73 research outputs found

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

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    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 TT and the entropic parameter qq. The normalized qq-expectation value and the physical temperature \Tph were employed in this study. The normalized qq-expectation value was expanded as a series of the value (1βˆ’q)(1-q), where the absolute value ∣1βˆ’q∣|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 qq is smaller than that at qβ€²q' for q>qβ€²q>q'. The sigma mass at qq is lighter than that at qβ€²q' for q>qβ€²q>q' at low physical temperature, and the sigma mass at qq is heavier than that at qβ€²q' for q>qβ€²q>q' at high physical temperature. The pion mass at qq is heavier than that at qβ€²q' for q>qβ€²q>q'. The difference between the pion masses at different values of qq 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 ∣1βˆ’q∣|1-q|, and the pion mass is also affected by the statistics of small ∣1βˆ’q∣|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

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    We studied the escort averages in microcanonical and canonical ensembles in the Tsallis statistics of entropic parameter q>1q>1. The quantity (qβˆ’1)(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 CVCEC_V^{\mathrm{CE}} at constant volume in the canonical ensemble and the entropic parameter qq: 0<(qβˆ’1)CVCE<10 < (q-1) C_V^{\mathrm{CE}} < 1. This condition gives the known condition when CVCEC_V^{\mathrm{CE}} equals the number of ingredients NN. 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 CVCEC_V^{\mathrm{CE}} and the quantity (qβˆ’1)(q-1) are related to the energy fluctuation and the difference. It was shown that the magnitude of the relative difference ∣(SRqCEβˆ’SRqME)/SRqME∣|(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 SRqMES^{\mathrm{ME}}_{\mathrm{R}q} is the R\'enyi entropy in the microcanonical ensemble and SRqCES^{\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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