The Physical Basis of The Direction of Time (The Frontiers ...

204 Appendix: A Simple Numerical Toy Model
equilibrium distributions could be constructed by using intervals of different
lengths.
ThefirstplotoftheMathematica notebook 7 below (plot1) shows the entropy
evolution for n P = 100 and n S = 20 during the first 2000 units of time.
The relaxation time scale is of the order 1000 units. Only integer times are
plotted in order to eliminate otherwise disturbing lattice effects of the model.
At some later time (see plot2), the coarse-grained representation no longer reveals
any information about the existence of a low-entropy state in the recent
past, although this information must still exist, since motion can be reversed
in this deterministic model.
One can now simply enforce a two-time boundary condition (Sect. 5.3.3)
by restricting all relative velocities ∆v i to integer multiples of n S divided by
a large integer (e.g., by rounding them off at a certain figure). Consequences
of this change are negligible for small and intermediate times, although the
evolution is now exactly periodic (on an interval of 200 000 units in the chosen
numerical example, which is a very small fraction of the statistical recurrence
time). Relative entropy minima may occur at simple rational fractions of this
interval (such as at 2/5 of 200 000 in plot3).
When using the coarse-graining dynamically, one obtains a master equation
for mean occupation numbers ¯n j (t) in an ensemble of individual solutions.
For v 0 = 0 it would read [see (4.45)]
d¯n j
= λ(¯n j−1 +¯n j+1 − 2¯n j ) , (6.20)
dt
where λ (in this case given by ∆v/8) is the mean rate for particles to move by
one unit in either direction. For v 0 ≠ 0, the result would hold in the center-ofmass
frame. The lower smooth curve of plot4 shows the resulting monotonic
increase of ensemble entropy, compared with the individual (fluctuating) solution
of plot1.
Evidently, the two curves agree only for about the first 2/∆v units of time.
This demonstrates that this Zwanzig projection is not very appropriate for
dynamical purposes, since some fine-grained (neglected) information remains
dynamically relevant: the individually conserved velocities lead here to relevant
correlations between position and velocity. The master equation, which
does not distinguish between individual particles, allows them even to change
their direction of motion. A slightly improved long-time approximation can
therefore be obtained by assuming all particles to diffuse in only one direction
(upper smooth curve of plot4) – equivalent to using a reference frame that
moves with a velocity −∆v/2 in the center-of-mass frame. It still neglects dynamically
relevant correlations resulting from the fact that the most distant
particles continue travelling fastest. These correlations remain dynamically
relevant during relaxation, that is, until most particles have diffused once
around the ring. This can be recognized in plot4.
The master equation (6.20) may also represent an ensemble of individually
stochastic (indeterministic) histories n j (t), described by a Langevin equation.
7 Programming aid by Erich Joos is acknowledged.