Irreversible growth of average energy

How does the energy spreading cause growth of the average energy?
It is a result of both the
diffusion term
being *energy-dependent*,
and of the extra drift (bias) due to transformation from to
(see Fig. 2.2b).
In the billiard context, the former effect is easy to visualize:
we can imagine that particles which happen to
have a slightly increased kinetic energy will hit the walls more often,
thus diffusing faster in energy.
We will see in Chapter 4 that the
dependence is
, independent of .
The latter effect is less intuitive.
Considering that
for billiards it is not hard to show that the ratio of the latter to the
former effect is , independent of energy.
Therefore which effect dominates depends on the system dimensionality.

We compute the rate of change of average energy, substituting
(2.21) and integrating by parts,

The first term is the reversible heating (easily verified using the microcanonical case ). The second is irreversible since it relies on diffusion; it can be written (by substituting for ) as

is a

It is interesting that Koonin and Randrup derived an equivalent expression (Eq. (2.23) of [120]), without explicitly considering energy spreading. If we start in a particular choice of ensemble (initial ), and diffusion has not yet caused to differ much from , then therefore takes special forms:

where the subscript means evaluation at the Fermi energy . The Fermi distribution has a constant phase-space density of , normalised to represent a single particle. For an application to many non-interacting fermions, see Section 2.2.4. The case of the canonical distribution, relevant to a thermal gas of non-interacting classical particles, is covered by Cohen (Section 4 of [46]) and I do not discuss it here. After long times, if the system remains isolated, it can be that is very different from , and the special forms (2.25) or (2.26) will have to be replaced by (2.24).

What I have presented is only the dissipation appearing to lowest order in (Jarzynski [107] discusses this by way of an expansion in powers of a `slowness parameter' ). There will be higher-order terms in the friction which become more relevant as approaches for example the limit (2.12). There are other ways that the classical spreading picture can break down. For instance for less than the correlation time the energy spreading should be much less than itself. Also for this same the average energy increase should remain small compared to the spreading. These set additional upper limits on [46].