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 $x(t)$. 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 ( $\propto \dot{x}^2$), 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 $\dot{x} = V$, and in the case of sinusoidal time-dependence $x(t)=A\sin(\omega t)$. 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]).

Figure 2.1: a) General picture: constant-energy contours in phase-space at initial time, and subsequent phase-space spreading about a moving energy shell. b) Invariance of phase-space volume $\Omega$ determines shell energy $E(x)$ in the adiabatic limit. This can be seen as contours of the surface $\Omega (E;x)$ in 3D. The elastic force $F(x)$ is the negative of the slope of $E(x)$. At any parameter $x$ and energy $E$, the shell weight $g(E;x)$ is the tangent slope in the constant-$x$ plane.