Complication with all simulations
In the demonstrations so far, we note that the potential
energy rises from near zero to say 14kT,
and we say that 14kT of work has been done.
This is a little sloppy, because some of this work could be "done"
by spontaneous fluctuations in the absence of any driver.
For example, if the energy U(Q) is a linear function, U(Q)=Q,
then at equilibrium the average value of U is 1kT.
Not a big deal.
If on the other hand we define U(Q)=log(Q/Q0),
the natural fluctuations of U are expected to be much bigger.
One way of thinking about this is that the smallest values of
U have very little measure dQ associated with them;
so while the Boltzmann factor favouring Q=Q0=0.001 over Q=Q1=1
is large (exp(log(1000))) = 1000, the interval (1.000,1.050)
is 1000 times bigger than the interval (0.001,0.00105).
The marginal distribution of U is
P(U) = exp(U) / (dU/dQ) .
For U = log(Q/Q0), (dU/dQ)=1/Q, so
P(U) is uniform!
The demonstrations thus far are therefore rather misleading.
We aim to make new simulations accompanied
by more precise statements.
The figures below clarify the situation.
Here the potential is
U(Q) = 1.9 log(Q/Q0) + 0.1 Q
(note the extra log(Q) compared with what has been simulated up to now).
The first three figures show a simulation in which the driver and piston are independent.
The driver (red line) wanders off; ignore it!
The piston's velocity is thermalized whenever it hits the wall at Q=Q0.
The piston spends a lot of time near to Q=Q0=0.001,
but huge excursions in energy do take place.
This figure shows the histogram of "work done":
The next three figures show a standard simulation, with thermalizations taking place
at the walls only. Piston and driver bounce off each other as normal.
Notice more work is typically done than was done in the earlier demonstrations.
The next four figures show standard simulations, with thermalizations taking place
every 1 time unit, for both particles (top 2) and for the driver alone (bottom 2).
Piston and driver bounce off each other as normal.
The final figure2 shows results for the case where the charged exit path potential is
included.
