The property of kappa-deformed statistics for a relativistic gas in an
  electromagnetic field: kappa parameter and kappa-distribution
The κ parameter for a relativistic gas
arXiv:0704.1437v1 
The property of κ -deformed statistics for a relativistic gas in an 
electromagnetic field: κ parameter and κ -distribution 
Guo Lina, Du Jiulin* and Liu Zhipeng  
Department of Physics, School of Science, Tianjin University, Tianjin 300072, China 
Abstract 
We investigate the physical property of the κ parameter and the κ -distribution in 
the κ -deformed statistics, based on Kaniadakis entropy, for a relativistic gas in an 
electromagnetic field. We derive two relations for the relativistic gas in the framework of 
κ -deformed statistics, which describe the physical situation represented by the 
relativistic κ -distribution function, provide a reasonable connection between the 
parameter κ , the temperature four-gradient and the four-vector potential gradient, and 
thus present for the case 0≠κ  one clearly physical meaning. It is shown that such a 
physical situation is a meta-equilibrium state, but has a new physical characteristic. 
PACS: 05.90.+m; 05.20.Dd; 03.30.+p 
Keywords: generalized statistics; relativistic gas 
                                                        
* Corresponding author: jiulindu@yahoo.com.cn 
In recent years, some new statistics have been put forward to generalize the classical 
Boltzmann–Gibbs (BG) one, such as the nonextensive statistics based on Tsallis entropy 
(q-entropy)[1], the so-calledκ-deformed statistics based on theκ-entropy given by 
Kaniadakis [2] besides others. We mention that the concept ofκ-distribution was 
introduced about four decades ago and also discussed recently by Leubner [3], which was 
actually equivalent to the nonextensive q-distribution, but different from Kaniadakis one. 
These statistics are studied all dependent on one parameter. For instance, the 
nonextensive statistics depends on the parameter q different from unity and it will recover 
BG one if we take the parameter to be unity; theκ-deformed statistics depends on the 
parameterκ different from zero and it will recover BG one if we take the parameter to 
be zero. Then it is naturally important question for us to ask what should these parameters 
stand for and under what physical situations should these statistics be suitable for the 
statistical description of a system. 
For the nonextensive statistics, we know that the parameter q different from unit is 
related to the temperature gradient, T∇ , and the long-range potentials, ϕ , of the 
systems such as self-gravitating system [4,5] and plasma system [6], by ϕ∇−∇ )~ qT 1( . 
Thus it can be reasonably applied to describe thermodynamic properties of the systems 
under an external field when they are in the nonequilibrium stationary-state. And this 
characteristic for the nonextensive statistics has recently received the experiment support 
by the helioseismological measurements [7]. As potential astrophysical examples with 
regard to the above theoretical investigation, a new application of the nonextensive 
theory for both dark matter [8], stars [9] and hot plasma [3, 10] density distributions in 
clustered astrophysical structures was investigated. In this situation, the potential gradient 
is parallel to the temperature gradient, thus fitting perfectly well into the discussions [4-7]. 
For theκ-deformed statistics, we actually know little about the physical property of the 
parameterκdifferent from zero and the corresponding statistics although it has been 
studied by Kaniadakis and his co-authors in many papers. In this letter, we study the 
physical properties in the case of parameterκ≠ 0 for a relativistic gas under an 
electromagnetic field. 
The general form of theκ-entropy proposed by Kaniadakis can be written by theκ
-distribution function, f , [2] as 
     321 )()( afaSafK += κ ,                                      (1) 
being , which can have different expressions after having given 
specific choices for the parameters a
∫−= ffvdfS κκ ln )( 3
i with i=1,2,3, where ai may depend on the position x. 
For instance, a particular case of K(f) is 
( ) ( )
2 1 2 1
S d v f f
κ κ κ κ κ
= − −  + − 
∫ 1 κ− ,                    (2) 
which recovers to the Boltzmann-Gibbs entropy in the limitκ→0, where z is a positive 
real parameter that needs to be specified. Based on Kaniadakis entropy, theκ-deformed 
statistics can be defined by the so-calledκ-exponential andκ-logarithm functions as 
( ) ( )
2 2exp 1f f f
κ κ κ= + + ,                       (3) 
( )ln
=  .                            (4) 
As one may check, we have expκ(lnκ(f) )= lnκ(expκ(f )) = f and the above functions could 
reduce to the standard exponential and logarithm ones if we letκ→0. Actually, thisκ
-framework can lead to a class of one parameter deformed structures with interesting 
mathematical properties[11].  
We consider a relativistic gas containing N point particles of mass m enclosed in a 
volume V, under the action of an external four-force field Fµ. The temporal evolution of 
the relativistic distribution function f(x, p) is governed by the following relativisticκ
-transport equation [11,12]: 
( )fp f mF C f
µ µ κ
∂ + =
,                          (5) 
where µ = 0, 1, 2, 3, the particles have four-momentum =  in each 
point  of the space-time, with their energy E/c = 
µpp ≡ ),/( pcE
),( rctxx =≡ µ 2c22 m+p , and 
denotes the differentiation with respect to time-space coordinates and 
the relativisticκ-collisional term. According to the generalized H theorem [11,12], 
the relativistic version ofκ-distribution function in this framework approach to 
( )1 ,tc− ∂ ∇µ∂ =
( ) ( )
2 2, exp 1f x p B B
κ θ κ θ κ= = + + θ ,                (6) 
where B may depend on x, and 
µβαθ pxx )()( +=                                (7) 
with ( )xα  a scalar and µβ  a four-vector, arbitrary space and time-dependent 
parameters. And the relativisticκ-collisional term vanishes and Eq.(5) becomes 
p f mF
∂ + =
.                                   (8) 
The distribution (6) can also be obtained by maximizing K(f) under the right constraints. 
Let us now consider the external four-force field as the Lorentz one, namely, 
( ) ( ) ( ), /F x p Q mc F x pµν ν= −                             (9) 
with Q the charge of the particle and F µν the Maxwell electromagnetic tensor. In this case, 
Eq.(8) is 
         1 0
p f Qc F p
µ ν µ
− ∂∂ − =
                                   (10) 
or, equivalently, it can be written as 
pFQcfp .                                 (11) 
And theκ-distribution function, Eq.(6), [11, 12] reads 
, exp
u p c QA x U
f x p B
−  − +  =
  
u p c QA x U u p c QA x U
k T k T
µ µ µ µ
− −      − + − +      = + + 
      
,  (12) 
where kB is Boltzmann constant, Uµ is the mean four-vector velocity of the gas with 
, T(x) is the temperature field, u is the Gibbs function per particle, and A2cUU =µµ
µ(x) 
is the four-vector potential. In the limitκ→0, it reduces to the well known relativistic 
expression [13] correctly, 
, exp
u p c QA x U
f x p B
−  − +  =
  
                   (13) 
From Eq.(12), we have  
( ) ( )
u p c QA x U u p c QA x U
f x p B
k T k T
µ µ µ µ
µ µκ κ κ κ
− −      − + − +      = + + 
      
, (14) 
( )12 2 2
u p c QA U
f B B
µκ κ κκ
= + f κ  .                      (15) 
Substitute Eq.(15) into Eq(11) and after a series of calculations, we get the equation 
( ) ( )
2 2 2
2 2 2 22 1
2 2 22 2
B B B
p B p B p B up U p B uc QA
k T k T k T
µ κ µ κ µ κ µ µ κ µUµ µ µ µ µ
κ −∂ + ∂ − ∂ − ∂
( ) ( )
2 2 2 21 1
B B B
p B p U p B p U c QA U p B c QA U
k T k T k T
µ κ µ µ κ µ µ µ κ µ
µ µ µ µ µ µ
κ κ κ− −+ ∂ + ∂ + ∂
2 2 2
2 2 2 2 2 2
2 22 2 22 2
p T p T p Tup U uc QA Uu
B B B
T T Tk T k T k T
µ µ µµ µ
µ µ µ
µ µκ κ κκ κ κ
−∂ ∂ ∂
− + +
( ) ( )
( ) ( )
2 2 2 2 11
2 2 2 2 2 2
2 22 2 22
p T p U p T p T c QA Up U c QA U
B B B
T T Tk T k T k T
µ µ µ µ µµ µ
µ µ µ µµ µκ κ κκ κ κ
−−∂ ∂ ∂
− − −
2 21 12
2 2 2 2 2 2
B B B
c Qp A U U p T c Qp A UQ u
B B F p B
k T c k T T k T k T
µ µ µ µ µ
µ µ µ µκ κ µν κ
νκ κ κ
− − ∂ ∂ 
− − −       
2 2 2 22 2
B B B B
p T p U c Qp A U p T c QA U c Qp A U
T k T k T T k T k T
µ µ µ µ µ µ µ µ1
µ µ µ µ µ µ µκ κκ κ
− − −∂ ∂ ∂ ∂
+ + µ
2 2 2 22 2
B B B B
p T U p T p U Uu Q Q
B F p B F p
T k T c k T T k T c k T
µ µ µ
µ µ µ µ µκ µν κ µν
ν νκ κ
2 2 2 22 2
B B B B
p T c QA U U c Qp A U UQ Q
B F p B F p
T k T c k T k T c k T
µ µ µ µ
µ µ µ µ µ µκ µν κ µν
ν νκ κ
− −∂ ∂
− + 0= .(16) 
We consider that Eq. (16) is the identically null for any arbitrary p and the sum of the 
coefficients of every power for p in this equation must be zero. Thus, when we consider 
the coefficients for the forth-power terms of p in Eq. (16), we obtain the relation, 
 ∂ ∂ 
=      
 .                                      (17) 
The coefficients of the third power of p is 
2 2 1 2 2
2 2 22 2 2
B uU B U c QA U B
Tk T k T k T
µ µκ κ µ µ κ
µ µ µ µ
− ∂ + ∂ + 2
2 2 2 2
222 2
T U c QA U T U c Q A U
T T k Tk T
µ µ µ µ µκ κκ κ
− −∂ ∂ ∂
− + µ  
T U UQ
T k T c k T
µ µ µκ µνκ
+ 0= .                                             (18) 
Substitute Eq.(17) into Eq.(18), we get 
( ) 0T A F Uν µνµ µ µ∂ ∂ − =                                    (19) 
The coefficients of the second power of p read 
( ) ( )
2 2 2
2 2 2 22 1
2 2 22
B B B
B B B uc QA U B c QA U
k T k T k T
κ κ κ µ κ µ
µ µ µ µ µ
κ − −∂ + ∂ − ∂ + ∂ 1 µ  
( ) ( )
22 2 2 112
2 2 2 2 2 2
2 22 2 22
c QA UT T Tuc QA Uu
B B B
T T Tk T k T k T
µµ µ µµκ κ κκ κ κ
−−∂ ∂ ∂
− + − 2
2 21 12
2 2 2 2 2 2
B B B
c Q A U U T c Q A UQ u
B B F B
k T c k T T k T k T
µ µ µ µκ κ µν κκ κ κ
− − ∂ ∂ 
− − −       
µ µ∂  
2 2 2 22 2
B B B
T c QA U c Q A U T Uu Q
T k T k T T k T c k
µ µ µ µ µκ κκ κ
− −∂ ∂ ∂
+ + µµν  
2 2 2 22 2
B B B
T c QA U U c Q A U UQ Q
B F B F
T k T c k T k T c k T
µ µ µ µ µκ µν κκ κ
− −∂ ∂
− + 0
µµν = .        (20) 
Substitute Eqs.(17) and (19) into Eq.(20), we find 
( ) ( )
2 222 2
k T A F U
µ µκ ∂ = ∂ − µ
 ,                             (21) 
which easily leads to the relation 
µµ κ FAQTkB −∂=∂ .                                  (22) 
Furthermore, using the electromagnetic tensor expression, F A Aµν νµ ν= ∂ − ∂
µ , we can 
write Eqs.(19)and (22), respectively, as 
( )( ) 0=∂∂ µνµ AT                                         (23) 
νµ κ AQTkB ∂=∂ , or 
νµκ AQTkB ∂∂= .                     (24) 
where and  are the relativistic expressions of the temperature gradient and 
the external potential gradient, respectively. From Eq.(24) we find that the parameter 
κ is different from zero if and only if the quantity Tµ∂  is not equal to zero, thus 
showing that 0≠κ  represents the physical situation of the relativistic gas with a 
time-spatial inhomogeneous temperature field and under the electromagnetic field. This is 
a nonequilibrium stationary-state. 
But such a nonequilibrium relativistic gas should have one additional characteristic: 
from Eq.(23) we find that the temperature four-gradient Tµ∂  must be vertical to the 
four-potential gradient . µν A∂
If take κ = 0, one has = 0, which just is the physical situation represented by 
the standard distribution (13). The non-relativistic versions can be written directly from 
Eqs.(23) and (24) by 
       0=∇⋅∇ ϕT ; ϕκ ∇=∇ QTkB  ,                             (25) 
which are in agreement on the relations derived recently for the parameter κ 0≠  in a 
nonequilibrium system [14]. Thus Eqs.(23) and (24) extend these classical relations to the 
special relativistic case. 
In summary, we obtain two relations, Eqs.(23) and (24), of the parameter κ  in the 
κ -deformed statistics for a relativistic gas in an electromagnetic field. They describe the 
physical situation represented by the relativisticκ-distribution function (12) and provide 
a reasonable connection between the parameter κ , the temperature four-gradient and the 
four-vector potential gradient, so presenting the case of the parameter 0≠κ  one clearly 
physical meaning. Such a physical situation is a nonequilibrium stationary-state, but has a 
new characteristic given by Eq.(23). 
Additionally, from Eq.(17) it follows that when B is a constant also the temperature 
turn out to be constant, which is independent on the parameter . Therefore the 
equilibrium state emerges also in the case of a relativistic gas described by the 
-distribution based on the entropy K(f) in presence of an external field. Conversely 
when B depends on x, the system admits the above nonequilibrium stationary-state. From 
the view of this point, the physical situation described by the κ -distribution may be 
called a metaequilibrium state. 
Additional remarks: Most recently, Plastino et al reported a general discussion on the 
generalized entropy and the maximum entropy approach [15]. 
Acknowledgements 
This work is supported by the project of “985” program of TJU of China and also by 
the National Natural Science Foundation of China, No. 10675088. 
References 
[1] C.Tsallis, J. Stat. Phys. 52(1988)479. 
[2] G. Kaniadakis, Physica A 296 (2001) 405. 
[3] M.P.Leubner, Astrophys. J. 604(2004)469. 
[4] J.L.Du, Europhys.Lett. 67(2004)893. 
[5] J.L.Du, Astrophys.Space Sci. 305(2006)247. 
[6] J.L.Du, Phys.Lett.A 329(2004)262. 
[7] J.L.Du, Europhys.Lett. 75(2006)861. 
[8] M.P.Leubner, Astrophys. J. 632(2005)L1. 
[9] J.L.Du, New Astron. 12(2006)60. 
[10] S.Shaikh, A.Khan and P.K.Bhatia, Z. Naturforsch. 61A(2006)275. 
[11] G. Kaniadakis, Phys.Rev.E 66(2002)056126; Phys.Rev.E 72(2005)036108. 
[12] R. Silva, Eur.Phys. J. B 54(2006)499; cond-mat/0603177. 
[13] R. Hakim, Phys. Rev. 162(1967)128; H. D. Zeh, The Physical Basis of the Direction 
of Time, Spring-Verlage, Berlin-Heidelberg,1992. 
[14] L.N. Guo and J. L. Du, Phys. Lett. A 362(2007)368. 
[15] A.R. Plastino, A. Plastino, and B.H. Soffer, Phys. Lett. A 363(2007)48. 
