Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion

The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.

Figure 1

Pictorial view of the initial condition taken in a two-dimensional cell of 256${}^2$ points with periodic boundary conditions. The outward velocity (or the density) takes distinct values inside and outside the circle of radius 10, while the remaining variables are taken homogeneously throughout the cell.

Figure 2

The thermal and the kinetic energies for the Maxwell-Boltzmann distribution, with the circle of radial velocity as initial condition, are shown here for the first 200 time steps, using both the d2q17 and d2q37 lattices. For time step 118 the traveling front wave hits the side of the unit cell.

Figure 3

The deviation of the temperature from its initial value is shown here for the first 5000 time steps using the circle of velocity as initial condition. The splitting among the FD, MB, and BE distribution curves is very small in comparison to the d2q17 and d2q37 lattice splitting and for this reason is not observable. The inset shows for a particular time-step window the splitting of the temperature deviation among the three distributions for the d2q37 lattice.

Figure 4

The kinetic energy of the FD, MB, and BE distributions using the d2q17 and d2q37 lattices is shown here for the first 1000 time steps. The initial condition used is of the circle of density. The inset shows in a particular time-step window the splitting of the kinetic energy among the three distributions for both d2q17 and d2q37 lattices.

Figure 5

The thermal energy of the MB distribution using the d2q17 and d2q37 lattices is shown here for the first 1000 time steps. The initial condition used is that of the circle of density. The inset shows in a particular time-step window the splitting of the kinetic energy among the three distributions for the d2q17 lattice.

Figure 6

The temperature deviation from its initial value is shown here for the case of a circle of density as initial condition for the first $20000$ time steps. The curves for the three distributions and the two lattices are clearly seen here. The inset shows the first 20 time steps.