Propagation of non-linear waves in hot, ideal, and non-extensive quark-gluon plasma

We study the propagation of energy density perturbation in a hot, ideal quark-gluon medium in which quarks and gluons follow the Tsallis-like momentum distributions. We have observed that a non-extensive MIT bag equation of state obtained with the help of the quantum Tsallis-like distributions gives rise to a breaking wave solution of the equation dictating the evolution of energy density perturbation. However, the breaking of waves is delayed when the value of the Tsallis q parameter and the Tsallis temperature T are higher.

. As far as our knowledge goes, these expressions were not used in any other studies, and our finding is expected to have applications in studies pertaining to thermodynamics of quantum Tsallis gas.
This article is organized as follows. In the next section, we describe the mathematical model containing some discussions about the relativistic Euler's equation, the conservation equation, the non-extensive equation of state, and the derivation of the thermodynamic variables to be used in the equation of state. Section III contains a detailed account of the evolution equation and its solution. Section IV highlights the results, and we summarize as well as conclude in Section V.

II. MATHEMATICAL MODEL
Since here we derive the nonlinear evolution equation for energy density perturbation in hot QGP using Tsallis statistics and relativistic hydrodynamics, we need to evaluate the required thermodynamic variables and the equation(s) of state. As the basic dynamical equations of the system we use the relativistic Euler's equation and continuity equation for entropy density. We consider hot QGP produced at LHC, where the baryon number density is zero at the central rapidity region and evaluate the energy density and pressure appearing in Euler's equation using Tsallis-MIT bag equation of state.

A. Relativistic Euler equation
Throughout this article we will follow natural units i.e, c = 1, = 1, k B = 1. As discussed in [17], the correct description of nonlinear wave structures in QGP including quantum effects should be given by relativistic hydrodynamics of colored superfluids which is too complicated and still in an initial stage. Admitting this fact, we stick to classical relativistic hydrodynamics to study the nonlinear wave structures in hot QGP following Tsallis distribution.
The velocity field is given by, v = v(x, t)x with a magnitude v in the x-direction. We use the one-dimensional relativistic Euler's equation [38,39] which is written as, where ≡ (x, t) is the energy density and P ≡ P (x, t) is the pressure. Our calculation will be aimed at quark-gluon plasma produced at the LHC energies where at the central rapidity region the net baryonic number density is zero, and hence we encounter vanishing chemical potential. This leads to the fact that the particle and the anti-particle distributions are identical. Now, defining the entropy four-current as s µ = su µ , where u µ ≡ (γ, γ v) is the four-velocity, the continuity equation for the entropy density s can be derived as (for details see [17]), where we have used the Lorentz factor as γ = 1/ √ 1 − v 2 .

B. Non-extensive MIT bag equation of state
From Eqs. (1) and (2), we observe that we have four unknown quantities appearing in two equations. Hence, we require two additional equations which may be provided by the equations of state. In the present work we consider the non-extensive MIT bag model [36,37] equations of state which enable us to express the pressure and the entropy density in terms of the energy density. In this model, quarks and gluons follow the quantum Tsallis fermionic ('f ') and bosonic ('b') single particle distributions [40] given by, where E p = p 2 + m 2 is the single particle energy of a particle of mass m, µ is the chemical potential, q is the Tsallis parameter, and T is the Tsallis temperature. Tsallis single particle distributions can be obtained from the Tsallis entropy defined by [41], where p i are the probabilities of the micro-states. This entropic form is non-additive, i.e. the total entropy of the system is not a summation of those of its sub-parts. This situation may appear due non-ideal plasma effects, fluctuation etc [42,43]. Extremizing the Tsallis thermodynamic potential with respect to p i one arrives at the expression for the Tsallis probabilities and the single particle distribution [44]. It is shown in Ref. [44] that the transverse momentum spectra obtained from the single particle distribution is expressed in terms of an infinite summation and the zeroth term coincides with the phenomenological classical Tsallis distribution widely used in the literature. The forms of the Tsallis quantum distributions used in literature are similar to (but not always exactly the same as) the zeroth order approximated term when one uses the factorization approximation used in Ref. [40]. In this work we restrict ourselves within this approximated version of the Tsallis quantum distribution. We reserve the analysis with the exact distributions for future.
Assuming an ideal gas of quarks and gluons, we write down the expressions for the thermodynamic variables like the net baryonic number density ρ B , energy density and pressure P .
where B is the bag constant which represents the pressure of the vacuum [45], and the subscript 'b'('f ') signifies the bosonic (fermionic) contribution of the thermodynamic variables. The factor 2 in front of the fermionic parts lets us consider both particles and anti-particles. According to the bag model, quarks and gluons are assumed to stay in a spherical cavity (the "bag") in QCD vacuum and the constant B takes care of the confinement property. ρ f (ρ f ) signifies the particle (anti-particle) number density and as already mentioned in the previous section, the net baryonic number at the central rapidity region at the LHC energies is zero. Hence, we have zero baryonic chemical potential which leads to ρ f =ρ f . For the energy density and pressure, we identify that the bag variables have contributions from the massless bosons (gluons) and from the massive (mass << temperature) fermions (up and down quarks) and they can be expressed in terms of n b and n f in the following way: where g is the degeneracy factor and i = f, b.

Energy density and pressure for massless bosons
The closed analytic expressions of the energy density and pressure of a non-extensive massless bosonic gas are given by, and they are related by b = 3P b . In Eqs. (9) and (10) ψ (0) is the digamma function [47]. We defer the detailed calculations till the appendix of the paper.

Energy density and pressure for massive fermions
We evaluate the fermionic thermodynamic variables up to O(m 2 T 2 ). We consider the plasma to be consisted of the up and down quarks having masses of 5 to 10 MeV and we have checked that for a wide range of the q and T parameter values appearing in the phenomenological studies of high-energy collisions, the O(m 2 T 2 ) approximation works very well when the mass of the particle is around 10 MeV. In some papers it has been suggested that the O(m 2 T 2 ) contribution helps one account for the non-perturbative effects [46] in QGP. The closed analytic expressions for the thermodynamic variables for a non-extensive gas of massive fermions up to O(m 2 T 2 ) are given by the following equations: where Φ is the Lerch's transcendent [47]. For this section also, we defer the detailed calculations till the appendix.

Calculation of bag and P bag
Using Eqs. (9)-(12), we write down the expressions for the energy density and pressure in the non-extensive bag model, When written out long-hand, i, and P i, (i = f, b; = 1, 2) are given by, In terms of the above variables, bag, and P bag, ( = 1, 2) are given by, Once we get the pressure, the entropy density is given by the partial derivative of the pressure with respect to the temperature T , i.e. s bag = ∂P bag /∂T . Hence we obtain, It has been verified that in absence of the baryonic chemical potential, the pressure, energy density and the entropy density obey the following relationship, bag + P bag = T s bag .

Equations of state
In order to find out the equations of state, we express the pressure P and the entropy density s as functions of the energy density . We solve Eq. (13) for the temperature T in terms of the bag model energy density bag , and put the solution in Eqs. (14), and (20) to express P bag , and s bag as functions of bag . Solving Eq. (13) for a real and positive value of T and denoting the solution with T sol we obtain, where R( bag ) = m 4 2 bag,2 + 4 bag,1 ( bag − B). The expressions of P bag ( bag ) and s bag ( bag ) are given by,

III. NONLINEAR EVOLUTION EQUATION OF ENERGY DENSITY PERTURBATION
In this section, we derive the evolution equation of energy density perturbation of the QGP system using the well known Reductive Perturbation Theory (RPT) [48] which helps one to deal with the perturbation which can not be neglected with respect to the mean value. For this procedure, we consider the two dynamical equations i.e, the relativistic Euler's equation (Eq. 1) and the entropy conservation equation (Eq. 2). We expand the dependent variables in terms of a perturbation parameter σ following the RPT. Finally combining Eqs. (1) and (2) and solving the system of equations order by order we arrive at the space-time evolution of a perturbation of the energy density in quark-gluon plasma. Now, we put P bag ( bag ), and s bag ( bag ) in Eqs. (1) and (2) and in the resulting equations express ( bag − B) in terms of temperature using Eq. (13). Hence we obtain, where E 1 , C 1 and C 2 are given by, Now we express Eqs. (24), and (25) in terms of dimensionless variables: where 0 is a reference energy density, and c s is the velocity of sound. We expressˆ andv in terms of a small parameter σ following RPT as,ˆ We also change the independent variables from (x, t) to (ξ, τ ) which are related by, where R is the characteristic length scale of the problem. Using the expansion (30) in Eqs. (24) and (25) and equating the coefficients of different powers of σ to zero, we get the system of differential equations for the perturbations.

A. Calculation at O(σ)
At the lowest order, i.e at O(σ) of Eqs. (24), (25), we get the following relations, Solving (32) we can determine the speed of sound c s as,

B. Calculation at O(σ 2 )
At the next order i.e, at O(σ 2 ) we get the nonlinear evolution equation for the energy density perturbation 1 . Using the relations between v 1 and 1 from Eq. (32) and Eq. (33) we finally get, which is our main result of our work which estimates the evolution of the first order energy density perturbation. Coming back to the x-t space, we get, whereˆ 1 = σ 1 is the first order perturbation term in energy density. This differential equation is similar to the form obtained in Ref. [17], differing only in the coefficient of the last non-linear term. For a constant T = T con , the equation (35) turns to be of the form of inviscid Burger's equation [49] the details of which is discussed in the next section. If we put, m = 0, bag,2 = 0, and bag,1 = 37π 2 /30, as used in Ref. [17] for the Boltzmann-Gibbs statistics of massless fermions and bosons, we exactly get back the equation derived in there.

C. Breaking wave solutions
For a constant T = T con , the coefficient of the nonlinear term of Eq. (34) can be written as, bag,2 + 8m 2 T 2 con bag,1 bag,2 + 4T 4 Hence Eq. (34) becomes, which is nothing but the inviscid Burgers equation [49]. For a linear initial condition, S. Chandrasekhar found the explicit exact solution [50] of (37) as, where a, b are arbitrary free parameters. Explicit solutions of Eq. (37) for other relevant initial conditions are not known as far as our knowledge goes. In the x-t frame the form of the exact solution (38) becomes, where σ, R are the small parameter and characteristic length of the system respectively ( defined in the previous section) and c s is given by Eq. (33). The solution (39) behaves as a breaking wave the energy of which dissipates quickly with distance unlike solitons. For a more general initial condition, the solution of is given by u (χ ≡ x − f (u)t) [51]. Assuming an initial (t = 0) energy perturbation distribution of the solution of Eq. (35) can be written as,ˆ where

IV. RESULTS AND DISCUSSION
For a fixed value of the reference energy density ( 0 = 0.16 GeV f m −3 ) and mass m = 10 MeV (which is of the order of the down quark mass), the final solution depends on the amplitude A of the initial wave, its width B, the Tsallis entropic parameter q, and the Tsallis temperature T .
In Fig. (1), we plot the solutions of Eq. (35) on the x-t plane. For the sake of a better understanding, we present The solutions, however, depend on the width and the amplitude of the waves as well. It is apparent from Fig. (3), that beyond A = 1 (when t = 15 f m, B = 0.5 f m, q = 1.08, and T = 140 MeV), the solutions start becoming multiplevalued. This means, the more the amplitude of the initial energy density perturbation is, the more it becomes prone to be a breaking wave. On the other hand, a narrower initial energy density perturbation is more likely to give rise to a breaking wave solution (as seen in Fig. 4).
The temperature T , and the Tsallis parameter q also influence the solutions so that moving from a lower to a higher temperature value makes functions single-valued. Hence, the lower the Tsallis temperature is, the more likely it is to get a breaking wave solution (as seen in

V. SUMMARY AND CONCLUSION
In summary, we have studied the evolution of the first order energy density perturbation in hot, ideal, and nonextensive quark-gluon plasma. To take into account the fluctuating ambiance, we have used a non-extensive version of the MIT bag model and find a breaking wave solution. However, we observe that the breaking of the energy density perturbation is influenced by the temperature and the Tsallis q parameter which according to Ref. [31] is related to the relative variance in a scale variable i.e. temperature. We observe that the breaking is favoured by decreasing values of both temperature and the q parameter. This may have implications in the LHC phenomenology as the quark-gluon plasma medium formed in this region may have higher initial temperature and q value in comparison  with that formed at the experiments having lower beam energies. So, the resulting wave may behave as a stable one even after a long time.
We have also verified that the at a particular instant a wave with higher amplitude and a smaller width is more prone to breaking which is intuitively understandable. During the analysis we have also provided the closed analytical formulae for the Tsallis thermodynamic variables of an ideal gas of massless bosons and very light fermions. This can be seen as an extension of an earlier work for the classical Tsallis distribution by one of us [52]. These results can be used in the studies pertaining to the thermodynamics of quantum Tsallis gases. However, an unapproximated expression for the thermodynamic variables may be used in the future studies to examine whether that affects the present conclusions. However, the present work relies on some simplifications, like one-dimensional expansion. and the extension to higher dimensions should be done. It will also be interesting to study the present work under the light of some other types of equations of state inspired by the mean field model for example. We reserve those studies for future. The pressure of a gas of massive fermions is given by, To find out the pressure of a gas of massive fermions up to O(m 2 T 2 ), we first expand the integrand in Eq. (46) in a series of mass m, and retain only the O(m 2 T 2 ) term. Hence, the pressure can be approximated as, where P (0) is the massless term and P (2) is the O(m 2 T 2 ) term. P (0) can be evaluated using the steps detailed in the previous section. When P is expanded in a series, P (2) is given by, The first term on the r.h.s of Eq. (48) contains a single power of the Tsallis Fermi-Dirac distribution and can be evaluated following the steps detailed in the previous section. The second term, which contains a square of the Tsallis distribution, can be expressed in terms of a partial derivative of the Tsallis distribution with respect to momentum p. Then carrying out the integration by parts, one may be able to express it in terms of a single power of the Tsallis distribution. Afterwards, following the steps already discussed, one may arrive at the expression given by the second term in Eq. (12).