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In the calculations presented here, we have chosen D = 7, so the 6 dimensional
phase-space is divided into 76 boxes. Because we are dealing with a small
number of events (typically 20) in the ensembles, the precise values of Sum
depend on the choice of grid. We have, however, verified that Sum behaves
qualitatively similar to what is shown here over a range of grid-sizes, varying
D between 2 and 9.
To give an idea of the dynamics and the timescales, we show in Fig. 1 the
scattering rate.
| Figure 1. The scattering rate (number of scatterings per particle per time unit) for the four
cases mentioned in the text. In the implosion-explosion event ('in') practically all particles
scatter around the time t ~= 6 fm/c. This is the time when the system is maximally
compressed. Then, as the expansion begins, the scattering rate decreases until t ~= 15 fm/c,
when interactions have essentially ceased. In the '100out7' case the particles start to hit
the container wall at t ~= 5 fm/c (the scatterings against the container walls are not counted
here), and the peak in the scattering rate at t . 20 fm/c results from particles moving back
after hitting the wall and scattering against other particles still on their way out.
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Fig. 2 shows how the pseudo-entropy behaves in each of the four cases.
In the 'th20' case, the pseudo-entropy Sum ~= 1 as long as the particles are in
equilibrium at fixed volume inside the container. Then at t = 20 fm/c, when
the container is removed and the system starts to expand, the pseudo-entropy
decreases, reflecting the fact that the system goes out of equilibriumb.
In the case '100out7', where the particles are started in an extreme non-
equilibrium situation, the pseudo-entropy is low (Sum= 0.8 is a low value in
this context), but increases towards Sim = 1 as the scatterings equilibrize the
system. By comparison with Fig. 1 one can see that the first jump in Sum at
t ~= 5 fm/c is due to particles scattering against the container wall (when
particles hit the wall their velocity is reversed, so in this process many new
states in phase-space are being populated), and the second "jump" around
t ~= 20 fm/c is due to the many particle-particle scatterings around this time.
The interesting case is the implosion-explosion ('in') scenario, since here
we do not know in advance if the system reaches a state of equilibrium or
not. From the fact that the pseudo-entropy in Fig. 2 reaches a value Sum . 1,
the same value as the known equilibrium case 'th20', we infer that the system
is in a state of equilibrium around t ~= 6 fm/c (which is also the time of
maximum compression). We have checked that the speed distribution of the
particles becomes nearly Maxwellian from t ~= 6 fm/c with a temperature
in the compressed state of 47 MeV, which is also the theoretical value of
the temperature assuming that all of the initial flow energy is converted to
thermal energy.
Another interesting feature of the pseudo-entropy is that it seems to decay
in a characteristic fashion when the system expands from a state of equilib-
rium, see Fig. 3.
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Figure 2. The pseudo-entropy for the four cases described in the text. This variable seems
to quantify the degree of equilibrium in the system, Sum = 1 characterising an equilibrium
state.Comment: a: in the limit D --> infinity in an ensemble of infinitely many events
b:The particles stay almost thermalized, though, in the sense that they retain their Maxwell-
Boltzmann velocity distribution. But they are certainly not in equilibrium, since this means
that the phase-space distribution is independent of time.
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Conclusions
In this note we address the problem of thermodynamic equilibration in the
context of heavy-ion collisions. We have defined a variable inspired by the
entropy which, at least for the cases we have considered, seems to characterize
the degree of equilibrium in an a priori highly non-equilibrium process such
as an explosion. Now, more theoretical work needs to be done in order to
understand why Sum behaves in this seemingly interesting way.
Acknowledgements
This work was in part supported by the Danish Natural Science Research
Council. GN thanks the Niels Bohr Institute for kind hospitality and The
Leon Rosenfeld Foundation for financial support during the preparation of
this work. JPB thanks the Nuclear Theory Group at GSI, where part of this
work was done. INM thanks The Humbold Foundation for financial support.
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Figure 3. The pseudo-entropy in a log-log plot, together with the functions: 1.52 t-0.2
and 1.87 t-0.2. The decrease of Sum during the initial expansion of the system seems to
follow a power law when a state of equilibrium was present, in contrast to the case '50/50'
(intended to simulate the expansion from a not-fully thermalized state) which does not
show this behavior. At later times Sum decreases less rapidly and turns over to approach a
finite limiting value, one sees the beginning of this behavior at the curve 'in'.
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