73 research outputs found

### Chiral phase transition in the linear sigma model within Hartree factorization in the Tsallis nonextensive statistics

We studied chiral phase transition in the linear sigma model within the
Tsallis nonextensive statistics in the case of small deviation from the
Boltzmann-Gibbs (BG) statistics. The statistics has two parameters: the
temperature $T$ and the entropic parameter $q$. The normalized $q$-expectation
value and the physical temperature \Tph were employed in this study. The
normalized $q$-expectation value was expanded as a series of the value $(1-q)$,
where the absolute value $|1-q|$ is the measure of the deviation from the BG
statistics. We applied the Hartree factorization and the free particle
approximation, and obtained the equations for the condensate, the sigma mass,
and the pion mass. The physical temperature dependences of these quantities
were obtained numerically. We found following facts. The condensate at $q$ is
smaller than that at $q'$ for $q>q'$. The sigma mass at $q$ is lighter than
that at $q'$ for $q>q'$ at low physical temperature, and the sigma mass at $q$
is heavier than that at $q'$ for $q>q'$ at high physical temperature. The pion
mass at $q$ is heavier than that at $q'$ for $q>q'$. The difference between the
pion masses at different values of $q$ is small for \Tph \le 200 MeV. That is
to say, the condensate and the sigma mass are affected by the Tsallis
nonextensive statistics of small $|1-q|$, and the pion mass is also affected by
the statistics of small $|1-q|$ except for \Tph \le 200 MeV.Comment: 9 pages, 6 figure

### Relation between the escort average in microcanonical ensemble and the escort average in canonical ensemble in the Tsallis statistics

We studied the escort averages in microcanonical and canonical ensembles in
the Tsallis statistics of entropic parameter $q>1$. The quantity $(q-1)$ is the
measure of the deviation from the Boltzmann-Gibbs statistics. We derived the
relation between the escort average in the microcanonical ensemble and the
escort average in the canonical ensemble. Conditions arise by requiring that
the integrals appeared in the canonical ensemble do not diverge. A condition is
the relation between the heat capacity $C_V^{\mathrm{CE}}$ at constant volume
in the canonical ensemble and the entropic parameter $q$: $0 < (q-1)
C_V^{\mathrm{CE}} < 1$. This condition gives the known condition when
$C_V^{\mathrm{CE}}$ equals the number of ingredients $N$. With the derived
relation, we calculated the energy, the energy fluctuation, and the difference
between the canonical ensemble and the microcanonical ensemble in the
expectation value of the square of Hamiltonian. The difference between the
microcanonical ensemble and the canonical ensemble in energy is small because
of the condition. The heat capacity $C_V^{\mathrm{CE}}$ and the quantity
$(q-1)$ are related to the energy fluctuation and the difference. It was shown
that the magnitude of the relative difference
$|(S^{\mathrm{CE}}_{\mathrm{R}q}-S^{\mathrm{ME}}_{\mathrm{R}q})/S^{\mathrm{ME}}_{\mathrm{R}q}|$
is small when the number of free particles is large, where
$S^{\mathrm{ME}}_{\mathrm{R}q}$ is the R\'enyi entropy in the microcanonical
ensemble and $S^{\mathrm{CE}}_{\mathrm{R}q}$ is the R\'enyi entropy in the
canonical ensemble. The similar result was also obtained for the Tsallis
entropy.Comment: 12 page

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