The collision integral in (2.20) has the following constants of
motion
In this article we will be dealing with the isotropic case only, and,
for simplicity, neglect the spin degree of freedom. Therefore we
simplify the collision integral by averaging it over all
angles. First, we change variables from particles momentum
to the particle kinetic energy
We introduce
and rewrite the kinetic equation as
However, the thermodynamic equilibrium is not the most general steady (equilibrium) solution of the kinetic equation and indeed in some cases has little relevance. The solutions we are most interested in are those which describe the steady state reached between ranges of frequencies where particles and energy are added to or removed from the system. These regions, where there is no pumping or dumping, are called "windows of transparency" or "inertial ranges". In particular, we have in mind the following situation. Particles and energy are added to the system in a narrow range of intermediate frequencies about . Particles and energy are drained from the system in a range of frequencies about and for . Because of conservation of energy and particles in the inertial ranges between and and between and where there is no pumping or damping and because the relations between particle number and energy density , we will find that a net flux of energy to the higher frequencies must be accompanied by a net flux of particles to lower frequencies as it might be expected by analogy with classical wave turbulence. The presence of sources and sinks drives the system away from the thermodynamic equilibrium. Therefore, in the windows of transparency, and , the system can also relax to equilibrium distributions corresponding to a finite flux of particles and energy flowing through these windows from the sources to the sinks. These are the new solutions of the QKE. The number of such finite flux solutions corresponds to the number of conserved densities (here two, and , or and ) of the QKE.
To demonstrate the existence of such solutions, we rewrite the KE in
the following form:
The relevant equation kinetic equation, which includes the presence of sources and
sinks is
(3.10) |
We are particularly interested in the solutions for which particles per unit time and units of energy per unit time are fed to the system in a narrow frequency window about . We will assume that the flux of particles passing through the left (right) window ( ) is () and the flux of energy though the right (left) window is (). We will also assume that the sinks consume all the particles and energy that reach them. Then (see Figure 1)
Figure 1
The first two relations in (3.14) express
conservation of particles and energy. The second two express the fact
that, in order to maintain equilibrium, the rate of particle
destruction at is the rate of energy destroyed there divided by
the energy per particle. Likewise the amount of energy destroyed at
(which absorption, in the context of application discussed in
the section 5, will be due to semiconductor lasing) must be
times the number of particles absorbed there. Solving (3.14)
we obtain
Solutions to (3.16), (3.17) have not been investigated even in the classical case. In the classical case, Zakharov (see, e.g., [22],[23]) had found the pure Kolmogorov solutions , which turns out to have power law behavior . Likewise, in the bosonic case, several authors have attempted to find power law solutions which essentially balance the quadratic terms in with a finite energy flux. However, in the differential approximation, there are no power law solutions.
In many cases it may be that whenever , may not be all that much smaller than . In particular, in order to exploit these solutions in the context of semiconductor lasers, it is advantageous to have close enough to to minimize energy losses (the ratio ) but far enough away to facilitate pumping unimpeded by Pauli blocking. We return to this application after introducing an enormous simplification for which gives a very good qualitative description of the collision integral.