We discuss the Tsallis entropy in finite N-unit nonextensive systems by using the multivariate q-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the q-average (q: the entropic index). The Tsallis entropy obtained by the q-average has an exponential N dependence: Sq(N)/Ne(1q)NS1(1) for large N(1/(1q)>0). In contrast, the Tsallis entropy obtained by the normal average is given by Sq(N)/N[1/(q1)N] for large N(1/(q1)>0). N dependences of the Tsallis entropy obtained by the q- and normal averages are generally quite different, although both results are in fairly good agreement for |q1|1.0. The validity of the factorization approximation (FA) to PDFs, which has been commonly adopted in the literature, has been examined. We have calculated correlations defined by Cm=(δxiδxj)m(δxi)m(δxj)m for ij where δxi=xixi, and the bracket stands for the normal and q-averages. The first-order correlation (m=1) expresses the intrinsic correlation and higher-order correlations with m2 include nonextensivity-induced correlation, whose physical origin is elucidated in the superstatistics.

1.
C.
Tsallis
,
J. Stat. Phys.
52
,
479
(
1988
).
2.
C.
Tsallis
,
R. S.
Mendes
, and
A. R.
Plastino
,
Physica A
261
,
534
(
1998
).
3.
C.
Tsallis
, in
Nonextensive Statistical Mechanics and Its Application
, edited by
S.
Abe
and
Y.
Okamoto
(
Springer-Verlag
,
Berlin
,
2001
), p.
3
.
4.
E. M. F.
Curado
and
C.
Tsallis
,
J. Phys. A
24
,
L69
(
1991
);
E. M. F.
Curado
and
C.
Tsallis
,
J. Phys. A
25
,
1019
(
1992
) [corrigenda].
5.
S.
Martinez
,
F.
Nicolas
,
F.
Pennini
, and
A.
Plastino
,
Physica A
286
,
489
(
2000
).
7.
C.
Tsallis
and
U.
Tirnakli
,
J. Phys.: Conf. Ser.
201
,
012001
(
2010
).;
e-print arXiv:0911.1263 [cond-mat].
8.
C.
Tsallis
,
M.
Gell-Mann
, and
Y.
Sato
,
Proc. Natl. Acad. Sci. U.S.A.
102
,
15377
(
2005
).
9.
F.
Caruso
and
C.
Tsallis
,
Phys. Rev. E
78
,
021102
(
2008
).
10.
G.
Wilk
and
Z.
Wlodarczyk
,
Physica A
376
,
279
(
2007
).
12.
S.
Abe
and
G. B.
Bagci
,
Phys. Rev. E
71
,
016139
(
2005
).
13.
S.
Abe
,
Astrophys. Space Sci.
305
,
241
(
2006
).
14.
16.
S.
Abe
,
J. Stat. Mech.: Theory Exp.
2009
,
P07027
.
17.
R.
Hanel
,
S.
Thurner
, and
C.
Tsallis
,
Europhys. Lett.
85
,
20005
(
2009
).
18.
J. F.
Lutsko
,
J. P.
Boon
, and
P.
Grosfils
,
Europhys. Lett.
86
,
40005
(
2009
).
19.
H.
Hasegawa
,
Phys. Rev. E
77
,
031133
(
2008
).
20.
H.
Hasegawa
,
Phys. Rev. E
78
,
021141
(
2008
).
23.
S.
Abe
,
S.
Martinez
,
F.
Pennini
, and
A.
Plastino
,
Phys. Lett. A
281
,
126
(
2001
).
26.
Q. A.
Wang
,
M.
Pezeril
,
L.
Nivanen
, and
A. L.
Méhauté
,
Chaos, Solitons Fractals
13
,
131
(
2002
).
27.
28.
30.
Liyan
Liu
and
Jiulin
Du
,
Physica A
387
,
5417
(
2008
).
31.
Z. -H.
Feng
and
L. -Y.
Liu
,
Physica A
389
,
237
(
2010
).
32.
G.
Wilk
and
Z.
Wlodarczyk
,
Phys. Rev. Lett.
84
,
2770
(
2000
).
33.
34.
C.
Beck
and
E. G. D.
Cohen
,
Physica A
322
,
267
(
2003
).
35.
C.
Beck
, in
Anomalous Transport: Foundations and Applications
, edited by
G.
Radons
,
R.
Klages
,
I. M.
Sokolov
, (
Wiley
,
New York
,
2008
).
36.
Although the expression for Zq(N) given by Eq. (22) is different from that given by Eqs. (36)–(38) in Ref. 20, both are equivalent.
37.
Lists of many applications of the nonextensive statistics are available at http://tsallis.cat.cbpf.br/biblio.htm.
38.
H.
Hasegawa
,
Phys. Rev. E
80
,
011126
(
2009
).
41.
A. K.
Rajagopal
,
R. S.
Mendes
, and
E. K.
Lenzi
,
Phys. Rev. Lett.
80
,
3907
(
1998
).
You do not currently have access to this content.