Physica A 392 (2013) 1972–1976
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Physica A
journal homepage: www.elsevier.com/locate/physa
A set of new three-parameter entropies in terms of a
generalized incomplete Gamma function
Somayeh Asgarani
∗
Department of Physics, Payame Noor University, PO Box 19395-3697, Tehran, Iran
article info
Article history:
Received 4 August 2012
Received in revised form 4 November 2012
Available online 23 December 2012
Keywords:
Generalized entropy
Generalized incomplete Gamma function
Nonextensive statistical mechanics
abstract
In this paper, by considering the first three Khinchin axioms (K1–K3) and neglecting the
fourth (K4), a set of new three-parameter entropies will be introduced which are expressed
in terms of a generalized incomplete Gamma function as S
d,c
1
,c
2
[p]∝
i
Γ (d + 1, 1 −
c
1
ln p
i
, 1 − c
2
ln p
i
). Also, its asymptotic behavior will be found and it will be shown that,
for some special values of parameters, some known entropies like Tsallis, Kaniadakis and
Abe entropies are recovered.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Nowadays, there are many of natural phenomena that appear to deviate from a standard Boltzmann–Gibbs (BG) distri-
bution. Those phenomena which show asymptotic power-law behavior can be seen in systems with long-range interaction,
long-time memory, or non-ergodic and non-Markovian systems. Many attempts have been made to find an appropriate en-
tropic form which can properly explain the macrostate of those systems. In this context, a series of generalized entropies has
been suggested over the past decades [1–15]. However, there is no systematic way of deriving the correct generalizations
of BG entropy. In this paper, we attempt to find a more generalized form of entropy from first principles.
As s known, the four Khinchin axioms [16] uniquely determine the BG entropy. Because some interacting systems do
not respect K4, it seems reasonable to consider only K1–K3. Recently, Hanel and Thurner [1], by assuming the first three
Khinchin axioms and neglecting the fourth, introduced a series of two-parameter entropies as
S
c,d
[p]∝
i
Γ (d + 1, 1 − c ln p
i
), (1)
where Γ (a, b) =
∞
b
t
a−1
exp(−t ) dt is an incomplete Gamma function [17] and the entropy is in the form
S =
i
g (p
i
). (2)
Using the first three Khinchin axioms, the parameters c and d characterize the scaling properties of the system in the large
size limit [1] as
f (z ) ≡ lim
x→0
g (zx)
g (x)
= z
c
(3)
h(a) ≡ lim
x→0
g (x
1+a
)
x
ac
g (x)
= (1 + a)
d
, (4)
∗
Tel.: +98 9132038900.
E-mail addresses: sasgarani@pnu.ac.ir, sasgarany@yahoo.com.
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.physa.2012.12.018