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

José A. Carrillo; Katharina Hopf; Marie-Therese Wolfram

doi:10.3934/krm.2020017

Article Contents

2020, Volume 13, Issue 3: 507-529. Doi: 10.3934/krm.2020017

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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
^* 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

Abstract / Introduction Full Text(HTML) Figure(7) / Table(5) Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: 35Q84, 35Q40, 35K20, 35B44, 65M06.

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Figure 1. Long-time behaviour for symmetric mass-supercritical initial datum (P1) (d = 1, γ = 2:9).

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Figure 2. Long-time behaviour for asymmetric mass-supercritical initial datum (P2) (d = 1, γ = 2:9).

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Figure 3. The mass-subcritical cases (P3) and (P4), d = 1, γ = 2:9, A = 1:5:

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Figure 4. Long-time behaviour in mass-subcritical case (P5) (γ = 1, d = 3).

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Figure 5. Long-time behaviour for mass-supercritical initial value (P6) (d = 3, γ = 1, ε = 10−12, δ = 0).

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Figure 6. Transient condensate in the mass-subcritical case (P7) (d = 3, γ = 1, ε = δ = 10−10).

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Figure 7. Spatial blow-up profile in (P7).

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Table 1. Convergence to reference solution at time T = 0:025.

timesteps	meshsize	L_x² 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

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Table 2. Convergence to reference solution (on space-time grid).

timesteps	meshsize	L_{t, x}² 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

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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

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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

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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

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References

[1]	L. Almeida, F. Bubba, B. 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 Abdallah, I. 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. Blanchet, V. 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. Cañizo, J. A. Carrillo, P. 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. Carrillo, A. 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. Carrillo, M. 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. Carrillo, A. Jüngel, P. A. Markowich, G. 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. Carrillo, P. 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. Carrillo, S. Lisini, G. 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. Carrillo, H. 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. Carrillo, J. 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. Dolbeault, B. 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. Escobedo, S. 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. Evans, O. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM J. Math. Anal., 37 (2005), 737-751. doi: 10.1137/04061386X.
[31]	F. Filbet, J. 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. Hu, Q. 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. Lacaze, P. Lallemand, Y. 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.