On Quantum Fokker–Planck Equation

Abstract

The quantum Fokker–Planck equation (QFPE) is revisited. Provided that the molecule is the Maxwellian molecule, the quantum Landau–Fokker–Planck equation is divided into characteristic four terms. The characteristics of three terms among four terms are investigated on the basis of Grad’s method, whereas the characteristics of the remained term, which is attributed to the collisional term of the QFPE proposed by Kaniadakis–Quarati, when the distribution function of the colliding partner is under the equilibrium state, are numerically investigated. The numerical result indicates that the time evolution of the distribution function obtained using such a remained term is instable, when the equilibrium or nonequilibrium state is given as initial data of the distribution function. Such an instability of the distribution function can be described by analyzing the propagation of the plane harmonic wave in one dimensional velocity space.

Appendix: Derivation of Terms \(\fancyscript{M}_1\), \(\fancyscript{M}_2\), \(\fancyscript{M}_3\) and \(\fancyscript{M}_4\)

In this appendix, we derive four terms $\fancyscript{M}_1$, $\fancyscript{M}_2$, $\fancyscript{M}_3$ and $\fancyscript{M}_4$ from the quantum Landau–Fokker–Planck equation, when the molecule is the Maxwellian molecule. The $ij$-element of $3 \times 3$ matrix $A(g)$, namely, $\left[ A\left( g\right) \right] _{ij}$, is rewritten using the peculiar velocity, when the molecule is the Maxwellian molecule, as

$$\begin{aligned} \left[ A\left( g\right) \right] _{ij}=\left( \varvec{C}-\varvec{C}_1\right) ^2\delta _{ij}-\left( C_i-C_{1,i}\right) \left( C_j-C_{1,j}\right) , \end{aligned}$$

(24)

where $C_{1,i}=v_{1,i}-u_i$ is the peculiar velocity of the colliding partner.

Substituting $\left[ A\left( g\right) \right] _{ij}$ in Eq. (24) into the gain term of the quantum Landau–Fokker–Planck equation for the Maxwellian molecule in Eq. (1) ($G_{\text{ QFPE }}$), we obtain

$$\begin{aligned} G_{\text{ QFPE }}&=\frac{F}{4}\frac{m}{\hat{h}^3} \nabla _{\varvec{v}} \cdot \nonumber \\&\quad \!\times \!\int _{\fancyscript{V}^3_1} \left[ \left( \varvec{C}\!-\!\varvec{C}_1\right) ^2\delta _{ij}\!-\!\left( C_i\!-\!C_{1,i}\right) \left( C_j\!-\!C_{1,j}\right) \right] f\left( \varvec{v_1}\right) \left( 1\!-\!\theta f\left( \varvec{v_1}\right) \right) \nabla _{\varvec{v}} f\left( \varvec{v}\right) d\varvec{v}_1\nonumber \\&=\frac{F}{2}\Biggl [\frac{1}{2}\fancyscript{A} \frac{\partial }{\partial v_j} \left( C^2\delta _{ij}-C_i C_j\right) \frac{\partial f\left( \varvec{v}\right) }{\partial v_i}+\frac{1}{2}\fancyscript{B}_{ij} \frac{\partial ^2 f\left( \varvec{v}\right) }{\partial v_i \partial v_j}+\fancyscript{B}\frac{\partial ^2 f\left( \varvec{v}\right) }{\partial v_i^2}\nonumber \\&\quad -\frac{1}{2} \frac{\partial }{\partial v_j} \left\{ \left( \delta _{ij}\!-\!1\right) \delta _{ik}\delta _{jl} \,+\, 2\delta _{ij}\left( 1\!-\!\delta _{ik}\right) \left( 1 - \delta _{il}\right) \right\} \left( \Gamma _k C_l \,+\, \Gamma _l C_k\right) \frac{\partial }{\partial v_i} f\left( \varvec{v}\right) \Biggr ],\nonumber \\ \end{aligned}$$

(25)

where $\fancyscript{A}$, $\fancyscript{B}_{ij}$, $\fancyscript{B}$ and $\Gamma _i$ are defined by Eqs. (3–6), respectively.

Substituting $\left[ A\left( g\right) \right] _{ij}$ in Eq. (24) into the loss term of the quantum Landau–Fokker–Planck equation for the Maxwellian molecule in Eq. (1) ($L_{\text{ QFPE }}$), we obtain

$$\begin{aligned} L_{\text{ QFPE }}&=-\frac{F}{4} \frac{m}{\hat{h}^3} \nabla _{\varvec{v}} \cdot \nonumber \\&\,\,\, \times \int _{\fancyscript{V}^3_1} \left[ \left( \varvec{C}\!-\!\varvec{C}_1\right) ^2\delta _{ij}\!-\!\left( C_i\!-\!C_{1,i}\right) \left( C_j\!-\!C_{1,j}\right) \right] \! f\left( \varvec{v}\right) \left( 1\!-\!\theta f\left( \varvec{v}\right) \right) \nabla _{\varvec{v}_1} f\left( \varvec{v}_1\right) d\varvec{v}_1,\nonumber \\&=\frac{F}{2} \rho \frac{\partial C_i f\left( \varvec{v}\right) \left( 1-\theta f\left( \varvec{v}\right) \right) }{\partial v_i}. \end{aligned}$$

(26)

From Eqs. (25) and (26), we readily obtain Eqs. (2–6).