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

  • * Corresponding author: K. Hopf

    * Corresponding author: K. Hopf 

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

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

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

    Citation:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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