Chapter 2: Quantum and classical theories of ergodic dissipation

I review the linear response theories of quantum and classical dissipation in general systems. Notation will be introduced which is necessary for fully appreciating the three chapters which follow, in particular the significance of the correlation spectrum (`band profile') for finding heating rate. In the final part I discuss and demonstrate correspondence between the two.

A system is `driven' by changing a parameter on which the Hamiltonian depends, in some prescribed function of time--the parameter is not a dynamical variable, unlike in some of the original studies. The Hamiltonian is assumed to be classically completely chaotic for all parameter values reached (this contrasts the nuclear dynamics example [118,29,158,149]). Dissipation is defined as irreversible growth of average energy. Rather than requiring a `bath' (for instance, harmonic oscillators in the quantum case [65], or the surrounding gas in the case of classical Brownian motion) which has infinite degrees of freedom, dissipation is a result of ergodicity in closed system with a small number of degrees of freedom (as little as 2). Dissipation can be thought of as a `friction force' felt by the agency responsible for changing the parameter, which arises from ergodic motion in the system.

The theory of dissipation in an ergodic system
has generally taken many forms appropriate to
various subfields (for instance, nuclear dynamics [118]).
Much of what I present in this chapter
relates to the work of Cohen[46,47],
who has attempted
to unify the field, outline a general theory of dissipation
and its various regimes of applicability.
In the picture of Cohen
the quantum and classical languages
are intertwined.
The first two parts of this chapter
(and the associated Appendix A)
can be read as a tutorial
introduction to this field.
For this reason I will present the quantum and classical cases in
separate languages, which only connect once the existence of
*stochastic energy spreading* has been established.
Therefore the following classical and quantum reviews can be read
somewhat independently of one another, although the connections are many.

- Review of classical dissipation in a general system
- Adiabatic invariance of phase-space volume
- Energy spreading--constant velocity case
- Energy spreading--periodic driving case
- Fokker-Planck equation for energy evolution
- Irreversible growth of average energy
- Microcanonical averages and trajectory averages

- Review of the linear response theory of quantum dissipation
- Basis choice, perturbation theory and the FGR
- Generalized force-force correlation and band profile
- Stochastic energy spreading
- Heating rate example--non-interacting fermions
- Relation to susceptibility
- Regime of applicability of linear response

- Quantum-classical correspondence