Heating rate example--non-interacting fermions

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 $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ should be replaced by $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$, and the the classical $g(E)$ divided by $(2\pi \hbar)^d$.

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' $f(E)$ is the appropriate initial distribution of single-particle states: the many-particle wavefunction is a Slater determinant of these states. At low temperatures $f(E)$ 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 $f$. What follows is therefore identical to the classical case. The normalization is such that $N = \int g(E) f(E)$ gives the number of particles in the system.

The diffusion equation is $\dot{f} = (D_{{\mbox{\tiny E}}} f')'$ where a prime means $E$-derivative. The rate of increase in ensemble-averaged energy is

$$\langle \dot{E} \rangle \; = \; \int dE \, E \dot{f}(E) g(E) \; = \; -\int dE \, (E g(E))' D_{{\mbox{\tiny E}}} f'$$

$$= g(E_{{\mbox{\tiny F}}}) D_{{\mbox{\tiny F}}} ,$$ (2.52)

where the boundary term in the integration by parts vanishes, $f'$ is replaced by a negative delta-function, and a constant density of states $g(E_{{\mbox{\tiny F}}})$ can be assumed near the Fermi energy. The diffusion rate at the Fermi energy I call $D_{{\mbox{\tiny F}}}$. Heating of the Fermi gas can be interpreted as friction on the parameter $x$. The friction coefficient $\mu$ is defined by $\langle \dot{E}\rangle = \mu \langle \dot{x}^2 \rangle$. Comparing (2.52) and (2.51) gives

$$\mu(\omega) \; = \; \frac{g(E_{{\mbox{\tiny F}}})}{2} \tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega).$$ (2.53)

In the zero-frequency limit, $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)\rightarrow \tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(0) \equiv \nu_{{\mbox{\tiny E}}}$ which is the (dc) fluctuations intensity of the observable $\mathcal{F}$. This resulting relation $\mu = \mbox{\small$\frac{1}{2}$}g(E_{{\mbox{\tiny F}}}) \nu_{{\mbox{\tiny F}}}$ is an example of a fluctuation-dissipation relation. Different relations arise in different ensembles--for instance the canonical ensemble produces the traditional fluctuation-dissipation relation at fixed temperature [127,65,46].