Introducing a quantum kinetic model using the generalized Boltzmann equation in the complex phase space

In the present work, using the generalized Boltzmann equation of the first author [Phys. Rev. E 94, 023316 (2016)], a quantum kinetic model in the complex phase space is proposed. Employing the Chapman-Enskog analysis and applying the Wick rotation (using complex-valued relaxation time), it is shown that the present model recovers the time-dependent Schröedinger equation, while preserving the main features of the conventional lattice Boltzmann models, e.g., simplicity of implementation, second-order accuracy (in space), and convenience in parallel programming. The present results are numerically verified by simulating three benchmark problems.

Figure 1

Probability distribution function ${\left|\psi \right|}^2$ for free propagation of a 2D quantum wave packet, for $y=y_0$, at three different times. The solid lines denote the numerical results, and the symbols are the analytical results.

Figure 2

Comparison between $X\left(t\right)$, $Y\left(t\right)$, ${\mathrm{\Delta }}_x\left(t\right)$, and ${\mathrm{\Delta }}_y\left(t\right)$ profiles and associated analytical solutions given by Eqs. (33) and (34). The solid lines represent the numerical results, and the symbols denote the analytical results.

Figure 3

Comparison between $X\left(t\right)$ and associated analytical profiles. The solid line represents the numerical results, and the symbols denote the analytical results.

Figure 4

Contours of the real and imaginary parts of the dimensionless wave function, ${\psi }^{*}$; (bottom) real part, (top) imaginary part.

Figure 5

The real and imaginary parts of the dimensionless wave function, ${\psi }^{*}$, (solid lines) along the $x^{*}$ axis and at $y^{*}=0.5$, as compared with the exact solutions (symbols).

Figure 6

Relative $L_2$ norm error versus the lattice spacing in a log-log scale at $t^{*}=1$.