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, or in quantum dots, 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 ), 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
The rate of increase in ensemble-averaged energy is