medium Airton Deppman Compsyst - Rio de Janeiro, October, 2013 Nonextensive thermodynamics of hadronic medium Airton Deppman

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1 of ic ic matter: of ic Compsyst - Rio de Janeiro, October, 2013

2 of ic ic matter: Hagedorn s A ic system is considered as an ideal gas of s a a T. The partition function is ln[1 + Z(V 0, T )] = V 0T 2π 2 n=1 1 n 2 0 ρ(m; n)m 2 K 2 (β n m) dm. The bootstrap idea is: Fireball is the static equillibrium of a system composed by fireballs, which on their turn are... (goto ) The partition function can also be written as Z(V 0, T ) = E 0 σ(e )Z 0 (E )de According to the bootstrap principle, both forms of partition function must be asymptotically identical with ln[σ(e )] = ln[ρ(m)]

3 of ic I ic matter: Tsallis entropy can be written form (J. Stat. Phys. 52 (1988) 479.) W S q = k p i ln q p i, (1) i=1 where we defined the q-logarithm function ln q x = x 1 q 1 1 q (q R). (2) The q-exponential function is defined as the inverse of the q-logarithm e x q [1 + (1 q)x] 1/1 q (q R), (3)

4 of ic ic matter:

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8 of ic ic matter:

9 of ic ic matter: e Z q (V o, T ) = ln[1 + Z q (V o, T )] = V o 2π 2 The bootstrap principle Z q (V o, T ) = 0 = exp 0 σ(e)[1 + (q 1)βE] n=1 1 n 0 dm 0 q (q 1) de dp p 2 ρ(n; m) [1 + (q 1)β p 2 + m 2 ] nq (q 1), σ(e)[1 + (q 1)βE] { Vo 2π 2 β 3/2 At the same time we must have 0 q (q 1) de } dm m 3/2 ρ(m)[1 + (q 1)βm] 1 q 1 1 ln[σ(e)] = ln[ρ(m)]

10 of ic ic matter: Mass spectrum and density of states The self-consistency principle is satisfied if and m 3/2 ρ(m) = γ m [ 1 + (qo 1)β o m ] 1 qo 1 = γ m [1 + (q o 1)m] β o q o 1 σ(e) = be a[ 1 + (q o 1)E ] βo q o 1 Using properties of Γ(z) function it results that for (q o 1) 0, ( ) a+1 1 Z q (V o, T ) bγ(a + 1) β β o Then both expression for the partition function Z q converge if a + 1 = α = γv o 2π 2 β 3/2 Limiting : β o and entropic index: q o. A. Deppman, Physica A 391 (2012)

11 of ic Comparison with experiment ic matter:

12 of ic Evidences from e + e collisions ic matter: Bediaga (2000).

13 of ic Evidences from pp collisions ic matter: T (MeV) T: Hagedorn-Tsallis s (TeV) (a) Chinellato: Au+Au: 62 GeV Chinellato: Au+Au: 200 GeV Chinellato: p+p: 200 GeV This work I. Sena and AD Eur. Phys. J. A 49 (2013) 17 q q Chinellato: Au+Au: 62 GeV Chinellato: Au+Au: 200 GeV Chinellato: p+p: 200 GeV This work s (TeV) (b)

14 of ic q and T as a function of m ic matter: J Cleymans e D Worku: J. Phys. G: Nucl. Part. Phys. 39 (2012) (c) (d)

15 of ic ic matter: T o = (60±7) MeV q o =1.103±0.007

16 of ic Hadron mass spectrum ic matter: L. Marques, E. Andrade and AD, Phys. Rev. D 87, (2013)

17 of ic Correlation between T and q ic matter: Wilk e Wlodarczyk: Cent. Eur. J. Phys. 10 (2012) T eff = T o (q 1)c T H = (192±15) MeV c =-(950±10) MeV

18 of ic ic matter: 1 γ(q 1) ln[z(v, β)] = V 2π 2 β 3/2 ( 2 F 1 [ 1 q 1, 1 q 1, 1 (β β o )M(q 1) q q 1, ) 1 q 1 1 (q 1)(β β o )M pressure:p = T V ln[z(v, β)] entropy density: s = p T energy density: ε = T 2 ln[z(v, β)] V T trace anomaly: a(t ) = ε 3p T 2 = T ( ) p T T Σ T 3 T MeV Ξ T T MeV ]

19 of ic ic matter: Map between non-extensive and extensive quantities Mapa entre T e τ: τ(t o ) = τ o τ(0) = 0 τ(t 1 + T 2 ) = τ(t 1 ) + τ(t 2 ) The function satisfying this conditions is: τ(t ) = kt = τ o T o T From here we get the following relations: p T 4 = k 1 p k 3 (k 1 T ) 4 = π k 3 τ 4 s = σ ε = k 1 ɛ a = k 3 α,

20 of ic Comparison to lattice-qcd ic matter: P T T MeV T MeV C 2 T MeV a Ε T TMeV

21 of ic ic matter: Finite chemical potential In collaboration with E. Megías and D. P. Menezes Defining the partition function as V i g i ξ i log(z q(v, τ, µ)) = V i g i ξ i dεi (2π) 3 log (+) q dεi (2π) 3 log ( ) q where ξ = 1 for bosons and ξ = 1 for fermions. The occupation number is obtained by resulting n i (V, τ, µ) = ) ( e q ( ) (x) ξ i ( e q ( ) (x) ), x i < 0 e q (+) (x)+1 e q (+) (x), x i > 0 (4) n i = τ µ i log[z q (V, τ, µ)] (5) [ q dεi (2π) ξ 3 i + e q (β(ε i µ i ))], x i > 0 [ q 2 dεi (2π) ξ 3 i + e q (β(ε i µ i ))], x i < 0 Entropy results to be the same proposed by Conroy, Miller and Plastino (PLA 374 (2010) 4581). (6)

22 of ic ic matter: Phase transition to deconfined state - Preliminary The condition for phase transition is E / N =1 GeV (Cleymans and Redlich - PRC60, ) 0.20 TGeV ΜGeV BoltzmannGibbs q 1.02 q 1.12 q 1.12 x 2.75

23 of ic ic matter: PGeV q 1.12 Pressure as a function of - Preliminary BoltzmannGibbs Μ B 0.8 GeV TGeV

24 of ic ic matter: Relation between pressure and energy Neutron stars stability? - Preliminary

25 of ic ic matter: It is possible to obtain a self-consistent for fireballs non-extensive. leads to a effective, T o, and a entropic index, q o. data for p T -distributions give support for the existence of T o and q o. The mass-spectrum formula describes very well the known ic states (mesons and barions). It is possible to find a connection between extensive and non-extensive. Thermodynamics resulting from the non-extensive self-consistent are in agreement with lattice-qcd results. The has been extended to µ 0, and the phase transition line has been found. There are indications that the neutron star stability can be achieved in.

26 of ic ic matter: Thank you!