FiniteSize Effects and Critical Behavior of the QCD Deconfinement Phase Transition

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Finite-Size Effects and Critical Behavior of the
QCD Deconfinement Phase Transition
M. Ladrem, A. Ait-El-Djoudi and G.
YezzaLaboratoire de Physique des Particules et
Physique StatistiqueEcole Normale
Supérieure-Kouba, Algiers, Algeria Quark Matter
2002, Nantes - France
Presented by Amel Ait-El-Djoudi
2
Introduction
Phase transitions are infinitely sharp only in
the thermodynamical limit .
However, real systems and the systems we simulate
are finite Limited reaction zone in which
the formation of the Quark Gluon Plasma in a
relativistic heavy ion collision is assumed to
take place Lattice QCD calculations are
performed in finite lattices In general, finite
size effects lead to a possible mixed phases
system and a smoothing of the transition
How to sign a possible phase transition in
a finite system ? We can extract the true
critical behavior of infinite systems from
calculations in finite systems, by studying how
some thermodynamic quantities vary with the size
of the system by a Finite Size Scaling (FSS)
analysis.
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Finite size effects on the Deconfinement Phase
Transition
Phenomenological model used in 1 based on the
coexistence of the hadronic and QGP phases in the
total system volume Fractional
volumes and
, characterized by the parameter
, pure hadron phase

, pure QGP phase The mean
value of an intensive thermodynamical quantity
with
the Partition Function (PF) of the total
system. The PF of a pion gas
1 C. Spieles, H. Stocker and C. Greiner, Phys.
Rev. C57 (1998) 908.
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For the QGP, the color singlet PF is where
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Let us examine the behavior of some thermodynamic
quantities of the system, with temperature at
for a temperature driven phase transition ,
and with chemical potential at a fixed
temperature for a
density driven phase transition. We consider the
two lightest quark flavors, and a bag constant
. The mean value of the
order parameter (the hadronic volume fraction) is
given by
The mean values of the energy and entropy
densities are respectively
with
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Figure 1 Variations of (a) the order parameter,
the energy density normalized by , and the
entropy density normalized by , versus
temperature at for different system
sizes, and (b) the order parameter, the energy
density and the entropy density versus chemical
potential for various
system sizes.
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Figure 2 Critical behavior of the
susceptibility , the specific heat
and the second derivative of the
order parameter ( for a
temperature driven DPT).
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Figure 3 (left) Susceptibility as
a function of temperature for different volumes,
and (right) fit of the results for the maxima of
the susceptibility to a power law of the volume.
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For a temperature driven DPT, the specific heat
is defined as
Figure 4 Variations of (left) the specific heat
density with temperature for
different system volumes, and of (right) the
maxima of the specific heat with volume.
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Figure 5 Plot of the shift of the critical
temperature versus inversed volume.
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The width of the temperature region over which
the transition is rounded can be defined by the
gap
i. e. , and the temperatures
at which the second derivative of the order
parameter reaches its maxima
Figure 6 (left) Variations of the second
derivative of the order parameter versus
temperature for different volumes, and (right)
data of the width of the critical region versus
volume.
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Influence of the finiteness of the system
size on the behavior of thermodynamical
quantities near criticality The
sharp transition observed in the thermodynamic
limit, signaled by discontinuities in the first
derivatives of the thermodynamic potential at a
critical temperature or chemical potential, is
rounded off in finite volume and the variations
of the thermodynamic quantities are perfectly
smooth on the hole range of temperature or
chemical potential. Determination of the
critical exponents for the temperature driven
DPT. Our results are in good agreement with the
analytical ones
found in 2. These results are
characteristic of the first order phase
transition, as predicted by the FSS theory.
A further FSS analysis can be carried out for the
density driven DPT (See poster on panel N 92 in
the session of Monday).
2 Poster presented at the present conference
by Dr M. LADREM (Panel N 174 in the session of
Monday)