.___
BB
20 January
cQ-223
1997
PHYSICS
EL..‘SEVIER
LETTERS
A
Physics Letters A 224 (1997) 326-330
A note on the q-deformation-theoretic aspect of the generalized
entropies in nonextensive physics
Sumiyoshi Abe
College of Science and Technology, Nihon University, Funabashi. Chiba 274. Japan
Received 3 1 May 1996; accepted for publication 24 October 1996
Communicated by C.R. Doering
Abstract
We show that a connection between the generalized entropy and theory of quantum groups, recently pointed out by
Tsallis [Phys. Lett. A 195 (1994) 3291, can naturally be understood in the framework of q-calculus. We present a new
entropy which has q +-+q -’ invariance and discuss its basic properties.
There is a growing
interest in extending Boltzmann-Gibbs
thermodynamics
by generalizing
the
concept of entropy to nonextensive
physics. This is
primarily concerned with understanding
the statistical and multifractal properties of systems with longrange interactions.
One way to accommodate
the
scale invariance to the formalism is to employ for
the definition of entropy a quantity such as (pijy,
where q is a real number and pi is the probability
that the system under consideration
is found in the
ith configuration.
An example has been given by
Tsallis [l], who made the following postulate for
entropy,
$p
-
&( lmJ41)
i
where q is a real parameter and is assumed to be
positive here. The summation is taken over all possible configurations.
The Boltzmann constant is set
equal to unity. Sz is related to Rdnyi’s entropy [2,3],
Sf = -(q
- l>-’
In E&P~)~, as (q - l>S,T +
0375-9601/97/$17.00
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PI1 SO375-9601(96)00832-S
exd - (q - 1 ).$I = 1. Moreover, in the case when
2, it becomes Fermi’s quadratic entropy 131:
Sr = SF = Cipi(l - pi>. The parameter q describes
the deviations of Re’nyi’s and Tsallis’s entropies
from the standard Shannon entropy [3-61,
q =
S=
- Lpi
In pi.
(2)
In the limit q --f 1, both RCnyi’s and Tsallis’s entropies converge on the Shannon entropy. Recently
Tsallis’s entropy has found a lot of applications in
various research areas, including stellar polytropes,
random walks, the travelling salesman problem, the
hydrogen-atom
specific heat and the cosmic background radiation. (See Ref. [7] as well as review
article [S].)
In a recent paper [9], Tsallis gave an intriguing
discussion of the similarity of the postulate (1) for
the entropy to q-deformed physics. Unlike the Shannon and RCnyi entropies, S: does not possess additivity. Suppose a system can be divided into two
independent subsystems 2, and &, . The probability
0 1997 Elsevier Science B.V. All rights reserved.
S. Abe/Physics LettersA 224 (1997) 326-330
327
distribution for the total system is factorized as
p(‘) 8 p (” in an obvious notation. In this case, Sl is
calculated to be
The idea is to deform the ordinary differential d/da
in this equation to the Jackson q-differential
d’/d(a; q) [17], which is defined by
s:[ p(l) ‘X+2’] = s:[ p”‘] + s,‘[ p(2)]
d'f(
a) = f(vPf(4
qa-a
d(a; s>
+(1
-q)s,T[pqs$7(*q.
(3)
This equation makes manifest that extended thermodynamics based on ST is nonextensive unless q = 1.
On the other hand, consider a q-deformed system
whose certain physical quantity is given by
where A is the corresponding physical quantity (e.g.,
energy) of the undeformed system reproduced in the
limit q + 1. For the system composed of two independent systems 2, and &, having A”’ and Ac2),
the total amount is, if the quantity is extensive,
simply the sum A = A(‘) + Ac2), whereas
[A’” + A’*‘]; = [ A”‘]; + [ A’2’];
+(q-
l)[A”‘];[A’*‘];.
(7)
’
and becomes the ordinary differential in the limit
1. From this definition, it is clear that Tsallis’s
entropy is expressed as follows,
q --j
d’
S4’= -
d(a;
cc
s)
i
”
(I
.)/
(8)
n=,’
There is another type of deformation which frequently appears in studies of quantum groups [lo161. The q-differential operation in this type is given
by [1*1
d.f(ff) =f(wFf(qPa)
d(a; 4) -
qa-q-la
(9)
’
which also approaches the ordinary differential as
1. In contrast to Eq. (4), the expression used in
the theory of quantum groups is often of the form
q +
(5)
Thus, one sees a striking similarity between Eqs. (3)
and (5). This, in turn, suggests the possibility of a
consistent q-deformation-theoretic approach to constructing a class of extended thermodynamics.
Now, in the area of quantum groups, the formalisms are often required to possess invariance
under the interchange q c, q- ’ [lo- 161. Apparently
there is a huge ambiguity in making Tsallis’s entropy
symmetric, and therefore some guiding
V-V-’
principle is needed. In this Letter, we show using
q-calculus that Tsallis’s postulate for the entropy, in
fact, contains the q-deformation structure in itself. In
a sense, this observation turns out to be more fundamental than the above-mentioned similarity. Based
on this idea, we then present a new generalized
entropy, which can be interpreted as the q f, q- ’
symmetric modification of Tsallis’s entropy. Its basic
properties are also discussed.
Our basic idea consists of rewriting the Shannon
entropy as follows,
(10)
For A = A(‘) + At2), this yields
[ A(‘) + A’z’], = [ A(“lq x f{qA(2)+ q-A’*‘}
+ [ A@$ x +{qA”’ + q-A”‘]
= [ A(‘)lg + [ A(‘$ + +( q
-
1)
x ([ A(‘)],[ A(~)]; + [ A(~)],[ A(‘)]:)
+$(q-’
-
l){[ A”‘],[ AC2’];-’
+ [ Ac2)lq[ A(‘)];-I}.
(11)
Now replacing the ordinary differential in Eq. (6) by
d/d(a;
q),
we obtain the following new quantity,
d
Sg”= d(a;
= _ c
9)
C(
i
”
n
.)I
o=~
(Pi)4-((pi)4-’
q_q-’
’
(12)
i
s=-&~(Pi)”
’
L
a='
(6)
which has the manifest q w q-’ invariance. The
formalisms based on Eqs. (4) and (7) and based on
328
S. Abe /Physics
Letters A 224 (1997) 326-330
Eqs. (9) and (10) are occasionally referred to as the
mathematical and physical deformation theories, respectively, whose difference from each other has
been stressed and investigated in Refs. [ 15,161. In
this sense, Sy’ and Sg” can, respectively, be interpreted as the mathematical
and physical deformations of the Shannon entropy.
In what follows, we discuss the properties of Si
as a generalized entropy. Most of them will turn out
to result from those of Tsallis’s entropy. This is
because Sz can be expressed by Tsallis’s entropy as
follows,
SS = (4 - l)S,T - (4-l
4
4-V’
Henceforth
Si. (Regarding this function, it is worth mentioning
that, as the function s(x) = -x In x, q,,(x) is also
smaller than
1 - x: -(xq - xq-‘>/(q - 4-l) < 1
- x (x > 0). This can be shown using Theorem 42
in Ref. [20]. Note that the upper bound is independent of q.)
(ii) Additiuity. For the factorized probability distribution p”’ 8 pc2’, one immediately finds the following pseudo-additivity,
S,s[ p”‘@JP’]
- 1)$--l
(13)
the range of q in S: is restricted to
O<q<l,
since the whole of positive q can be compactified to
this range by virtue of the q * q- ’ symmetry.
(i) Concur@.
Tsallis’s entropy is known to be
concave for positive q [1,19]. Here we note that the
factors of SJ and Sz--I on the r.h.s. of Eq. (13) are
both nonnegative.
Therefore Si also possesses the
concavity. That is, for the combination of two probability distributions p: = A, pi + A, p: (A,, A, 2 0; A,
+ A, = I), it satisfies the following inequality,
S,~[P”] 2 A,S,S[p]
provided
+AP$[P’],
that the equality holds
1, we present plots
“-I>/( q - q- ’ ),
r/q(x)= -(x4-x
the concavity of vq(x), leading to
q = 1. In Fig.
= s;[ p”‘]
(14)
(151
if pi = pi or if
of the function
which illustrate
the concavity of
+ s,s[ p]
{s”[ p”‘]S,‘[
JP’]
+;(I
- q-‘){s,s[
+s;[
p’2’]syTI[
+ +< 1 - q)
+S,S[ p’“]sqT[
p”‘]S;-1[
p”‘]}.
p’“]]
JP’]
(16)
This relation is to be compared with Eq. (11). Clearly
Sz is nonextensive unless q = 1. S,S[p(‘) SJ+~)] can
become larger or smaller than S,“[ #‘)I+ S,S[~‘~‘1 in
a complicated manner. Eqs. (3) and (16) form the
extended closed relation concerning
the pseudoadditivity.
(iii) Generalized H-theorem. It is known that
Tsallis’s entropy satisfies the generalized H-theorem
for q > 0, if the time evolution of the probability
distribution is subject to the master equation [21]
z=
F(w’jls--~iPi)>
(17)
where Wij is the transition probability per unit time
from the jth configuration to the ith one. With [22]
or without 1231 the detailed balance Wij = Wji, Tsallis’s entropy obeys dSz/dt 2 0 (q > 0). Therefore,
due to the reason mentioned in (i), Sg” also satisfies
the generalized H-theorem,
X
Fig. 1. Plots of the function 77q(x)=-(xY-~4-‘)/(9-9_‘)
versusxforq=0.1,0.3,1.~p~,(~)~-~X~.
dSS
2
2 0.
dt
(18)
(iv) Equiprobability distribution. To see how .Si
measures the system degree of freedom, it is worth
S. Abe/Physics
Leffers A 224 (1997) 326-330
329
examining the special case of equiprobability
distribution. Let us assume a finite number of configurations W. Then the equiprobability
distribution
pi =
l/W (i= 1,2,...,W)
gives rise to
WI-Y
S,s[l/W]
= -
_
Pi
w’-Y-’
.
q-q-i
( 19)
For q in Eq. (13). this is a concave function of W
and diverges as W + cc. In Fig. 2, we present plots
of S$l/W]
for some values of q. Clearly the
Boltzmann
relation S = In W is recovered in the
limit q + 1.
(v) Power-law behavior. Finally we consider the
maximization of Sy” under the conditions of normalization as well as a given mean value of a certain
physical quantity Q. The relevant functional to be
minimized is
where LYand p are the Lagrange multipliers.
the maximization condition is found to be
4(
q-l(
Pi)‘-]
Then
p;)Y-‘-’
+a+pQi=O.
(21)
9-K’
It is generically a numerical problem to solve this
equation with respect to pi. When q = 1, one obtains
the standard exponential
distribution
pi = exp( - CY
- pQi), whereas for 0 < q < 1, the distribution
comes to exhibit the power-law behavior, in general.
In Fig. 3, we present, for some values of q, plots of
relation (21) using the variable ci = (Y,,+ pQi. (Here
8
-5
-7.5.
Fig. 3. Plots of the unnormalized distribution function p, of the
quantity
c, = (Y,,+ PQ, for 4 = 0.3, 0.5, 1. For q= 1, p, =
exp( - c,).
(~a stands for the value of (Y, which can be determined in principle by the normalization
condition.
However (Ye is a highly complicated function of q
and p, and is not specified in Fig. 3. Accordingly, in
Fig. 3, pi is not normalized and the variable ci
differs between all curves by additive constants, i.e.,
the values of (Ye.) Significant deviations from the
exponential
case (q = 1) due to the deformation
effects can be seen for small values of ci. In particular, a small q can lead to the points of inflection.
In conclusion, we have reconsidered the definition
of Tsallis’s generalized entropy in the framework of
q-calculus. We have presented the generalized entropy of a new type which possesses q ++ q- ’ invariance and have studied its basic properties. In this
the concept of the q-differential has played an essential role. It is known [24] that there is a close
connection between a q-deformation structure and a
lattice structure with an approximate lattice constant
( 1 - q (. A lattice structure means truncation of substructures whose scales are smaller than the lattice
constant. In Eqs. (8) and (12), the q-differentials are
applied to the exponents of the probability distributions, which can be related to the fractal dimensions
of systems having self-similarity. The physical meanings of ignoring smaller “scales”
are yet to be
clarified.
References
20
40
Fig. 2. Plots of the function
S,“= ,[l/W]=
In W.
60
$[l/
WI versus W for 4 = 0.1.0.3,
1.
[ 11 C. Tsallis, J. Stat. Phys. 52 (1988) 479;
2. Dankzy, Inform. Control 16 (1970) 36.
330
S. Abe/Physics
Letters A 224 (1997) 326-330
[2) A. RCnyi, Probability
theory (NorthHolland,
Amsterdam,
1970).
[3] G. Jumarie, Relative information (Springer, Berlin, 1990).
[4] C.E. Shannon and W. Weaver, The mathematical theory of
communication
(University of Illinois Press, Urbana, 1949).
[5] L. Brillouin, Science and information theory (Academic Press,
New York, 1956).
[6] A. Wehrl, Rev. Mod. Phys. 50 (1978) 221.
[7] A.R. Plastino and A. Piastino, Phys. Lett. A 174 (1993) 384;
D.A. Stariolo, Phys. Lett. A 185 (1994) 262;
P.A. Alemany and D.H. Zanette, Phys. Rev. E. 49 (1994)
R956;
T.J.P. Penna, Phys. Rev. E 51 (1995) RI;
L.S. Lucena, L.R. da Silva and C. Tsallis, Phys. Rev. E 51
(1995) 6247;
C. Tsallis, F.C. SB Barreto and E.D. Loh, Phys. Rev. E 52
(1995) 1447.
[S] C. Tsallis, Chaos solitons fractals 6 (1995) 539.
[9] C. Tsallis, Phys. Lett. A 195 (1994) 329.
[IO] V.G. Drinfel’d, Sov. Math. Dokl. 32 (1985) 254.
[I I]
[12]
[I31
[14]
[I51
[16]
[17]
[IS]
[19]
[201
1211
[22]
[23]
[24]
M. Jimbo, Lett. Math. Phys. 10 (1985) 63.
A.J. Macfarlane, J. Phys. A 22 (1989) 4581.
L.C. Biedenharn, J. Phys. A 22 (1989) L873.
V.I. Man’ko and R.V. Mendes, Phys. Lett. A 180 (1993) 39.
AI. Solomon and J. Katriel, J. Phys. A 26 (1993) 5443.
AI. Solomon, Phys. Lett. A 188 (1994) 215; 196 (1994) 29.
F. Jackson, Mess. Math. 38 (1909) 57; Quart. J. Pure Appl.
Math. 41 (1910) 193.
A.J. Bracken, D.S. McAnalIy, R.B. Zhang and M.D. Gould,
J. Phys. A 24 (1991) 1379.
G.A. Raggio, J. Math. Phys. 36 (1995) 4785.
G. Hardy, J.E. Littlewood and G. PBlya, Inequalities, 2nd Ed.
(Cambridge Univ. Press, Cambridge, 1952).
N.G. van Kampen, Stochastic processes in physics and chemistry (North-Holland,
Amsterdam, 1981).
A.M. Mark, Phys. Lett. A 165 (1992) 409.
J.D. Ramshaw, Phys. Len. A 175 (1993) 169.
A. Dim&is and F. Miller-Hoissen,
Phys. Lett. B 295 (1992)
242.