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| It is clear that we cannot use ordinary thermodynamics (or its well-known
extensions to small systems 1 or to small deviations from equilibrium 2) for
the description of the overall behavior of our model. First, it is not clear
that equilibrium prevails, even locally. Indeed we wish to investigate to what
extent equilibrium is reached in the course of an implosion-explosion process.
Second, our system has no fixed volume, it expands freely into the vacuum.
It is the combination of these two facts, no temperature and no volume, that
makes our approach di_erent from much previous work on the subject 3.
Equilibrium thermodynamics is linked to the motion of the individual
constituents making up the macroscopic system via the entropy 4,5. A natural
starting point for the investigation of the overall, i.e. the "thermodynamic",
behavior of our system is therefore to apply an expression similar to the
entropy, but in a way that makes sense in this highly non-equilibrium system.
To study one-body observables, we reduce the 6A dimensional phase-space
of the A particles to 6 dimensions in the standard way 4. |
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Then we introduce
a finite grid in the reduced phase-space, dividing each of the 6 axes into D
segments. Instead of working with a fixed grid in phase-space, which would
give us the usual entropya, we let the entire grid expand or contract along
with the swarm of points in phase-space in a uniform way: The outer grid
edge follows the outermost point, the boxes are of equal size, and the number
of boxes is kept fixed, thus the physical size (e.g. in units of h3) of each box
in phase-space varies with time. This is to deal with the no volume problem,
we mentioned above. We then introduce the pseudo-entropy as
Sum=-1/zSumipilog(pi) (2)
where
pi=number of points in box i / total number of points in phase space (2)
and z is a normalization constant. We choose z as the theoretical maximum
value of -Sumipilog(pi) so that Sum belongs [0, 1]. In the current set-up, for the case
of N points in a reduced phase-space divided into D6 boxes, z is the smaller
number of log(N) and 6 log(D). We refer to the quantity Sum as the pseudo-
entropy instead of entropy, since some important features of the entropy, e.g.
that it increases, are not retained in this formulation. Nevertheless, we shall
see that Sum has some nice properties, including that of characterizing the degree
of equilibrium. For a system of fixed volume in equilibrium, Sum is the usual
one-body entropy (apart from normalization).
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