Solutions and properties of the DQKE

Let us now rewrite the DQKE in the form:

$$\dot N_\omega = \frac{\partial^2}{\partial \omega ^2} {\cal W}[n_\omega ],$$

$${\cal W}^{\rm fermionic} [n_\omega ] = -I\left( { n_\omega ^4 \frac{\partial^2}{\partial\omega ^2}(\frac... ...^2 \frac{\partial^2}{\partial\omega ^2}(\ln (n_\omega ))}\right)\times\omega^s,$$

$${\cal W}^{\rm bosonic} [n_\omega ] = I\left( { n_\omega ^4 \frac{\partial^2}{\partial\omega ^2}(\frac{... ...^2 \frac{\partial^2}{\partial\omega ^2}(\ln (n_\omega ))}\right)\times\omega^s,$$

$${\cal W}^{\rm classical} [n_\omega ] = I \left( { n_\omega ^4 \frac{\partial^2}{\partial\omega ^2} (\frac{1}{n_\omega })} \right)\times\omega^s.$$

We can now use (3.11) to calculate the fluxes $P$ and $Q$ in terms of $n_\omega$ and its derivatives. We concentrate on the fermionic case. There,

$$Q=\frac{\partial {\cal W}}{\partial\omega } = I s \omega ^{s-1}\left(-n'^2(2n-1)-n n''(1-n)\right)$$

$$+ I \omega ^s\left(-2n'^3+n'n''(1-2n)+n n'''(n-1) \right)$$

$$P=\left({\cal{W}}-\omega \frac{\partial{\cal W}}{\partial \omega }\right) = I \omega ^s(1-s)\left(-n'^2(2n-1)-n n''(1-n)\right)$$

$$-I \omega ^{s+1} \left(-2n'^3+n'n''(1-2n)+n n'''(n-1) \right).$$

Let us make a change of variables $n=1/(G+1)$ and $n=1/(e^m+1)$. $n, G, m$ are functions of $\omega$ and $t$, ``dot'' is used to denote differentiation with respect to time, and ``prime'' with respect to $\omega$. Then

$$\dot G = (1+G)^2 \frac{I}{\Omega _0 k^{d-1} (dk/d\omega )} \frac{\partial^2}{\partial \omega ^2}\left(\omega ^s\frac{G'^2-G G''}{(1+G)^4} \right)$$

$${\rm or}$$

$$\dot m_\omega = - \frac{I}{\Omega _0 k^{d-1} (dk/d\omega )} \cosh^2 (\frac{m}{2}) \frac{\partial^2}{\partial\omega ^2}\omega ^s \frac{m''}{4 \cosh^4(\frac{m}{2})}$$ (4.5)

Since the stationary DQKE is a fourth order ODE, its solutions will have four free parameters. Indeed, assume a steady (equilibrium) state and integrate (4.9,4.11) twice to get

$${\cal W}[n_\omega ] = Q \omega + P,$$

$${\rm or}$$

$$G = \exp{(-\frac{\partial^{-2}}{\partial \omega ^2} \frac{(Q\omega +P)(G+1)^4}{I\omega ^s G^2})},$$

$${\rm or}$$

$$m''&= \frac{Q \omega + P}{I\omega ^s}\cosh ^4(\frac{m}{2}),$$ (4.6)

where $Q$ and $P$ are fluxes of particle number and energy. For $P=Q=0$, (4.12) trivially gives

$$n(\omega )=\frac{1}{e^{\frac{1}{T}( \omega -\mu)}+1},$$

the Fermi-Dirac distribution function. Therefore we observe that the Fermi-Dirac distribution function corresponds to a zero flux solution of the kinetic equation, consistent with our findings of the previous section.