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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
![\begin{displaymath}
\dot{Q}_{{\mbox{\tiny irrev}}}(t)
\; = \;
\mu V^2 .
\end{displaymath}](img208.gif) |
(2.23) |
is a friction coefficient, named so because
he agency moving
has to do work against a nonzero
average force.
We have ohmic dissipation,
corresponding to an average force
which is proportional to velocity.
The value of
depends on the
energy distribution
at time
, and has the general expression
![\begin{displaymath}
\mu(\omega) \; = \; -\frac{1}{2} \int_0^\infty
dE \,\tilde...
...ox{\tiny E}}}(\omega))
\hspace{0.4in} \mbox{general $\rho$} .
\end{displaymath}](img211.gif) |
(2.24) |
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].
Next: Microcanonical averages and trajectory
Up: Review of classical dissipation
Previous: Fokker-Planck equation for energy
Alex Barnett
2001-10-03