Review of classical dissipation in a general system

Here I review the mechanism for dissipation in
a classical ergodic system whose Hamiltonian is
dependent on an external parameter .
An example model to keep in mind
is that of a gas particle trapped inside a deformable chaotic cavity,
where heating of the particle (increase in
expectation value of the particle kinetic energy) can occur
due to time-dependent deformation (driving).
Note that
the cavity wall is ``cold'', *i.e. *there are no
thermal fluctuations in its velocity.
Heating rate in the cavity case will be discussed in detail in
Chapters 3 and 4.

The heating has two components:
I will only be concerned with the *irreversible*
part, which we will see is due to broadening of the distribution in energy.
The *reversible* part, which vanishes for any cyclic parameter change,
is due to solely to changes of phase space volume
and corresponds to adiabatic heating found in
the usual thermodynamic treatment of a classical gas.
In terms of the *generalized force* acting on the parameter,
the former corresponds to a friction force
(
), and the latter an elastic
force (gradient of a conservative potential).
Note that the notion of irreversibility relies on the assumption that
you are not allowed to ``look'' at the particle in order to decide how
to vary the deformation (clearly in that case, any heating is `reversible').

As is usual in classical statistical mechanics, a `state' of the system will mean a distribution in phase space. The expectation of quantity is then defined over this distribution (this will sometimes be referred to as an `ensemble average'). It is simplest to derive the energy spreading rate assuming an initial distribution which is microcanonical (uniform in a single energy shell). I will derive this for the case of the parameter changing slowly and non-periodically in time with velocity , and in the case of sinusoidal time-dependence . Energy spreading leads to an energy diffusion equation. The heating rate can then be found for arbitrary initial distribution.

This field is quite young: Koonin and Randrup [120] first derived the dissipation rate in the context of one-body nuclear viscous forces, in 1977, using classical linear response theory. My presentation generally follows that of the wave of activity since 1990, chiefly Wilkinson [201] (following on the work of Ott [153]), Jarzynski [106,107], and Cohen [47,46]. I will not consider the interesting case of more than one parameter, in which Berry's phase effects arise (see [27]).

- 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