MATHEMATICA MONTISNIGRI
Vol LVIII (2023)

TO THE DERIVATION OF RELATIVISTIC HYDRODYNAMIC
EQUATIONS FOR A RAREFIED NON-IDEAL GAS SYSTEM
OF HIGH-ENERGY PARTICLES IN THE FRAMEWORKOF TSALLIS STATISTICS
A.V. KOLESNICHENKO
Keldysh Institute of Applied Mathematics, Russian Academy of Sciencee-mail: kolesn@keldysh.ru, web page: http: keldysh.ru/kolesnichenko/person.htm

DOI: 10.20948/mathmontis-2023-58-4
Summary. The paper discusses the construction of non-extensive relativistic dissipative hydrodynamics of an anomalous hadronic fluid on the basis of relativistic kinetic equation, obtained
earlier in the context of the Tsallis statistics, characterized by the nonextensivity parameter q,
and taking into account correlation effects (by rejecting the standard hypothesis of molecularchaos) in the collision term for heavy ions. It is shown that some specific form of local thermal
equilibrium quark-gluon matter is described by a generalized version of the relativistic Yüttner
distribution. With the help of this distribution all thermodynamic parameters of state are definedin explicit form. Linear constitutive relations and transport coefficients such as shear viscosity,bulk viscosity and heat conductivity are derived from the linearized collision integral written in
the Anderson-Witting form and evaluated using a relaxation time approximation. The designed
non-extensive relativistic fluid dynamics is designed to simulate a wide range of phenomena inastrophysics, cosmology and high-energy physics.
1

INTRODUCTION

Recently, there has been increasing evidence that Tsallis's non-extensive statistical mechanics[1], used to describe the statistical and kinetic behavior of anomalous non-additive systems ofvarious natures, can be regarded as the most appropriate basis of theoretical framework for modeling a huge number of physical phenomena and processes. These phenomena include situations
when a system exhibits any kind of memory effect, experiences long-range correlations (when its
size is comparable to the range of forces acting in it), experiences some internal fluctuations, andthe phase space in which it functions is limited or has a fractal structure.
Recall that Tsallis's modified version of the Boltzmann-Gibbs (BG) entropy in the case of anon-extensive gas system is of the form [1]:Sq : kB  f q (r )lnq f (r) d  kB lnq fq,q [0, 2] ,where kB - Boltzmann constant; f (r ) , d - respectively, the partition function and the volume
element in phase space; q - the deformation parameter (entropy non-additivity parameter Sq ) associated with some additional degrees of freedom inherent in anomalous systems, which must be
determined a posteriori; ln q z - the deformed q -logarithmic function, the inverse of which is thedeformed q -exponent expq z. , Both these functions are defined as follows [3]:2010 Mathematics Subject Classification: 85А35, 91B50, 82C40.
Key words and Phrases: non-extensive Tsallis statistics, relativistic hydrodynamics, transport coefficients.

41

A.V. Kolesnichenko

lnq z : (1  q)1( z1q  1), ( z  0),
expq z : [1  (1  q)z]1/(1q) , z  (q  1)  ;1moreover expq (lnq z)  lnq (expq z)  z . It follows from the above formulas that for a system consisting of two subsystems A and B, the total non-additive entropy of Tsallis is determined by the
relation
Sq (A B)  Sq (A)  Sq (B)  (1  q)Sq (A)Sq (B) ,from which it is clear that for q  1 logarithmic extensive entropy measure associated with theclassical BG approach is recovered, Sq  SBG . When writing the second form of the Tsallis enq
tropy definition, averaging ... q :  (...) f d with a non-normalized distribution f q for any mi-

croscopic physical quantity, intrinsic to the Curado-Tsallis statistici) is used [3].
Tsallis entropy has been widely studied in both theoretical and applied contexts (see, e.g.,[2],[3]). In particular, the non-extensive statistical formalism has proven to be a useful construction for analyzing many astrophysical and cosmological phenomena [(see, e.g., [4-12]). Recently, another of the unexplored applications of non-extensive statistical Tsallis mechanics has beenfound, related to high-energy physics (see, e.g., [13], [14], [15], [16], [17]). It turned out that experimental particle spectra at the Relativistic Heavy Ions Collider and the Large Hadron Collider
can be successfully described by energy distributions according to the so-called extended TsallisPareto power law (typical for the canonical equilibrium distribution, associated with the qentropy-Tsallis measure) in a wide range of energies, sizes of colliding systems and produced
hadron varietiesii). In other words, mathematical modeling of the processes of secondary particle
production inside gas flows in hadronic relativistic heavy ion collisions (resulting in the formation of a new hadronic state of matter quark-gluon plasma (QGP)) leads to the most adequate
results only when using non-extensive statistical mechanics of Tsallis (or Renyi), while the conditions necessary for the application of classical BG statistics and relativistic kinetic theory (see,e.g., [8], [18], [19]; [20]. [21], [22]) are fulfilled in this case rather approximated.
The mentioned non-extensive distribution in high-energy physics is presumably connectedwith the fact that hadronizing systems are characterized by strong internal fluctuations and longrange correlations, which can be interpreted as a manifestation of some internal dynamical, nonequilibrium processes (like the decay of resonances or the emergence of temperature fluctuations). As a result, instead of the usual local thermal equilibrium (which takes place in all applications of class statistics) in the considered anomalous systems, a certain "stationary state" (theso-called non-extensive equilibrium depending on the deformation parameter) appears, which
implicitly takes into account various internal dynamical effects [15]-[25]. In works ([26]-[28]) iti

In nonextensive Tsallis statistical mechanics it is possible to average microscopic physical quantities by three
distributions, which have their advantages and disadvantages, defining quite different q-thermodynamics corresponding to certain thermodynamically anomalous systems. In this paper we will use the non-normalized CuradoTsallis averaging, since this is the only averaging that does not lead to a redefinition of the notion of q-system temperature. Only in this statistic, the temperature T is an intensional parameter rather than a functional Tq, as, for example, in averaging with the normalized Tsallis-Mendez-Plastino escort distribution f q/∫dΩ f q [3], [7].
ii)Hadrons are colorless composite particles (protons and neutrons) built from quarks and gluons (massless particles that are carriers of the strong color-color interaction between quarks).

42

A.V. Kolesnichenko

was also found that in anomalous relativistic systems such phenomena can be effectively described (without involving any considerations concerning the dynamical sources of various fluctuations) in the framework of the concept of non-extensive kinetics in the form of a generalizedversion of the Jüttner distribution [29], a general zero-order solution of the relativistic kinetic
equation [30].
For this reason, it became clear that an important step on the way to a successful solution of
the discussed problem is a careful construction of a non-extensive relativistic hydrodynamics,
which is fully consistent with the Tsallis statistical approach and on the basis of which an effective modeling of a wide range of dynamical phenomena in systems with high-energy relativistic
nuclear collisions is possible. In a number of foreign papers (see, e.g.,[16], [23], [25]), a relativistic kinetic theory has been developed in the q-non-extensive Tsallis context, in which a newinterpretation of the non-extensivity parameter q, as a quantitative measure of internal fluctuations characteristic of hadronizing systems, has been giveniii) .
At the same time it should be noted that almost in all known to the author works on the discussed problem there is no any detailed kinetic derivation of the non-extensive relativistictransport equation describing the effects connected with ultrarelativistic velocities of particle motion. At the same time, the formulations of the involved assumptions are usually made without
detailed explanations, and the given formulas are often given without appropriate mathematicaljustification. For this reason, the author of the synopsis [8] attempted a more detailed derivationof the non-extensive relativistic kinetic equation taking into account the inclusion of correlation
effects (by rejecting the Boltzmann' Stosszahlansatz hypothesis of molecular chaos) in the collision term. This generalized kinetic equation allows us to obtain the conservation laws of density,
momentum and energy for anomalous systems, to confirm the validity of the second law ofthermodynamics, according to which the entropy emergence is nowhere and never negative, andalso to construct non-extensive relativistic hydrodynamics describing a wide range of phenomena arising at high speeds (comparable to the speed of light) of macroscopic motion of particles
constituting the liquid.
The results presented in this synopsis are of practical interest both in nuclear physics and incosmological and astrophysical situations involving the effects of thermal conduction and neutrino gas viscosity, as well as in studies of the collapse of stars, accretion of black holes, and theearly Universe.
2

INITIAL DEFINITIONS OF BASIC MACROSCOPIC QUANTITIES

Some initial definitions. Let us first give statistical expressions for the basic macroscopic
physical quantities written in the language of non-extensive relativistic kinetic theory. In a nonhomogeneous medium of a relativistic gas consisting of identical particles, macroscopic quantities are functions of space-time coordinates x : x : (t , x), where the index  takes 4 values:   0,1, 2, 3 ; t - time. Further we will use the metric tensor g  = diag(1, 1, 1, 1) , where ,  0,1, 2, 3 , and the covariant differentiation operator will be denoted asiv).

iii)
Note that the assumption linking the parameter q to fluctuations is formalized in the form of a new branch ofstatistical mechanics called Beck-Cohen superstatistics (see [31], [32]).
iv)
In this paper, unless specifically stated, we will, for simplicity of calculations, use the units in which Planck's
constant ħ and the speed of light are equal to one.

43

A.V. Kolesnichenko

 :



    ,  : ( 0 , ) .
 x  t x 

(1)

The macroscopic state of a non-extensive relativistic gas is characterized by the 4-vector ofparticle flux Nq ( x) , the 4-energy-momentum flux q ( x) and the 4-entropy flux vector Sq ( x) .
These macroscopic quantities are defined in kinetic q -theory as statistical averages by means ofa non-normalized distribution function f q , where f (t , x , p0 , p) is the distribution function of thephase space of one particle (local probability density function), which is normalized by the number of particles in the system, i.e. it is defined in such a way that the product d  f d 3 x d 3 p
gives the average number of particles, which at the moment of time t are in the volume elementd 3 x with the center in a point x and have particle momenta within (p, p  p) . This function
depends on the space-time coordinates x : x  (t , x) and the energy-momentum 4-vector
p : p  ( p0 , p) : ( E, p) , where m  p p , p, E  p0 2

p  m2 are respectively the rela-tivistic mass, momentum and energy of the particle located in the element of the spatial volume
d 3 x at a point x at time t . With the help of f q ( x, p) the funtional field quantities N  ( x) ,( x) and S ( x) (here   0,1, 2, 3) are written in the covariant form as follows [8]:Nq (x) = p   q :=  p  f q ( x, p)d P , (  0,1, 2, 3) , 
q ( x)   p p  q : q

 p p f ( x , p)dP , ( ,   0,1, 2, 3) ,Sq ( x) :  kB  p  f q( x , p) ln q f ( x , p)  f ( x , p)  d P , (  0,1, 2, 3) .



(2)
(3)
(4)

Here d P : g d 3 p / p0 (2 )3 -invariant volume of space with momentum, where g  2 j  1 denotes the number of internal degrees of freedom ( j − spin of the particle); kB − Boltzmann constant. Concerning formula (4), let us immediately note the following: in the few literature on
nonextensive relativistic kinetic theory, there are various non-equivalent definitions of the 4vector q-entropy flux Sq ( x) , which lead to different thermodynamic relations (see, e.g.,[28],[33], [34]). In this paper, we will use the definition proposed in [34] because only it corresponds
to the statistical quantum q-entropy, which takes into account corrections concerning thermodynamic consistency in the case of relativistic quantum distributions of high-energy particles [35].
Hydrodynamic 4-velocity. An important concept in the construction of relativistic kinetic theory is the hydrodynamic 4-velocity u ( x) , which is used in the determination of a number ofphysical parameters that play an important role in the formulation of macroscopic conservation
laws. The 4-velocity u ( x) is usually given as a time-like vector with modulus equal to one at44

A.V. Kolesnichenko

each spatio-temporal point: u ( x) u ( x)  1 . With the help of the hydrodynamic velocity vector
the so-called tensor-projector is determined ( x) : g  u ( x) u ( x) ,(5)

which at convolution with an arbitrary 4-vector acts as a projection operator, annihilating the velocity parallel u ( x) part of this vector,   ( x) u ( x)  0 ; besides this projection operator ischaracterized by the following properties:      ,        ,   3 .
Since the hydrodynamic velocity u ( x) is a time-like vector, it is possible to introduce in eachspace-time point a local rest system (also called the associated coordinate system), denoted by
the index LR (local rest). The hydrodynamic velocity in this system has the following compo
nents: uLR  (1,0,0,0), as well as LR   LR  diag(0, 1, 1, 1) ,  LR  diag (0,1,1,1). To
fix the choice of hydrodynamic velocity u ( x) , we will further use the following convenientdefinition of it: u: N  / N  N , given by Eckart [36].
The covariant differentiation operator   can be decomposed with respect to the hydrodynamic 4-velocity u ( x) into time-like and space-like parts according to the relation   u D   ,(6)

where the symbol D : u   stands for the convection derivative in time (DLR   / t) , and thesymbol  :     is the gradient operator, which in the local coordinate system is purely spa-tial, u  0 (or with iLR   LRi   / x ).
Basic macroscopic parameters of a non-extensive relativistic fluid. With the help of basic
field quantities Nq ( x) , q ( x) , Sq ( x) and hydrodynamic velocity u ( x) it is possible to determine the following macroscopic parameters of a system of identical relativistic particles: particle density nq ( x) , energy density q ( x) , local hydrostatic pressure Pq ( x) , heat flux J q ( x) ,pressure tensor Pq ( x) and entropy density sq . Thus,(i) the particle density nq ( x) is given by the covariant expressionnq ( x) : Nqu ;(7)(ii) the energy density q ( x) is determined by the expressionq ( x) : eqnq : u qu ,(8)where eq − average energy per particle; hq ( x) : q  Pq  hq nq − enthalpy density, Pq ( x) − local static pressure;45

A.V. Kolesnichenko(iii) the heat flux J q ( x) , defined as the difference of energy W q ( x) : u  and enthal-py fluxes carried by particles, is given by the expression
Jq ( x) : (u hq N  ) ,(9)where hq  hq / nq is the enthalpy per particle; in the covariant formulation, the orthogonality
condition Jq u  0 for the heat flux takes place;(iv) the pressure tensor Pq ( x) is defined by the formulaPq ( x) :  q  ,( ,   0,1, 2, 3) ,(10)from which it follows that this tensor is symmetric because the energy-momentum tensor issymmetric; for further purposes it is convenient to break the pressure tensor into "reversible" and"irreversible" parts: Pq  Pq   Пq , where the tensor П  ( x) is called the viscous pressure
tensor;(v) the entropy density sq ( x) is defined as a scalarsq  sqnq : Sq u ,(11)where sq is the entropy per particle.
Given the definitions of energy density (8), heat flux (9) and pressure tensor, the following relations can be written [30]:q ( x)  u qu , J q ( x)  u q , Пq ( x)  Pq ( x)   q  .

(12)

Decomposition of the energy-momentum tensor. Using the definition (6) for the projective
operator  , the following identity can be proved (see, e.g., [8])q ( x) 

where
part:



q(0)  q(1) ,



 

q(0) ( x) − the reversible part: q(0): q0u u  Pq0  , and(13)
q(1) ( x) − the nonreversible


 

  
q(1) :  Jq u  J q u   П q . The decomposition (13) plays an important role in the deri-

vation of macroscopic conservation laws.
3

GENERALIZED RELATIVISTIC KINETIC EQUATION
The generalized kinetic equation describing the behavior of the scalar distribution functionqf ( x, p) in space-time has the form (see [8], [28], [34]):p  f q( x, p)  p u D   f q( x, p)  Cq  f , f1  , (   0,1,2,3 ),

where

46

(14)

A.V. Kolesnichenko

Cq  f , f1  1
 R [ f , f1 ] Wp ,p1 p,p1 d P1 d P d P1 .
2 q

(15)

− is the collision integral (collision term).
In deriving this equation in the non-extensive relativistic kinetic theory, along with the requirements of covariance, the same assumptions were made as in the purely relativistic approach,namely:(i) only two-particle collisions were considered;(ii) the value of transition velocity Wp ,p p,p , which is a measure of probability of colli1

1sion processes, was used;(iii) the assumption that the distribution function changes slowly in space-time, i.e. its changes at the characteristic length of interaction and during the characteristic time of particle interaction are negligibly small, is used;(iv) finally, most importantly, the q-expansion of the molecular chaos hypothesis is used,which allows us to take into account the influence of statistical correlations on the type of thecollision term.
The transition velocity Wp ,p p,p , depending only on the 4-impulses of the two particles
1

1

before and after the collision, is a Lorentz scalar. It is obvious that both arguments before andafter the arrow in this notation can be placed in an arbitrary order. In accordance with the accepted condition of slow change of the distribution function at distances and times of the order ofcharacteristic lengths and times of force interaction, it is assumed that the difference of spacetime coordinates x  (t , x) of colliding particles before and after the collision can be neglected.
The magnitude Wp ,p p,p is a function of ten scalar invariants, which can be constructed from11

the 4-impulses p , p1 , p , p1 . In addition, since the law of conservation of energy-momentum
must be satisfied, the ratio p  p1  p  p1 . It follows that the transition probability is symmetric for the time reversal of collision processes, i.e., the substitution p, p1  p, p1 leaves this expression unchanged. Thus, the principle of detailed equilibrium takes place
Wp ,p p,p  Wp,p p ,p  Wp ,p p,p .
1

1

1

1

1

1

(16)

In the collision integral (15), the quantity Rq [ f  f1] , describing in non-extensive relativistic
kinetics the average number of collisions of correlated particles, is written here in the following
form ([28], [33], [37]):
Rq [ f , f1 ]  expq  ln q f  ln q f1   expq  ln q f  ln q f1  .
(17)




Note that when q  1 , the relation (17) corresponds to the standard Boltzmann hypothesis ofmolecular chaos: lim Rq  R1   f  f1  f f1  . Thus the ansatz functionv) (17), which takes intoq1

account the influence of statistical correlations on the collision term in the non-extensive case, isv)

Recall that the ansatz-function is nothing but an essential assumption based on heuristic considerations, which
can be confirmed only after obtaining all corollaries from the constructed transport equation

47

A.V. Kolesnichenko

thus a q-extension of the molecular chaos hypothesis. In this case, the non-extensive parameter q
lies in the interval [0, 2].
Let us now define the following correlation (positive and symmetric in arguments) function

Hq  f , f1  : expq ln q f  ln q f1   0 ,



(18)

associated with two particles in the same spatiotemporal position x , but with different 4-vectorsof energy-momentum p and p1 , respectively. This formula, leading to the validity of the H theorem for statistically correlated particles, is postulated. For q  1 it returns the classical collision integral, where the number of binary collisions around spatiotemporal coordinates is proportional to the number of binary collisions H1  f , f1   f f1. around spatiotemporal coordinates x .
However, we will not discuss this problem in this paper.
Taking into account the function (18), the non-extensive relativistic equation of perenance(14) can be written in the form ofpu D f  p f  Cq  f , f1  
q


4

q





1
 Hq  f , f1   Hq  f , f1  Wp ,p1 p,p1 d P1 d P d P1 .
2(19)

MACROSCOPIC CONSERVATION LAWS. H-THEOREM

Some properties of the collision integral. It can be shown that there is a lemma according towhich (due to the fact that at collisions of particles the microscopic law of conservation of energy-momentum p  p1  p  p1 must be satisfied) the collision integral (30) has the following
property: : ( x, p) Cq  f , f1  d P  0 ,(20)

where ( x, p  ) is the summation invariant (linear combination of some scalar ( x) and 4vector p ):( x, p  ) : ( x)   ( x) p .

(21)

The space-time functions ( x) and  ( x) are arbitrary, with only one restriction: the functions
( x) must be additively conserved under collisions. Any additive momentum function is a linear

combination of the additive invariants of (21).
To prove lemma (20), we substitute the explicit expression (15) into (20):






1
  H  f , f1   Hq  f , f1  Wp ,p1p,p1 d P d P1 d P d P1 .
2 p q

This integral is obviously equal to the integral

48

(22)

A.V. Kolesnichenko







1
   Hq  f , f1   Hq  f , f1  Wp,p1 p ,p1 d P d P1 d P d P1 .
2 p

(23)

Performing a postal addition of these two integrals, we obtain






1
 [ p  p] Hq  f , f1  Hq  f , f1  Wp ,p1 p,p1 d P d P1 d Pd P1.
2

(24)

The integral (23) is symmetric with respect to p and p1 . Therefore






1
 [ p1  p1 ] Hq  f , f1  Hq  f , f1  Wp ,p1 p,p1 d P d P1 d P d P1 .
2(25)

Adding (25) and (26), we obtain 

1 [ p   p1   p  p1 ]
4 Hq  f , f1   Hq  f , f1  Wp ,p  p ,p d P d P1 d P d P1  0 .
1

(26)

1

Here it is taken into account that by virtue of the definition (21) of the function ( x , p  ) , theexpression  p   p   p  p goes to zero if we take into account the restriction on functions
1

1









( x) and conservation of energy-momentum in a binary collision: p  p1  p  p1 . This com-pletes the proof of lemma (20).
General form of the equations of motion of a non-extensive relativistic fluid. The transfer
equation (19) allows us to derive a general form of the laws of conservation of the number ofparticles and conservation of energy and momentum, which are valid at the macroscopic level.
Using lemma (20), let us set the function  ( x) equal to zero and all functions equal to one
arbitrary function . As a result, we obtain the expression

 Cq  f , f1  d P  0 ,(27)

which, taking into account the transfer equation (19), can be written as




q

q

 Cq  f , f1  d P  p  f ( x, p)d P    p f ( x, p)d P  0 .

(28)

Expression (28) with the help of the averaged 4-vector of particle flux Nq ( x) , defined by formula (2), can be rewritten as a macroscopic law of conservation of the number of particles
 Nq ( x)    p f q( x, p)d P   Cq  f , f1  d P  0 , (  0,1, 2, 3) .

(29)

If we now set the functions ( x) in (21) equal to zero, we obtain  p   ( x)p ; then fromlemma (20), taking into account the transport equation (19), we find49

A.V. Kolesnichenko



 

q

 p Cq  f , f1  d P   p p   f ( x, p)d P  0 .

(30)

Hence follows the macroscopic law of conservation of the energy-momentum 4-tensor

 q ( x)   pCq  f , f1  d P  0 ,where the tensor(  0,1, 2, 3) ,

(31)


q ( x) is defined by formula (2). In the case   0 of equation (31) is the lawof conservation of energy, and for   1, 2, 3 - it is the law of conservation of momentum.
Dynamical laws for density, momentum and energy. The non-extensive relativistic hydrodynamic equations follow directly from the laws of conservation of the number of particles and energy-momentum derived above.
Using the law of conservation of the number of particles (29) and the convection time derivative operator D  u   , we obtain the following noncontinuity equation for the density

nq ( x) [30]
D nq  nqu .

(32)

The equation of motion is obtained from the energy-momentum conservation law (31) byconvolving it with the projection operator  . Taking into account the energy-momentum tensor expansion (13), we obtain [30]:hq D u   Pq  Пq  hq1 Пq Pq   D Jq  Jq u  Jq u ,(33)

in which, as before, hq  hqnq  (q  Pq ) is the enthalpy density. From this equation, we see thatthe acceleration of the medium is due to pressure gradients and, in addition, to a number of termsof purely relativistic origin. If we neglect dissipative flows Пq and Jq , then the equation of
motion is reduced to the zero-order equation
D u  hq1  Pq ,(34)

which is valid for an ideal relativistic fluid. This correlation between acceleration and pressure
gradient plays a certain role in the derivation of suitable forms of hydrodynamic equations in ourcase, when only equations linear in the quantities related to the transport phenomena are considered.
The balance equations for the energy eq ( x) per particle of the system are derived from thelaw of conservation of energy-momentum (31) by taking into account relations (9), (10), as wellas the expansion (13) for the 4-tensor q ; as a result, we have [30]
nq D eq  Pq u  Пu   Jq  2 Jq D u .

50

(35)

A.V. Kolesnichenko

If we neglect dissipative terms in equations (33) and (35), we obtain the relativistic Euler equations
D nq  nqu ,D u  (hqnq )1  Pq ,nq D eq  Pq u .

(36)

H-theorem. In the first section by the formula (4) has been determined the macroscopic 4vector of entropy flux

Sq ( x)   kB  p  f ( x , t )q ln q f ( x , t )  f ( x , t )  d P ,and by formula (11) we introduced the entropy density sq ( x) : sq nq : Squ , where sq is the entropy per particle. We obtain here a formal expression for the entropy balance using the identity
nq u sq   (sq Nq ) , which is a consequence of the law of conservation (29) of the number ofparticles and the definition (1) for the differentiation operator on space-time coordinates,   . By
adding and subtracting the same quantity Sq , we rewrite this identity in the formnqu sq   (Sq  sq Nq )  Sq , which can be interpreted as a balance equation for the en-tropy per particle. Indeed, it can be rewritten asnqu sq   J q ,s  q ,

(37)

where
Jq,s ( x) : Sq ( x)  sq ( x) Nq ( x)− is the entropy flux (by definition), and the value 0  q : Sq , being a stylized law of increasing q-entropy, describes here the intensity of its source. The local mathematical expression(37) of the second law of relativistic thermodynamics does not contain any physical provisions
except for the law of conservation of the number of particles, and therefore has here a purely
formal character.
From the definition of the 4-entropy flux vector (4) and the kinetic equation (19) it follows
that for a non-extensive system the entropy increase q : Sq is determined by the formulaq ( x) :  Sq   kB  p   f q ( x , p) ln q  f ( x , p)  f ( x , p) d P q  kB  ln q f ( x , p)  p   f ( x , p)d P   kB  ln q f ( x , p)  Cq  f p , f p1  d P.







(38)

When writing this expression, the differentiation property of the deformed logarithm was used lnq z/ z  1 / zq (see, e.g., [8]). The entropy gain (38) with the help of the notation (20) can be

written as

q ( x)   k B

51

ln f ( x, p)  , q

A.V. Kolesnichenko

which, with the help of lemma (27), can be represented asq ( x) kB  ln f  lnq f p1  lnq f p  lnq fp1  
8  q p

 Hq  f , f1   Hq  f , f1  Wp ,p  p ,p d P d P1 d P d P1  0.
1

1

(39)

This result can be compared with the extensive relativistic expression given, for example, in[30]. As is known, the irreversible character of the thermodynamics arising from molecular collisions is restored if the above value is positively determined, i.e. q  0 . In this case, this condition is true due to the presence of two reasons:(i) first, the positive definiteness of the quantity

 ln f  ln f  ln f  ln f    H  f , f    H  f , f  
q p

q p1

q p

q p1

q

1

q

1

for any pair of distributions ( f p , f p ) and ( f p , f p ) ; the sign of the source of the four entropies is1

1completely determined by the sign of the non-extensive para-meter q , which, as was shown in[26], lies in the interval q  [0, 2] ;(ii) secondly, the successful definition of (4) for the 4-vector flux of the q -entropy.
Non-extensive equilibrium. Assuming that the partition function f ( x, p) tends to a certain
limit with time, the state of a non-extensive q-system will transition from strict (local) thermal
equilibrium (which is the prerogative of an ideal gas-gas medium) to a non-extensive stationary
state. This state includes some microscopic dynamical interactions and fluctuations, which are
generalized by the deformation parameter q [15], [38]. It should be noted that in the steady state
the entropy of the -system reaches its maximum value. A necessary condition for such a state isthat the entropy production value (39) goes to zero everywhere in space-timeq ( x)   k B

ln f   0 .
 q p

Let us now obtain as a consequence of the non-extensive relativistic H-theorem (38) the nonextensive stationary state condition, which is a generalized version of the relativistic Juttner distribution [30]. As in the classical case q  0 , equality is a necessary and sufficient condition for
the existence of both local and global equilibrium. The integral (39) is positively definite, whichis possible only if and only if the following equality holds

lnq f p lnq f p  lnq f p  lnq f p  0 ,1

1

where the impulses are related by the conservation law p  p1  p  p1 , which is true for elastic pairwise collisions.
Thus, the quantity   lnq f p is a summation invariant of the collision of molecules andtherefore must have the form lnq [ f p0 ]  0  0 p , where 0 and  0 are the collision invariants

52

A.V. Kolesnichenko

given earlier. The distribution function f p , for which this condition is satisfied, will be further
denoted by the symbol f p 0 . Thus, the equilibrium distribution function f p 0 is defined by the expressionf(0) ( p)  f p0  1  (1  q) 0  0 p 

1/(1q ) expq 0  0 p  .



(40)

Further we will identify the parameter 0 with the expression 0   0 u0 (where u0 - 4speed of the system as a whole; 0  1/ kBT , T − temperature field), and 0 we will put thevalue equal to 0  q0  q / kBT , where q − chemical potential. Here all thermodynamic
parameters are global, i.e., do not depend on x . Then expression (40) can be written in the formof the generalized Jüttner distribution function (on impulses) [29] for a homogeneous equilibrium relativistic medium
f p0 

1/(1q )1  (1  q)( 0  0 pu0 )  p u 
q
 0
 .
 expq 

kBT 



We will further assume that for inhomogeneous systems in thermodynamic equilibrium the
distribution function f(0 )( x , p) in space-time coordinates and momenta has the same functional
form (40) as for the case of global equilibrium of a complex q-system, but with parameters0 ( x) , 0( x) and u ( x) , depending on x . Then with the help of the local equilibrium function
f p0 

1  (1  q)( 0  0 pu0 )1/(1q )  p u q 0 expq 

kBT 



(41)

of a non-extensive relativistic q-system, one can determine explicitly all equilibrium thermodynamic variables.
5

EQUILIBRIUM STATE

Recall that the indices are raised and lowered using the pro-space-time metric g  . Note alsothat when solving the relativistic transport equation it is convenient to use the following tensors: a :  a , b : 1 (        )b  ,

 
 
2


 b : b   31   b    21 (        )  13     b  ,   1  
 


 2 (          )  31     A ,
A




p  p :     13    p p  .







53

(42)

A.V. Kolesnichenko

In addition, we note that any 4-vector A , in particular the 4- momentum of a particle p ,can be decomposed using an arbitrary hydrodynamic 4-velocity u  of the flow into two parts asfollows:p  Epu  p .
Here Ep : pu , is the energy of the particle in the local coordinate system (LR), and the summand p :  p is the momentum in the associated coordinate system [42].
First, let us determine the equilibrium number density nq0 . Given formulas (2), (7) and (28)
we findnq0 ( x) : Nq0u   Ep f0 ( , )d P 
q

10 .

(43)

Similarly, we determine the q-equilibrium energy density q 0 and hydrostatic pressure Pq 0 .
 Using the expansion (13) for the equilibrium energy-momentum tensor Tq0  qu u  Pq  ,we obtain the following expressions for the values q 0 and Pq 0 :q0 ( x)   Ep2 f0q ( , )d P 

20 ,

(44)

1
1
q
 
Pq0 ( x)   Tq
0     d P(  p p ) f0 ( , )  
3
3

21 .

(45)

In writing relations (43)-(45), the following moment integrals were used (see [20], [21],[38]):
2 k k! d P( Ep )n 2 k (  p p )k f0 ( , ) ,
0
(2 k  1)!

(46)

2 k k! qd P ( Ep )n-2 k (  p p )k f0 (  , ) ,nk (  , ) 
0
(2 k  1)!

(47)

2 k k! 
2 q 1d P( Ep )n-2 k (  p p )k f0 ( , ) ,nk (  , )  q
0(2 k  1)!

(48)

nk (  , ) 

where n, k  0 .
2 q 1q

q
Using the derivatives (f0 p /0 )  f p0 and (f p0 / 0 )  q f p000

nk (  , )   nk / 0 0 ,nk (  , )  nk / 0 0 .

In addition, the convection time derivatives D of the functions
pressed in terms of D0 and D0 :54, we obtainnk and(49)nk

can be ex-

A.V. Kolesnichenko

D

nk 

nk D 0 (n 1, k) D 0 ,

(50)

D

nk 

nk D 0 (n 1,k) D 0 .

(51)

It can also be shown (applying integration by parts) that the following recursive co-relations 1,0,0,0 ) [21, [38]:take place in the global rest system ( uLR m2 (n,k)  (2 k  3) (n  2,k 1) , (n  2,k) 01 (n 1,k 1)  01 (n  2 k) (n 1,k) , (n,k)21
 (n  2,k)  m (n,k)  (2 k  3)0 (n  2,k  2) ,1
1 (n,k)  0 (n 1,k 1)  0 (n  2 k) (n 1,k) ,2 (n  2,k)  m (n,k)  (2 k  3) (n  2,k 1) .
(52)To complete the description of the equilibrium state of the non-extensive system, we find theequilibrium entropy density sq0 ( x) , for which we substitute expression (40) into formula (4); asa result we obtain k  ( x) p  f ( x , p)  ( x)  p u ( x)   f ( x , p) d P .
Sq0 ( x)   kB  p f0 ( x , p)ln q  f0 ( x , p)   f0 ( x , p) d P 
q



q
0

B 0

q





0

(53)

The convolution of this expression with the velocity u ( x) yields the following equality forthe entropy density defined by formula (11):

sq0 ( x)  kB  d P Ep 0 ( x) q ( x)  u ( x) p  f0q ( x , p)  f0 ( x , p) 


20

T



q
T

10  kB 10 ,(54)

which, taking into account the relation following from formulas (45) and (50)10  0

21  0 Pq0

and formula (44), can be written in the form of the following fundamental thermodynamic equalityvi): T( x) sq0 ( x)  q0 ( x)  Pq0 ( x)  q ( x) nq0 ( x) . If we now take the covariant derivative of
this equality, we obtain the equationvi) Note that Pq0 ≠ kBTqnq0, since nq0 = J10 ≠ I10. However, if the number of particles is copreserved, the integrals
from I10 and J10 must lead to the same number of particles per unit volume, although the normalizations of the distribution functions are different.

55

A.V. Kolesnichenko

T  sq0   eq0  nq01  Pq0  Pq0 nq01  sq0 T  q ,which leads to two thermodynamic equations:(i) to the relativistic Gibbs equation (for entropy per particle)

T  sq0   eq0  Pq0  (1/ nq0 ) ,(55)

or in the form written for the entropy density sq0  sq0nq0 ,  sq 0  0   q 0 
0  1
 sq 0  q 0  Pq 0    nq 0  0   q 0   0   nq 0 ;nq 0  0(ii) to the relativistic Gibbs-Duhem equations:nq01  Pq0  q  sq0 T ,(56)

or in another form Pq0 nq0
0 0 q0  Pq0
00 .

The Gibbs relation (55) with the help of the operator D can be represented also in the form of 1 .
T ( x) D sq0 ( x)  D eq0 ( x)  Pq0 ( x) D 
 nq0 ( x) 



(57)

This relation is fundamental to finding an explicit form of the entropy production intensity ( x).
Finally, using recursive relations (52), we can use moment integrals (46)-(48) and after somealgebraic transformations to obtain the relations used below:

D 0 

1

D 
q0

30 D 0,D 0  

20

D 

1

q0

20 D  0

,

(58)

30

and also formulas for hydrostatic pressure Pq 0 :
Pq0 



1
q0  m2
3

(0,0)



(59)

for enthalpy per particle

hq 0  (q 0  Pq 0 ) / nq 0 

31

,

(60)

21

2
and for the square of relativistic speed of sound cqs
 (Pq0 / q0 ) (at fixed relativistic entropy

per particle, sq0 / nq0 )56

A.V. Kolesnichenko Pq0  1 
m2   1



3

20 
 0 


2
cqs



10 

30 00 hq0

20 10   .

(61)




Here m : p p is the relativistic mass of the particle,nk:

6(n 1,k) (n 1,k) 

2
nk .

(62)

DEVIATION FROM EQUILIBRIUM
Deviation of a non-extensive system from equilibrium. If the system is out-of-equilibrium,q

the distribution function differs from the q-equilibrium distribution, f pq  f p0 . This leads in thegeneral case to the appearance of additional terms in the determination of macroscopic parameters of the state and to the appearance of a non-zero source of entropy q  0 .
In Section 5, we considered an ideal non-extensive relativistic gas-gas system in which thermal equilibrium is preserved, i.e., any dissipative processes are absent. For equilibrium ideal fluids, the energy-momentum tensor is defined by the following general covariant expression




 

q0  q0u u  Pq0  . For a non-equilibrium medium, weak spatial and temporal gradients

lead to a change of the particle flux vector and the energy-momentum tensor by small additions
Nq and  q . As a result, all effects related to dissipation appear as small contributions to
the 4-vector N q and 4-tensor




 


q  q0u u  Pq0    q . Hence, the further problem is tofind explicit expressions for Nq and  q .
For convenience in performing the proper operations, we decompose the viscous stress tensor
Пq as follows:


П 
q  П q  Пq  .

(63)Here Пq is the viscous pressure tensor with zero trace for the q-system;Пq   31 Пq    31 Пq g   31 Пq,

(64)− is the volume viscous pressure, defined as one third of the trace of the viscous pressure tensor
taken with the minus sign.
Suppose now thatf p  f0 p  f p ,57( f p  f0 p ),(65)

A.V. Kolesnichenko

where through f p denotes a small deviation from q-equilibrium. This expression can be approximated by a series expansion of the function f p with respect to equilibrium, which in turn isdetermined by the summation invariant of the collision





(66)

 p  ...  f0 p  ( f0 p )q  p  ... ,(67)

 p 0  ln q f p 0   0  0 pu ;as a result we obtainf p (  )  f0 p (  0 ) f0 p ( p0 ) p0where the symbol  p : (   p0 ) denotes the corresponding small deviation. Comparing expressions (66) and (67) and restricting ourselves to terms linear in  p , we find
f p  ( f0 p )q  p .

(68)

Since in the non-extensive relativistic theory all macroscopic quantities are related to the distriqqq

bution function f ( x, p) , the deviation f pq : f pq  f p0 from the equilibrium distribution f0 can
also be found from the seriesqf pq   f0 p  ( f0 p )q  p  ...  ( f0 p )q  q( f0 p )2q1  p  ... ;as a result, in linear approximation we obtainf pq  q( f0 p )2q1  p .

(69)

For further purposes it is also necessary to have a series expansion by a small parameterf p0 of the function Hq [ f , f1 ]  expq [ l nq f  lnq f1 ]; as a result we will have
Hq [ f p , f p ]  Hq [ f p 0 , f p 0 ] 1

Hq [ f p 0 , f p 0 ]f p 01

1

f p 0 

Hq [ f p 0 , f p 0 ]f p 01

f p 0 ...,
1

(70)

1

where, by virtue of the differentiation property of the deformed logarithm, the following relations are validHq [ f p 0 , f p 0 ]f p 01

q

 H q [ f p 0 , f p 0 ] ( f p 0 ) q ,

(71)



(72)

1

1

1

Hq [ f p0 , f p 0 ]
f p0

1



q

 Hq [ f p 0 , f p 0 ] ( f p 0 ) q .
1

Substituting (71) and (72) into the series (70), we obtain58

A.V. Kolesnichenko





q

Hq [ f p , f p ] Hq [ f p0 , f p 0 ]  Hq [ f p0 , f p 0 ] (  p   p ) .
1

1

1

1

(73)

Thus, the collision integral in the transport equation (19) approximizes up to first order in deviations from q-equilibrium as

Cq [ f p ]  Cq [ f p0 ]  Cq [f p ] .



(74)



Since the summand Cq [ f p 0 ]  Hq  f p 0 , f p 0   Hq  f p 0 , f p 0  is equal to zero, and hence

1 

1 
Cq [ f p ]  Cq [f ] , the transport equation (19) in the case of small linear deviation of a nonextensive relativistic system from equilibrium ( f0p  f p ) takes the following form) :p  ( f p0 )q  Cq [f p ] .

(75)

Here the collision integral Cq [f p ] is defined by the expression

Cq [f ] :







q
1
Wp ,p p,p Hq [ f p0 , f p 0 ] 

1
1
1
2



  p   p   p   p d P1 d P d P1 .
1

1

(76)

Determination of macroscopic state parameters in the non-equilibrium case. Using the nonequilibrium distribution function f q ( x, p) to calculate the particle density, energy and total pressure of the system, the following expressions can be obtained:
nq ( x)  u Nq   (u p  ) f pq d P   Ep f pq d P ,q ( x)  uu uu  p p f pq d P   Ep2 f pq d P ,Pq ( x)   13   13  (  p p ) f pq d P .

(77)
(78)
(79)

At the same time, the particle number density nq and energy density  q are fully determined only by the equilibrium distribution function f pq :qnq   Ep f pq d P   Ep f p0 d P  nq 0 ,(80)

q   Ep2 f pq d P   Ep2 f pq0 d P  q0 ,(81)

while the isotropic pressure Pq is divided into two parts: hydrostatic pressure

Pq0   13  (  p p ) f p0 d P
q

59

(82)

A.V. Kolesnichenko

and the bulk viscous pressureПq   13  (  p p ) f p0 d P ,q(83)

so that Pq  Pq0  Пq . Note that relations (80) and (81) are satisfied if the so-called fit conditions
are imposed on the additives f pq
q

 Ep f p d P  0 ,q

 Ep f p0 d P  0 ,2(84)

which ensure that at any stage of the procedure for solving the transport equation, this solution
depends only on the parameters nq0 ( x), q0 ( x), u ( x) and their gradients.
In addition, it is clear that, since the moments of the equilibrium distribution lead to the disq

appearance of dissipative quantities, only deviations f p 0 from the equilibrium distributionqfunction f p 0 can describe dissipative processes. Consequently, the following relations are valid:Nq   Nq   p  f pq d P

Wq   u  p Ep f pq d P

 



q

 p Ep f p d P ,

 

q

q

 p f p d P ,q p p fp d P  p p  fp d P

(85)
(86)

(87)

which are valid up to the first order in deviations f pq of the non-equilibrium function f pq fromits equilibrium value f p 0 . It follows that the small additive  q to the equilibrium energyq

momentum tensor

q 0 has the form

 q  Wq u  Wq u  П 
q .

(88)

For completeness of the study of the deviation of the system from equilibrium, we calculate
the 4-flow entropy Sq [ f p ] up to first order in the deviations f p of the function f p from f p 0 .
The expression obtained in this case has the form
Sq [ f pq ]  Sq0  kB  p  f pq ln q f p  f p  d P







Sq0  kB 0 Jq   0 Nq  [ ( f pq )2 ] ,in which the expression for the equilibrium entropy is defined by the relationqq
Sq0  kB  p  f p0  0 f p0 pu  0 f p0  d P 



60

(89)

A.V. Kolesnichenko

 kB   0 Nq0  0
q 0 u  0 Pq 0u  .

(90)When writing (90), we took into account the expression already used above10  p u fp d P  0 21  0 Pq0 .

7
NON-EXTENSIVE RELATIVISTIC BGK MODELAND TRANSPORT COEFFICIENTS OF TRANSPORT
To solve the non-extensive relativistic transport equation for non-equilibrium processes, themethod of Grad's moments ([18], [21], [40], [41]), in particular, can be used. This method, however, suffers from the inability to evaluate some resultant integrals (even numerically), unless it
is restricted to decompositions over the equilibrium distribution. However, this problem can beovercome by the well-known BGK model (Bhatnagar, Gross, and Krook) using relaxation time , which allows us to restrict ourselves to equilibrium distributions only, although we have to
pay for it with lower accuracy. In this paper we propose to use the relaxation approach to the derivation of the closed-form non-extensive relativistic hydrodynamic equations developed for thepurely relativistic case in [40].
The transport equation in the BGK approximation, linear governing relations. Suppose thatthe nonequilibrium contributions from dissipative fluxes are negligibly small near equilibriumf pq  f p0 , i.e., that p   f pq  0. Since the approximate equality p  f pq p  f p0 , allqq

nonequilibrium moments vanish from the left-hand side of the transport equation (19), which in
the BGK approximation takes the following form [39]:qp  ( f p0 )q

f pq  f p0



 p u

puf pq



,

(91)

where  is the relaxation time of the non-equilibrium distribution function f pq to equilibrium.



1/(1q )Using the formula f0 ( x , p)  1  (1  q) 0( x)  0 ( x) pu ( x) for the generalized
equilibrium distribution function f0 ( x, p) , we obtain the flux term in equilibrium asp  ( f p0 )q  q p ( f p0 )q1  f p0  q( f p 0 )2 q 1 p    0  Ep p  0  0 p p  u ) .

Combining equations (91) and (92), we obtain2q 1

f pq   q f p0

  

 0 ( p p )  D 0  Ep D 0   Ep 3



61

(92)

A.V. Kolesnichenko h01  Ep1 p0 0  p p   .
Ep


(93)

1
Here the decomposition u  u D u       is used and the following notations
3

 
 
are introduced:  : ( u   u ) / 2 − covariant rotor,  : u −covariant divergence of

1
2hydrodynamic velocity.   u    (u  u  u ) − shear tensor.
23Note that in deriving expression (93), the conservation laws of the equilibrium system wereused to calculate the corresponding time derivatives, which can be written in terms of gradients.
Applying relations (50)-(52) at different values of n and k , we obtain as a result

where

nk:

D 0  n0  h0 20 

30 
/

20 ,

(94)

D 0  n0  h0 10 

20 
/

20 ,

(95)

D u   h01  0  0  / 0 ,



(96)

(n 1,k) (n 1, k)

2
nk .

q

q

Knowing the deviations f p 0 from the equilibrium distribution function f p 0 , we can find generalized linear relations in the q-system between gradients and irreversible flows defined by formulas (85)-(87). Thus, the linear relation between the stress tensor in the first approximationП  (formula (87)) and the shear tensor  , obtained using the relation (93) and the corresponding moment integrals (47), takes the form:П  ( x)   p p f pq d P 
2 q 1  0  q  d P f p0 Ep1 p p p p    (20



32 )



,

(97)

or


П ( x)      21 (u  u )  31    ,

(98)



where П ( x)      21 (u  u )  31    is the shear viscosity coefficient.
Similarly to the shear viscosity, a linear equation for the scalar viscous pressure Пq can be obtained and the bulk viscosity coefficient can be calculated as followsПq ( x)  

1
 q
q
2q1
d P( p p ) f p0   0  d P f p0 Ep1( p p )2 

3
3 3

62

A.V. Kolesnichenkoqnq0 
30  hq 0 20 
qnq0 
20  hq 0 10 

 5

3 0
20

32

2 q 1  d P f p0 (  p p ) 20  d P f p0 Ep (  p p ) 




20

20  nq0 21



2 q 130  hq0 20

n

q0 31





20  hq0 10  .

(99)Hence, the relativistic version of the classical Stokes result followsПq ( x)  uu  u ,(100)

where for the bulk viscosity coefficient the following representation takes place [38]:u  

nq020

 5  n1
 3 q0 q0

20 32 21

30  hq0 20

 
31

20  hq0 10

  0 .

(101)

Let us now find an expression for the energy flux Wq , defined by the formula (96); as a result, we obtain the following approximate relationWq ( x)   p Ep f pq d P
h0

q 
2q1
d P f p0 (  p p ) 

0
3

q 
2 q 1d PEp f p0 (  p p ) 
03

21  hq 0 31

0.

(102)

Finally, using (9), we define an expression for the heat flux, which in the case of the first order in the deviations f p of the function from f p takes the form f p 0
J q ( x)  hq 0 Nq 

hq 0  p  f pq d P

Substituting here (93), we obtain q
2 q 1Jq ( x)  (  0 )  hq 0   d P f p0 Ep1( p p  )  

 3


q 
2q 1
d P f p0 ( p p  )    21  h0 11  0 .

0

3

(103)

Thus, in the linear case under consideration, the Fourier law for the heat flux in a nonextensive medium takes the form:Jq ( x)  h0T 2 ( 21  h0 11 ) T  T D u 63

A.V. Kolesnichenko  T  ( hq nq )1T  Pq ,(104)

where    h0T 2 ( h0 11  21 )  0 is the heat conduction coefficient. When writing formula(104), the relation (96) and the zero-order equation of motion (34) were used.
Entropy balance based on the Gibbs relation. Let us now find the explicit form of the entropy balance equation (37) by substituting the continuity equations (32) and the energy equations(35) into the Gibbs relation (57); as a result, we obtain

D sq    Jq / T  q 

1 
T


 T

= Пqu  Пq u  Jq 


 T



 Pq   ,hn  


(105)

where
  21 (u  u )  23 u- is the symmetrized space-like and zero-trace part of the hydrodynamic velocity gradient,
known as the shear tensor.
If we now exclude thermodynamic flows from the relation (105) by means of linear laws, we
can obtain the following expression for entropy growth T  Pq   T  Pq 0.
T q  u  (  )    T
hq nq   T
hq nq 



2



(106)

It follows that the generalized linear laws are compatible with non-negative entropy growth. The
expression (106) for has a familiar form, but includes only the first approximation for the transfer coefficients. In this case, the only explicit dependence on the deformation parameter takes
place in the definitions of thermodynamic integrals (46)-(48). For this result to be consistent withthe relativistic thermodynamics of irreversible processes, the values of the transfer coefficients
approximated here must be in good agreement with the coefficients obtained in the framework ofa more complex theory. Conversely, the obtained first-order coefficients can serve as a good criterion of the correctness of the method of second-order dissipative hydrodynamics derived fromthe relativistic Boltzmann equation using fourteen moments with a linearized collision integral[21], [41].
8.

CONCLUSION

Modern high-energy particle gas pedals have already reached the energy range in which thematter of the early Universe was formed. In particular, heavy ion collisions at the RHIC and
LHC colliders produce nuclear matter of extreme temperature and density. This is the so-called
quark-gluon plasma (QGP), which existed shortly after the Big Bang. Its properties are measured
in ultra-relativistic heavy-ion collisions, where the energy density is high enough to form a QGP
in a short time. Unfortunately, due to the nature of the strong interaction, there is no method to
directly observe this matter. As has been found in experiments, characterizing the identified had-64

A.V. Kolesnichenko

ron spectra using the Boltzmann-Gibbs thermodynamic approach is insufficient. Instead, thespectra of particles iz measured in high-energy collisions are well described by non-extensive
Tsallis-Pareto distributions derived from non-extensive statistical Tsallis thermodynamics. In
this connection the purpose of the presented work is to construct q-modified relativistic hydrodynamics, on the basis of which it is possible to develop various adequate models in elementary
particle physics to describe a wide range of phenomena in space and in the processes of production of high-energy particles in the evolving fluid at modern gas pedals, especially in nucleusnucleus collisions.
In the paper on the basis of the generalized kinetic Boltzmann equation derived earlier, a nonextensive relativistic hydro-dynamics for dissipative fluid is constructed. Relativistic thermodynamics is presented here as a field theory. Linear constitutive relations and transport coefficientssuch as shear viscosity, bulk viscosity and thermal conductivity are derived from the linearized
collision integral written in the Bhatnagar-Gross-Krook form and evaluated using the relaxation
time approximation. These relations relate dissipative fluxes to thermodynamic forces linearlywith positive transfer coefficients depending on the deformation parameter via thermodynamic
integrals introduced in [21], [40].
At the same time, it is important to keep in mind that the approach to the con- struction of
non-extensive relativistic hydrodynamics considered here contains a well-known fundamental
drawback, which leads to parabolic differential equations (and, consequently, to infinite velocities of propagation of heat and viscosity fluxes) and to instability of the anomalous q-system,
which is not suitable for many phenomena in high-energy astrophysics connected with steep gradients of thermodynamic parameters or with their fast spatial and temporal changes. This effectis explained (as in classical hydrodynamics) by the fact that the traditional thermodynamic approach to their derivation is based on a too restrictive hypothesis concerning the relationship between flows and thermodynamic forces. In particular, in the case of the simple fluid consideredhere it was assumed that the entropy flux is simply proportional to the heat flux and that the entropy density is independent of the heat and viscosity fluxes. However, this assumption is valid
only in the case of first order deviations from equilibrium. Neglecting the second-order terms inentropy density and flux leads to errors of the same order in entropy production, except for thecase when the spatial and temporal gradients of heat and viscous fluxes are negligibly small, i.e.,in quasi-steady-state conditions. Unfortunately, the problem related to the hyperbolicity of thesystem does not disappear with the introduction of the deformation parameter More perfect, butalso much more complicated non-extensive relativistic hydrodynamic equations than those presented in this paper can be obtained by the Grad method, using the 14-moment approximation.
REFERENCE[1] C. Tsallis, “Possible Generalization of Boltzmann-Gibbs-Statistics”, J. Stat. Phys., 52(1-2), 479-487(1988). (a regular updated bibliography is accessible at http: /tsallis. cat.cbpf.br/biblio.htm).
[2] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press, New York. 2004.
[3] C. Tsallis, Introduction to Nonextensive Statistical Mechanics. Approaching a Complex World,
Springer, New York. 2009.
[4] A.V. Kolesnichenko, “Thermodynamics of the Bose Gas and Blackbody Radiation in Non-Extensive
Tsallis Statistics”, Solar System Research, 54(4), 420-431 (2020).

65

A.V. Kolesnichenko[5] A.V. Kolesnichenko, “Jeans Instability of a Protoplanetary Circular Disk Taking into Account theMagnetic Field and Radiation in Nonextensive Tsallis Kinetics”, Solar System Research, 55(2), 132-149(2021).
[6] A.V. Kolesnichenko, “Jeans Instability of a Protoplanetary Gas Cloud with Radiation in Nonextensive
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Received August 10, 2023

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