi: 10.3934/krm.2018041

[13]

T. Colin, Géraldine Ebrard, Gérard Gallice. Semi-discretization in time for nonlinear Zakharov waves equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 263-282. doi: 10.3934/dcdsb.2009.11.263

[14]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629

[15]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845

[16]

Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017

[17]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306

[18]

Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851

[19]

Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519

[20]

Peter Benner, Tobias Breiten, Carsten Hartmann, Burkhard Schmidt. Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations. Journal of Computational Dynamics, 2020, 7 (1) : 1-33. doi: 10.3934/jcd.2020001

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (305)
  • HTML views (100)
  • Cited by (0)

[Back to Top]