Once the stochasticity of energy spreading is established,
the quantum and classical pictures of dissipation coincide.
Therefore the reader is referred to the general diffusion equation and
expressions for heating rate in
Sections 2.1.4 and 2.1.5 of the classical review.
In these expressions, the classical
should be replaced by
,
and the the classical
divided by
.
However I will for introductory purposes present here the simple example of
non-interacting
fermions in a driven chaotic system.
This is relevant to nucleons in a deforming chaotic
nucleus [118,120,29], and to electrons in
small irregular metal particles[201], or in quantum dots[15],
subjected to external driving fields.
The simplicity arises because all the heating occurs
at the Fermi edge.
The Fermi `occupation function'
is the appropriate initial distribution
of single-particle states: the many-particle
wavefunction is a Slater determinant
of these states.
At low temperatures
is close to a unit step-function,
and subsequent diffusion smoothes out the step
as shown in Fig. 2.4.
One might ask how the exclusion principle affects the
evolution of the many-particle system: the answer is that at all
future times the many-particle
wavefunction is
correctly given by a Slater determinant of the single-particle states
(this follows from unitarity of time-evolution, also see [201]),
as long as any external interaction can be ignored.
Thus we can get the heating rate by evolving the single-particle
distribution function
.
What follows is therefore identical to the classical case.
The normalization is such that
gives the number of
particles in the system.
The diffusion equation is
where a prime means
-derivative.
The rate of increase in ensemble-averaged energy is