We now investigate numerically the DQKE to gain an intuitive understanding of its properties.
We begin by studying the time independent solutions of
(4.9) and solve (3.16) as an initial value problem with
given by
(see (4.9)),
In semiconductor lasers, it is the finite temperature effect of broadening the Fermi-Dirac distribution that contributes to inefficiency. If one could operate at , then one could simply choose the chemical potential, related to the total carrier number (see, e.g., [24]), so that the distribution cuts off immediately after the lasing frequency. However, the finite temperature broadens the distribution and means that one has to pump momentum values which play no role in the lasing process. The effect of the finite flux is to make effectively smaller.
We next solve (3.17) for the steady state solutions in the
larger momentum region with
:
We then consider the time evolution of the distribution function as given by the DQKE. The fundamental property of the kinetic equation that any distribution function relaxes to its thermodynamical equilibrium value in the absence of forcing (pumping/damping) is also true for the DQKE, as illustrated on Figure 3(a). There an initial distribution function, shown by a thin solid line, relaxes to the FD function, shown by a thick solid line, through several intermediate states shown by dashed lines. Since there is no forcing to the system, we take "fluxless" boundary conditions in (4.10) on the boundaries, so that no particles or energy cross the boundaries. The distribution relaxes to the FD distribution roughly by the time , which can be estimated as , where is the frequency where distribution approaches zero value. To check that the final distribution is indeed FD, we calculate and verify that it is a linear function. The total particle number and energy are conserved in our numerical runs to an accuracy of .
We then address the question of what is the steady (equilibrium) solution when the system has some external forcing. To model the forcing, we specify some positive flux of on the boundaries, and wait for the distribution to reach a new equilibrium, a hybrid state with a constant flux of , a zero flux, energy and particle number (see Figure 3(b)). The more the flux of number of particles is, the more the final distribution is bent in the manner of Figure 2 and according to (4.12). The total particle number and energy are the same for all curves on Figure 3(b).