Speed of Sound in a System Approaching Thermodynamic Equilibrium

Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 61 (2016) 842

Arvind Khuntia1, Pragati Sahoo1, Prakhar Garg1, Raghunath Sahoo1∗, and Jean Cleymans2 1Discipline of Physics, School of Basic Science, Indian Institute of Technology Indore, Khandwa Road, Simrol, M.P. 453552, India 2UCT-CERN Research Centre and Department of Physics, University of Cape Town, Rondebosch 7701, South Africa

Introduction uses Experimental high energy collisions at 1 fT (E) ≡ . (1) RHIC and LHC give an opportunity to study E−µ the space-time evolution of the created hot expq T ± 1 and dense matter known as QGP at high initial energy density and temperature. As the where the function expq(x) is defined as initial pressure is very high, the system un- ( dergo expansion with decreasing temperature [1 + (q − 1)x]1/(q−1) if x > 0 exp (x) ≡ and energy density till the occurrence of the fi- q [1 + (1 − q)x]1/(1−q) if x ≤ 0 nal kinetic freeze-out. This change in pressure with energy density is related to the speed of (2) sound inside the system. The QGP formed and, in the limit where q → 1 it re- in heavy ion collisions evolves from the initial duces to the standard exponential; QGP phase to a hadronic phase via a possi- limq→1 expq(x) → exp(x). In the present con- ble mixed phase. The speed of sound reduces text we have taken µ = 0, therefore x ≡ E/T to zero on the phase boundary in a first or- is always positive. In Eqn. 1, the negative der phase transition scenario as the specific sign in the denominator stands for BE and heat diverges. The temperature dependence the positive stands for FD distribution. of speed of sound in a medium is well estab- For an ideal gas with zero chemical po- lished but the effect of temperature fluctua- tential, the temperature dependent speed of tions is less explored, particularly in the case sound, cs(T ) is given by of heavy-ion collisions. Since, in non-extensive c2(T ) = ∂P = s(T ) , statistics, the Tsallis parameter (q) is related s ∂ V CV (T ) to the temperature fluctuations [1], we have Where, s = ∂P is the entropy density explored it to estimate the speed of sound us- ∂T V ∂ ing Tsallis statistics in hadronic medium. and CV (T ) = ∂T V is the specific heat at constant volume. Speed of Sound in a Physical Results and discussion Hadron Resonance Gas Figure 1 shows the speed of sound for A hadron resonance gas consists of mesons hadron resonance gas taking different mass and baryons obeying Bose-Einstein (BE) and cut-offs of hadrons for q = 1.05 . The mass Fermi-Dirac (FD) statistics, respectively. The cut-off, M is introduced as the highest mass Tsallis form of the Fermi-Dirac and Bose- of the resonances contributing to the hadron Einstein distributions as proposed in Refs. [1] resonance gas. This shows that, adding more massive resonances to the system the speed of sound decreases near TH . To explore this, we have studied the square ∗Electronic address:[email protected] of the speed of sound as a function of the

Available online at www.sympnp.org/proceedings Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 61 (2016) 843

This behavior, however, vanishes for higher values of q. This indicates the softening q = 1.05 Pion Gas 0.3 M < 0.5 GeV of equation of state and a possible phase transition. Interestingly, this critical value 0.25 of q is close to the value one obtains in the

2 s M < 1 GeV C analysis of pT spectra in p + p collisions at 0.2 high energies [1]. Again for higher q-values M < 1.5 GeV the criticality of speed of sound shifts towards 0.15 M < 2.5 GeV the lesser value of temperature.

0.1 0 0.2 0.4 0.6 0.8 1 Conclusion T/TH 2 The speed of sound cs has been studied for FIG. 1: Speed of sound for q=1.05 for hadron systems those deviate from thermalised Boltz- resonance gas with different cut-off on mass. mann systems as a function of temperature taking different q values using Tsallis non- extensive statistics. Taking higher q-values 2 M < 2.5 GeV in non-extensive statistics, cs increases near 0.3 q=1.005 q=1.01 q=1.03 T as compared to the extensive Boltzmann q=1.05 q=1.07 q=1.09 H q=1.1 q=1.13 q=1.15 statistics. It is also observed that the criti- 0.25 cality effect appears at lesser temperature for

2 s higher values of “q”, which indicates that if C 0.2 there are temperature ﬂuctuations inside the system then the critical behavior, if any, possi-

0.15 bly the boundary of the phase transition shifts towards lesser temperature in the phase dia-

0.1 gram or the phase transition is achieved earlier 0 0.2 0.4 0.6 0.8 1 as compared to the systems described by ex- T/T H tensive statistics. Our results leave open the possibility that there exists a special value of q FIG. 2: Speed of sound for different values of the where a phase transition is no longer present. non-extensivity parameter, q, for a mass cut-off of Taking a mass cut-off of 2.5 GeV in the phys- M < 2.5 GeV. 2 ical resonance gas, we have studied the cs as a function of system temperature, for differ- scaled temperature for different values of q, ent q-values and found that for q-values higher taking a mass cut-off of 2.5 GeV. This is than 1.13, criticality disappears. shown in Figure 2. It could be inferred from the figure that with progressive increase of References the q-values, when the system goes away [1] A. Khuntia, P. Sahoo, P. Garg, R. Sahoo from equilibrium, the speed of sound slowly and J. Cleymans, arXiv:1602.01645 [hep- decreases to a minimum value up to q = 1.13. ph][Eur. Phy. J. A(Press)].

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