ConstantinoTsallis · 2017. 6. 1. · 0 20406080 100n 0 50 100 150 200 S q (n) q=-0.2q= 0 q=+0.2...
Transcript of ConstantinoTsallis · 2017. 6. 1. · 0 20406080 100n 0 50 100 150 200 S q (n) q=-0.2q= 0 q=+0.2...
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Constantino TsallisCentro Brasileiro de Pesquisas Fisicas, BRAZIL
C. T., M. Gell-Mann and Y. Sato, Proc Natl Acad Sci (USA) 102, 15377 (2005)
L.G. Moyano, C. T. and M. Gell-Mann, Europhys Lett 73, 813 (2006)
S. Umarov, C. T., M. Gell-Mann and S. Steinberg, cond-mat/0603593, 0606038, 0606040
P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)
L.F. Burlaga and A.F.-Vinas, Physica A 356, 375 (2005).
A. Rapisarda and A. Pluchino, Europhysics News 36, 202 (EPS, Nov/Dec 2005)
Erice, August/September 2006
COMPLEXITY AND NONEXTENSIVE STATISTICAL MECHANICSTHEORY, EXPERIMENTS,
OBSERVATIONS AND COMPUTER SIMULATIONS
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UBIQUITOUS LAWS IN COMPLEX SYSTEMS
ORDINARY DIFFERENTIAL EQUATIONS
ENTROPY Sq (Nonextensive statistical mechanics)
PARTIAL DIFFERENTIAL EQUATIONS (Fokker-Planck, fractional derivatives, nonlinear, anomalous diffusion, Arrhenius)
STOCHASTIC DIFFERENTIAL EQUATIONS (Langevin, multiplicative noise)
NONLINEAR DYNAMICS (Chaos, intermittency, entropy production, Pesin, quantum chaos, self-organized criticality)
CENTRAL LIMIT THEOREMS (Gauss, Levy-Gnedenko)
q-ALGEBRA
CORRELATIONS IN PHASE SPACE
GEOMETRY (Scale-free networks)
LONG-RANGE INTERACTIONS (Hamiltonians, coupled maps)
SIGNAL PROCESSING (ARCH, GARCH)
IMAGE PROCESSING
GLOBAL OPTIMIZATION (Simulated annealing)
q-TRIPLETTHERMODYNAMICS
FURTHER APPLICATIONS (Physics, Astrophysics, Geophysics, Economics, Biology, Chemistry, Cognitive psychology, Engineering, Computer sciences, Quantum information, Medicine, Linguistics …)
AGING (metastability, glass, spin-glass)
SUPERSTATISTICS (Other generalizations)
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Enrico FERMI
Thermodynamics (Dover, 1936)
The entropy of a system composed of several parts is very oftenequal to the sum of the entropies of all the parts. This is true if the energy of the system is the sum of the energies of all the parts and if the work performed by the system during a transformation is equal to the sum of the amounts of work performed by all the parts. Notice that these conditions are not quite obvious and that in some cases they may not be fulfilled. Thus, for example, in the case of a system composed of two homogeneous substances, it will be possible to express the energy as the sum of the energies of the two substances only if we can neglect the surface energy of the two substances where they are in contact. The surface energy can generally be neglected only if the two substances are not very finely subdivided; otherwise, it can play a considerable role.
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Ettore MAJORANAThe value of statistical laws in physics and social sciences.Original manuscript in Italian published by G. Gentile Jr. in Scientia 36, 58 (1942); translated into English by R. Mantegna (2005).
This is mainly because entropy is an addditive quantity as the other ones. In other words, the entropy of a system composed of several independent parts is equal to the sum of entropy of each single part. [...]Therefore one considers all possible internal determinations as equally probable. This is indeed a new hypothesis because the universe, which is far from being in the same state indefinitively, is subjected to continuous transformations. We will therefore admit as an extremely plausible working hypothesis, whose far consequences could sometime not be verified, that all the internal states of a system are a priori equally probable in specific physical conditions. Under this hypothesis, the statistical ensemble associated to each macroscopic state Aturns out to be completely defined.
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//
: -
:
, ( )( )
(
ii
i
i
E kTE kT
ii
The values of p are determined by the followingif the energy of the system in the i th state is E and if thetemperature of the system is T then
ep where
dogm
Z T eZ T
this a
a
l
−−= =∑
1).
. ;
"
ii
i We shall giveno justification for thi
st constant is taken so that p
This choice of p is called the Gibbs distributioneven a physicist like Ruelle
disposes of this question ass dogma
de
=∑
".ep and incompletely clarified
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ENTROPIC FORMS
Concave
Extensive
Lesche-stable
Finite entropy production per unit time
Pesin-like identity (with largest entropy production)
Composable
Topsoe-factorizable
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C.T., M. Gell-Mann and Y. Sato Europhysics News 36 (6), 186 (2005) [European Physical Society]
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( , ) qS N t versus t
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LOGISTIC MAP:
21 1 (0 2; 1 1; 0,1,2,...) t t tx a x a x t+ = − ≤ ≤ − ≤ ≤ =
(strong chaos, i.e., positive Lyapunov exponent)
V. Latora, M. Baranger, A. Rapisarda and C. T., Phys. Lett. A 273, 97 (2000)
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1
1 1
11
(0) 0
( )lim
(
( )( ) lim(0)
)
t
tx
We verify
w
Pesin like ide
here
S t
ntity
Kt
andx tt e
K
xλ
ΔΔΔ
λ
ξ
→∞
→
≡
=
−=
≡
-
q = 0.1
q = 0.2445
q = 0.5
S (t)q
t
N = W = 2.5 106
a = 1.4011552
x = 1 - a xt +1 t
2
# realizations = 15115
0
10
20
30
40
50
0 20 40 60 80
(weak chaos, i.e., zero Lyapunov exponent)
C. T. , A.R. Plastino and W.-M. Zheng, Chaos, Solitons & Fractals 8, 885 (1997) M.L. Lyra and C. T. , Phys. Rev. Lett. 80, 53 (1998) V. Latora, M. Baranger, A. Rapisarda and C. T. , Phys. Lett. A 273, 97 (2000) E.P. Borges, C. T. , G.F.J. Ananos and P.M.C. Oliveira, Phys. Rev. Lett. 89, 254103 (2002) F. Baldovin and A. Robledo, Phys. Rev. E 66, R045104 (2002) and 69, R045202 (2004) G.F.J. Ananos and C. T. , Phys. Rev. Lett. 93, 020601 (2004) E. Mayoral and A. Robledo, Phys. Rev. E 72, 026209 (2005), and references therein
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CASATI-PROSEN TRIANGLE MAP [Casati and Prosen, Phys Rev Lett 83, 4729 (1999) and 85, 4261 (2000)](two-dimensional, conservative, mixing, ergodic, vanishing maximal Lyapunov exponent)
G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)
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[G. Casati, C.T. and F. Baldovin, Europhys Lett 72, 355 (2005)]
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0 20 40 60 80 100n0
50
100
150
200
Sq(n) q=-0.2
q= 0
q=+0.2
(a)4000 4000 1000
100
[ 0 0.99993]
W cellsN initial conditions randomly chosen in one cellAverage done over initial cells
q linear correlation
= ×=
= → =
CASATI-PROSEN TRIANGLE MAP [Casati and Prosen, Phys Rev Lett 83, 4729 (1999) and 85, 4261 (2000)](two-dimensional, conservative, mixing, ergodic, vanishing maximal Lyapunov exponent)
0 0
00
( ) lim 1
t
n
Also eS nwith
n
λξ
λ →∞
=
= =
q - generalization of Pesin (- like) theorem
G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)
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( , ) qS N t versus N
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( N = 0 )
( N = 1 )
( N = 2 )
1 1
1
11
1 1 2 2
1 1 2
1
3 6
×
× ×
× ×
1 3
1 1 1 1 4 1 2 1 2 4
1 1 1 1 1
1
1 3 3 1
1 4 6 4 5 2 0 3 0 2 0
( N = 3
1 5
1
)
( N = 4 )
( N
= 6
5 ) 1
×
× × × ×
× × × × ×
×
1 1 1 1 1
5 1 0 1 0 5 1
3 0
6 0 6 0 3 0 6
1 ( )NΣ
× × × × ×
= ∀
HYBRID PASCAL - LEIBNITZ TRIANGLE
Blaise Pascal (1623-1662)Gottfried Wilhelm Leibnitz (1646-1716)Daniel Bernoulli (1700-1782)
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11- pp
1- p2
p1
212p κ+ (1 )p p κ− −
(1 )p p κ− − 2(1 )p κ− +
A B
(N=2)
2 2
( 0) ( 1)
1 (
( 2)
11 1
11- )
[ ] [ (1 ) ] [(1 ) ]2 1p p
p p p
N
pNN κ κ κ
= ×= × ×
= × − −×+ − +×
EQUIVALENTLY:
-
100
90
80
70
60
50
40
30
20
10
01009080706050403020101
NS p q=1.0
q=0.9
q=1.1
(b) 20
10
020101
NS p
q=1.0
q=0.9
q=1.1
(c)
100
90
80
70
60
50
40
30
20
10
01009080706050403020101
NS p
q=1.0
q=0.9
q=1.1
(a)
1. ., 1 SY
STEM (
S) ( )i e such that
qS N N N∝ →∞
=
.0
1/ 2
NNp p
with p
Stretched exponentialα
α
⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= =⎝ ⎠
,0
1/ 2
N
Np p
N independen
with p
t coins
⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟=⎝ ⎠
,0
1
1N
Leibnitz triangle
pN
⎛ ⎞=⎜ ⎟+⎝ ⎠
(All three examples strictly satisfy the Leibnitz rule)C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
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C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
Asymptotically scale-invariant (d=2)
d+1
(It asymptotically satisfies the Leibnitz rule)
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. .,1 S
YSTEM( ) ( )
S qi e such that
qS N N N∝ →∞
≠
11qd
= −
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p q=0.0
q=-0.1
q=+0.1
(a)
(d =1) (d = 2) (d = 3)
(All three examples asymptotically satisfy the Leibnitz rule)
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p
q=1/2
q=1/2-0.1
q=1/2+0.1
(b)
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p
q=2/3
q=2/3-0.1
q=2/3+0.1
(c)
C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
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Continental Airlines
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0 10 20 30 40TIME
0
100
200
300
400
500
600
EN
TR
OP
Y
0 10 20 30 400
2
4
6
0.0 0.5 1.00.0
0.5
1.0
q=1
q=0.05
q=0.2445
q=0.5
(a)
(b)
q
R
a=1.40115519
t
q=1
(c)
S p
(d)
S p
q=0.8
q=0.2445
t
q=0.5
q=0.05
q=1.2
q =1sen q
-
1( ) ( )
(
( ) ( ) ( )
( )
,
. ., ,
) ( ) ( )
( ) ( 1)
(
q q q q
BG B
q
A B A Bij i
G BG
q
j
B
q
independent
qS A B S A S
If A and B are
i e if p p
B S A S Bk
S A S B if
pthen
whereas
But i
q
especiallyf A and glB ar
S A B A S
e
S B
+
−+ = + +
≠ +
=
+ =
≠
+
( ) ( )
)
( )
,
( ) ( )
( )q q q
BG BG BG
then
wher
obally correlated
S A B S A S Bea
A S Bs
S B A S+
+ ≠ +
= +
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NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS (CANONICAL ENSEMBLE):
1
1
1
1
1[ ]
1
1
Wqi
iq i
Wqi iW
ii qW
qii
i
qi
Extremization of the functional
with the constraints an
pS p k
q
p Ed
yi
p
el
p U
dse
p
=
=
=
=
−≡
−
= =
=
∑
∑∑
∑
( )
1
1
( )
, ,
q i qW
E Uq q qW
q ii
i
q qiE U
q
energy Lagwith andrange parameter ep
β
β
ββ β − −=
=
− −
≡ ≡ ≡∑∑
Z
Z
-
'
'
' '
1
'
1 (1 )
1 1( )
1( ) ln
,
ln ln
(
q i
q
WEq
q q qiq q
q
q
q q q q q q q q q q
i
q
Eq
iWe can rewrite
with and
And we can
Z eq U
Si T
T U k
ii F U T
p
S Z Z
rove
with
U
i
h e
Z
w er
ep
β
β
ββ
β
β
ββ
−
=
−
≡ ≡+ −
∂= ≡∂
≡ − = − = −
=
∑
Z
2
2
( . .,
) ln
( )
-
!)
q q q
q q qq
i e the Legendre structure of Thermod
ii U Z
S U Fiv C T T
T Tynamics is q invariant
T
β∂
= −∂
∂ ∂ ∂≡ = = −
∂ ∂ ∂
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NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS
Nonextensive Statistical Mechanics and Thermodynamics, SRA Salinas and C Tsallis, eds, Brazilian Journal of Physics 29, Number 1 (1999)
Nonextensive Statistical Mechanics and Its Applications, S Abe and Y Okamoto, eds, Lectures Notes in Physics (Springer, Berlin, 2001)
Non Extensive Thermodynamics and Physical Applications, G Kania-dakis, M Lissia and A Rapisarda, eds, Physica A 305, Issue 1/2 (2002)
Classical and Quantum Complexity and Nonextensive Thermodynamics, P Grigo-lini, C Tsallis and BJ West, eds, Chaos, Solitons and Fractals 13, Issue 3 (2002)
Nonadditive Entropy and Nonextensive Statistical Mechanics, M Sugiyama, ed, Continuum Mechanics and Thermo-dynamics 16 (Springer, Heidelberg, 2004)
Nonextensive Entropy - Interdisciplinary Applications, M Gell-Mann and C Tsallis, eds, (Oxford University Press, New York, 2004)
Anomalous Distributions, Nonlinear Dynamics, and NonextensivityHL Swinney and C Tsallis, eds, Physica D 193, Issue 1-4 (2004)
News and Expectations in ThermostatisticsG Kaniadakis and M Lissia, edsPhysica A 340, Issue 1/3 (2004)
Trends and Perspectives in Extensive and Non-Extensive Statistical MechanicsH Herrmann, M Barbosa and E Curado, eds, Physica A 344, Issue 3/4 (2004)
Complexity, Metastability and Nonextensivity, C Beck, G Benedek, A Rapisarda and C Tsallis, eds, (World Scientific, Singapore, 2005)
Nonextensive Statistical Mechanics: New Trends, New Perspectives, JP Boon and C Tsallis, eds, EurophysicsNews (European Physical Society, 2005)
Fundamental Problems of Modern Statistical Mechanics, G Kaniadakis, A Carbone and M Lissia, eds, Physica A 365, Issue 1 (2006)
Complexity and Nonextensivity: New Trends in Statistical Mechanics, S Abe, M Sakagami and N Suzuki, eds, Progr. Theoretical Physics Suppl 162 (2006)
-
Recent minireviews:(Europhysics News, Nov-Dec 2005, European Physical Society)
http://www.europhysicsnews.com
Full bibliography:(28 August 2006: 1953 manuscripts)
http://tsallis.cat.cbpf.br/biblio.htm
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(0) 0
( ) lim sup
( )
( ) sup lim
(0)
( )
qq t
x
q q
q tq
It can be pq generalized Pesin lik
roved that
whereS t
Kt
and
x ttx
e identit
w
yK
ith
eΔλΔ
Δ
λ
ξ
→∞
→
⎧ ⎫≡ ⎨ ⎬
⎩ ⎭
⎧ ⎫≡ =⎨ ⎬
⎩ ⎭
−= −
min
1mi
ma
n max
x
1 1 1 ln ( )1 | | ( 1)1 ( ) ( ) ( )
1 1 1 ln 1 1 ln
2
2
n
1
lz F
t t
Fq
zx a x zq z z
n
z
qa
qd
αα α
α λα α
+
⎡ ⎤= − ⇒ = − =
= − = =− −
−⎢ ⎥−⎣ ⎦
-
1
1
( 1; 0Generic pitchfork bifurcations:
Generic tangent bifurcations:
-
)
(
1; 0)
( ) | |
| |
zt t tt
zt tt
The fixed point map is a q exponential witq z
h
and the
z
s
z
e
bx x b sign x x
x x b x b
+
+
> >
>
=
>
= +
= +
1
:
-
( 1; 0 2; 1)
3
12
1 | |
se
sen
n
tt
Exam
nsitivity to the initial conditions is a q expo
ple The logistic family of mapsa
has
z for pitchf
nential wit
or
q
h
k
q
x a x ςς
ς ς+
−
>
=
≤ ≤
=
−
>= −
5( ), 3 ;3
32 ( ), 2 .2
sen
sen
bifurcations hence q and q
z for tangent bifurcations hence q and q
ς
ς
∀ = =
= ∀ = =
A. Robledo, Physica D 193, 153 (2004)
-
1
1
11
1
1 ln (ln ln )1
[1 (1 ) ] ( )
:
( 1 (1 ) 0;
:
- :
q
q
x x xqq
DEFINITIONS
q logxx x x
q
e q
arithm
q exponential
x e e
if q x
−
−
−≡ =
−
≡ + − =
+ − >
−
)vanishes otherwise
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D. Prato and C. T, Phys Rev E 60, 2398 (1999)
q-GAUSSIANS:2
2( / )1-1
1( ) ( 3)1 ( -1) ( / )
qx
qq
p x qq x
e σ
σ
−∝ ≡ <⎡ ⎤+⎣ ⎦
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q - CENTRAL LIMIT THEOREM:
2 ( , ) [ ( , )] (0 2; 3)| |
qp x t p x tD qt x
γ
γ γ−∂ ∂
= < ≤ <∂ ∂
C.T., Milan J. Math. 73, 145 (2005)
independent variables; divergent variance; Levy attractor
independent variables; finite variance; Gaussian attractor
globally correlated variables;finite q-variance; q-Gaussian attractor
q - CENTRAL LIMIT THEOREM (conjecture)
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q - CENTRAL LIMIT THEOREM (q-product and de Moivre-Laplace theorem):
11 1 1
1
:) )
[ ln ( ) ln ln (1
1
)(ln )(ln )
ln ( ) n ln
]
l
q
q q q q q
q qq
q q q q
Propertiesiii
whereas x y
x y x
x
y
x y x yx y
q
x y
y x y
− − −
= +
⎡ ⎤⊗ ≡ + −⎣ ⎦
⊗ =
+
⊗ =
−
+
The q- product is defined as follows:
[L. Nivanen, A. Le Mehaute and Q.A. Wang, Rep. Math. Phys. 52, 437 (2003); E.P. Borges, Physica A 340, 95 (2004)]
The de Moivre-Laplace theorem can be constructed with
,0 1/ 2
NNp p with p
Leibnitzand
rule
= =
-
0 0.2 0.4 0.6 0.8 1q
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
q e
qe=2-1/q
0.5 0.6 0.7 0.8 0.9 1q
0
0.2
0.4
0.6
0.8
1
q e
0 0.2 0.4 0.6 0.8 1
x2
-0.4
-0.3
-0.2
-0.1
0
ln-4
/3[p
(x)/
p(0)
]
N=50N=80N=100N=150N=200N=300N=400N=500N=1000
0 0.02 0.04 0.06 0.081/N
0.41
0.42
0.43
0.44
β(N)
L.G. Moyano, C. T. and M. Gell-Mann, Europhys. Lett. 73, 813 (2006)
q - CENTRAL LIMIT THEOREM: (numerical indications)
,0
11 1
,0
. .
1 1 1 1 ... ( )
( 1)
,
( 1/ 2)
q qN
q qN
We q generalize the de Moivre Laplace theorem with
i
N termsp p p p
p N p N i
e
w th p− −
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⊗ ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎡ ⎤= − − =⎦
−
⎣
−
[Hence q 2 – q (additive duality) and q 1/q (multiplicative duality) are involved]
(q = 3/10)
de Moivre - Laplace
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q - GENERALIZED CENTRAL LIMIT THEOREM: (mathematical proof)
S. Umarov, C.T. and S. Steinberg [cond-mat/0603593]
1[ ( )] (nonline
q-Fourier transform:
q-correlation
[ ]( ) ( ) = ( )
[ ( )]
ar
: [ ( )]
!)
X Y
qix
f xixq q q qF f f x dx f x dx
Two random variables X with density f x and Y with density f yare said
e eξ
ξξ∞ ∞
−∞ −∞
−⊗≡ ∫ ∫
- [X+Y]( ) = [X]( ) [Y]( ) ,
. .,
( ) ( ) ( ) ,
( ) ( , ) ( ) ( , )
q q q q
X Y X q Y
X Y
q q qiyiz ix
q q q
q correlated ifF F F
i e if
dz f z dx f x dy f y
with f z dx dy h x y x y z dx h x z x dy h
e e e ξξ ξ
ξ ξ ξ
δ
∞ ∞ ∞
+−∞ −∞ −∞
∞ ∞ ∞ ∞
+ −∞ −∞ −∞ −∞
⊗
⊗⊗ ⊗ ⊗⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= + − = − =
∫ ∫ ∫
∫ ∫ ∫ ∫
- 1 , . ., ( , ) ( ) ( )
( , )
( , )
.
X Yq correlation means independence if q i e h
z y y
where h x y is the joint densix y f x f y
global correlati
t
on
y= =
−
1 , ( , ) ( ) ( )X Yif q hence h x y f x f y⎛ ⎞⎜ ⎟≠ ≠⎝ ⎠
-
2 2(1 )
1
1
1
1
22
1 3
3 8
121
13(3 ) (1 )
2(1 )
qqq
q
q
tq qq F
qwhere qq
qandC
qif q
ourierTransform
qq qq
with C
Ce e βββ ω
ββ
πΓ
Γ
π
− −
−−⎡ ⎤ =⎢ ⎥⎢ ⎥⎣ ⎦
+=
−−
=
⎛ ⎞⎜ ⎟−⎝ ⎠ <
⎛ ⎞−− − ⎜ ⎟−⎝ ⎠
=
−
1
32( 1)
1 311
1
if q
qq
if qq
q
πΓ
Γ
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪=⎨ ⎬⎪ ⎪⎛ ⎞−⎪ ⎪⎜ ⎟⎪ ⎪−⎝ ⎠ <
-
1 0
- - ( )1( ) (- ,3)3-
2 (1 )( ) ( ( )
-
)
( 0, 1, 2,...; )2 (1
Closure:
Iterat
(the
ion
same as in
:
)
n n n
qThe q Fourier transform of a q Gaussian is a z q Gaussia
z qq
q n qq z q z z q n q
n wit
qn
h
q−
+= ∈ ∞
+ −≡ ≡ = = ± ± =
+ −
11
(i)
R.S. Mendes and C.T. [Phys Lett A 285, 273 (2001)] when calculating marginal probabilities!)
(the same as in L.G. Moyano, C.T. and
(1) 1 ( ), ( ) 1 ( ),1(ii) 2 ,
n
nn
q n q q qn
q
he ce
q
±∞
−+
= ∀ = ∀
= −
( ) 2
M. Gell-Mann (2005)!) (the same as in A. Robledo [Physica D 193, 153 (2004)] for pitchfork and
(1 )(iii) 2 =0, 2, 4,...
tangent bifurcations!)
(
1 (1 )
the sa
m myiq m qn m q q
melds
q+ −
= ± ± ≡ =+ −
me obtained in C.T., M. Gell-Mann and Y. Sato [Proc Natl Acad Sci (USA) 102, 15377 (2005)], by combining additive and multiplicative dualities, and which was conjectured only to be a possible explanation for the NASA-detected q-triangle for m = 0, 1!)±
-
,
1 1
( 0, 1, 2,...)
nn
q q
n
α
α α= +
− −
= ± ±
S. Umarov, C.T., M. Gell-Mann and S. Steinberg (2006), cond-mat/0606040
ALGEBRA ASSOCIATED WITH qALGEBRA ASSOCIATED WITH q--GENERALIZED CENTRAL LIMIT THEOREMS:GENERALIZED CENTRAL LIMIT THEOREMS:
-
, | |
- ( 0, 0, 0 2)
( , )
-
( )
q
bq
A random variable X is said to have a
if its q Fourier transform has the
q stable dist
form a
ribution
b
L x
a e αα
ξ
α
α− > > < ≤
S. Umarov, C. T., M. Gell-Mann and S. Steinberg (2006)
cond-mat/0606038
cond-mat/0606040
1[ ( )],, , ,
1,2
1,
,2
| |
. .,
( ) = ( ) =
)
)
[ ](
)
( ) ( ) ( )
( ) ( ) (
( ) ( ) (
qix
L xqq
xqqi
q q q q
qq
bq
stable Levy distribution
q Gaus
i e if
L x dx L x dx
s n
L
a
F
i
L x G x Gaussian
L x L x
L x G x
e e a eξ
αξα αα
α
αα
ξξ
α
−∞ ∞
−∞ −∞
⊗−
−
−
≡
≡
≡
≡
∫ ∫
-
1 [ ]q independent= 1 ( . ., 2 1 1) [ ]q i e Q q globally correlated≠ ≡ − ≠
1
, ( )
( ) (
Classic CL
)
T
with same ofx Gaussian G x
f xσ=F
<
( 2)Qσ
α
∞
=
(0 2)
Qα
σ< <
→∞
1/[ (2- )] -CENTRAL LI MIT THEOREMS: ( -
) ( )
q SCALED ATTRACTOR WHEN SUMMING NCORRELATED IDENTICAL RANDOM VARIABLES WITH SYMMETRIC DISTRIBUTION
xNq f x
α →∞F
2
1
, | |
( )
| | (1, )L ( )
( ) / | | | | (1, )
lim (
( ) L
1,
)
)
(
c
c
c
with same xasymptotic behavior
x Levy di
G xif x x
xf x C x
i
stri
f x xwit
b on x
h
u
x
ti
α
α
α
α αα
αα→
+
→∞
⎧ ⎫⎪ ⎪⎩ ⎭
∞
=
=
∼ ∼
F
Levy-Gnedenko CLT
2
3 11
3 11
( 1)/( 1)
3 1
1,
[ ( )] / [ ( )] ( )
( ) | | ( , 2)
( )
( ) / | | | | (
( )
( )
-
Q Q
c
cq
q q
qq
Q
q q
q
q
with same dx x f x dx f x of f x
G x if x x q
f x C x if x
x
x q
G x Ga
G
sian
x
us
σ
−+
−+
+ −
−
+=
⎡ ⎤≡⎣ ⎦>
≡
∫ ∫
∼ ∼
F
1
S. Umarov, C. T. and S. Steinberg
, 2)
(2006
lim ( ,2) ) [cond-mat/
060 5 3]
9
3
q cwith x q→
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
= ∞
( )
,
,
2 31
1
2
1
3
,
,1
(1 ) / (1 )( ) / | |
( )
( )
,
~
,
Lq
q
qq
q
q
q
q
qw i th L
o
x L s ta b le
f x C x
d i s t r ib u t io n
x L s t a b le d i s t rr
w i th
i t i o n
L
b u
αα α αα
α αα
α
α
α
α α
α α α
−
− ++
−
+
−−
+
+
++
+ + −
=
∼
=
F
F
( * )
, ,11
2 ( 1 ) / ( 1 )
S . U m a r o v , C . T . , M . G e l l -M a n n a n d S . S t e in b e r g ( 2 0 0 6 ) [ c o n d -m a t /0 6 0 6 0 3 8 ] a n d [ c o n d -m a t /0 6 0
(
6
~
0
) / |
4 0 ]
|qq
q qf x C xα αα
α α
+ −
+ − −∼
-
1 2 1 2 1
( )
( , ,..., )
( , ,..
.
? :
,
)
N N N
N body joi
It appea
nt
rs to be no proof available yet
dx h x x
WHAT
x h x x
IS IT q CORRELATIO
probability
N
x −
−
=
−
∫
. ., !
i e scale invariance
-
BOLTZMANN-GIBBS STATISTICAL MECHANICS(Maxwell 1860, Boltzmann 1872, Gibbs ≤ 1902)
Entropy
Internal energy
Equilibriumdistribution
Paradigmaticdifferential equation
1
W
B G i ii
U p E=
= ∑1
lnW
B G i ii
S k p p=
= − ∑
/iEi BGp e Zβ−=
1
jW
EBG
jZ e β−
=
⎛ ⎞≡⎜ ⎟
⎝ ⎠∑
( 0 ) 1
d y a yd xy
⎫= ⎪ ⇒⎬⎪= ⎭
-1/τtTypical relaxation of observable Ο
λtSensitivity toinitial conditions
Z p(Ei)-βEiEquilibrium distributiony(x)ax
SBG → extensive, concave, Lesche-stable, finite entropy production
( 0 ) 0
( )lim(0 )
t
x
x t ex
λξΔ →
Δ≡ =
Δ
/( ) ( )(0) ( )
tO t O eO O
τ−− ∞Ω ≡ =− ∞
axy e=
-
NONEXTENSIVE STATISTICAL MECHANICS(C. T. 1988, E.M.F. Curado and C. T. 1991, C. T., R.S. Mendes and A.R. Plastino 1998)
Entropy
Internal energy
Stationary statedistribution
Paradigmaticdifferential equation
tTypical relaxation of observable Ο
tSensitivity toinitial conditions
EiStationary state distribution
y(x)ax
11 /( 1)
Wq
q ii
S k p q=
⎛ ⎞= − −⎜ ⎟
⎝ ⎠∑
( ) /q i qE Ui q qp e Zβ− −=
( )
1
E Uq j qW
q qj
Z eβ− −
=
⎛ ⎞≡⎜ ⎟
⎝ ⎠∑
(0) 1
qdy a ydxy
⎫⎪⎬⎪⎭
=⇒
=
Sq → extensive, concave, Lesche-stable, finite entropy production
qsen
sen
tqeλξ =
/ qrelrel
tqe
τ−Ω =
[ ]1
11 (1 ) qqa x
q a xy e −+ −= ≡
1 1/
W Wq q
q i i ji j
U p E p= =
= ∑ ∑
(typically 1)senq ≤
1 /r e lq
τ−
senqλ
(typically 1)relq ≥
statqβ− ( )statq iZ p E
C. T., Physica A 340,1 (2004)
(typically 1)statq ≥
-
Prediction of the Prediction of the q q -- triplet:triplet: C. T., Physica A 340,1 (2004)
-
L.F. Burlaga and A. F.-Vinas (2005) / NASA Goddard Space Flight Center; Physica A 356, 375 (2005)
[Data: Voyager 1 spacecraft (1989 and 2002); 40 and 85 AU; daily averages]
SOLAR WIND: Magnetic Field Strength
0.6 0.2senq = − ±
3.8 0.3relq = ± 1.75 0.06statq = ±
-
( 2 ) ( 1/ )
1 2
(
)
relsen
Playing with additive dualityand with multiplicative dualityand using numerical results related to the q generalized central limit theorem
we conject
q qq q
qq
ure
→ −→
+
−
=
!
1 2
1 1 3 2
( )
statrel
statsen
stat
stat
and
hence
Burlaga and Vinas NASA most precise value of the q tripl
hence onl
et
y one independe
qq
qqq
t
is
n
q−
+ =
−− =
−
( 0.6 0.2 !)(
1.75 7 / 4 0.5 1/ 2
4 3.8 0.3 ! ) sen
re
sen
rel l
hencea
consistent with qconsistent withnd
qq q
= − ±= ±
= == − = −=
C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
-
2 2 /(3 )
2 2
2
1/(1 )2 2 /(3 ) / ( )
2 2
( , ) [ ( , )] [ ( ,0) (0)] ( 3)
( , ) 1 (1 ) / ( ) ( )
( . .,
)
q
q
qq x tq
The solution ofp x t p x tD p x q
t xis given by
p x t q x t e D
hence
x scales like t e g x t
with
Γ
γ γ
δ
Γ Γ−
−
−− −
∂ ∂= = <
∂ ∂
⎡ ⎤∝ + − ≡ ∝⎣ ⎦
∝
2 3
( . ., 1 1 , . ., )e g q i e normal diffu
q
sionγ
γ
= ⇒ =
=−
PREDICTION:
C.T. and D.J. Bukman, Phys Rev E 54, R2197 (1996)
-
Hydra viridissima: A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada Physica A 293, 549 (2001)
q=1.5
-
1.24 0.12
3
slope
hence is satisfiedq
γ
γ
= ±
=−
-
Defect turbulence:K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
-
K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
21.5 4 / 3 3
q and are consistent withq
γ γ≈ ≈ =−
-
XY FERROMAGNET WITH LONG-RANGE INTERACTIONS:
A. Rapisarda and A. Pluchino, Europhys News 36, 202 (European Physical Society, Nov/Dec 2005)
-
XY FERROMAGNET WITH LONG-RANGE INTERACTIONS:
A. Rapisarda and A. Pluchino, Europhys News 36, 202 (2005) (European Physical Society)
-
Silo drainage: R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)
(intermediate regime)
(fully developed regime)
q=3/2q=1
-
4 / 32
3
slope
hence is satisfiedq
γ
γ
=
=−
R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)
(outlet size 3.8 d)
-
COLD ATOMS IN DISSIPATIVE OPTICAL LATTICES:
Theoretical predictions by E. Lutz, Phys Rev A 67, 051402(R) (2003):
(i) The distribution of atomic velocities is a q-Gaussian;
(ii) 0
0
where recoil energy
potential depth
441 RREq
UE
U
= ≡
≡
+
-
Experimental and computational verificationsby P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)
(Computational verification:quantum Monte Carlo simulations) (Experimental verification)
0
441 REqU
= +
-
HADRONIC JETS FROM ELECTRON-POSITRON ANNIHILATION:
I. Bediaga, E.M.F. Curado and J.M. de Miranda, Physica A 286 (2000) 156
Hagedorn
Beck (2000): q=11/9
-
(Phenomenological model for collisions in a diluted gas with probability rof forming clusters of q correlated particles)
Monte Carlo
Single-parameter fitting
-
Connections withasymptotically scale free networks−
-
(1) Locate site i=1 at the origin of say a plane
(2) Then locate the next site with
(3) Then link it to only one of the previous sites using
2
( )
1/ ( 0) GG Gr distance to the baricenter of the pre existing cluster
P r α α+
≡ −
∝ ≥
4) Repeat
A
( )
( )
/ ( 0) Ai i Ai
i
k links already attached to site i
r distance to site i
k rp α α≡
≡
∝ ≥
GEOGRAPHIC PREFERENTIAL ATTACHMENT GROWING NETWORK:
THE NATAL MODELD.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva, Europhys Lett 70, 70 (2005)
-
G
( 1; 1; 250)A Nα α= = =
D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)
-
/
1 / ( 1 )
P (k ) /P (0 )=
1 / [1 ( 1) / ]
kq
q
e
q k
κ
κ
−
−≡ + −D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)
-
0.526
q=1+(1/3) ( ) A
G
e α
α
−
∀
Barabasi-Albert universality class
D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)
= 0 .0 8 3 + 0 .0 9 2 Aκ α
( )Gα∀
-
ASTROPHYSICS
-
A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)
(Band Q: 22.8 GHz) (Band V: 60.8 GHz) (Band W: 93.5 GHz)
1.045 0.005 (99 % confidence level)q = ±(Data after using Kp0 mask)
-
A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)
1.045 0.005 (99 % confidence level)q = ±
-
Connections with conservativeand Hamiltonian systems
-
0 1 2 3 4 50
1
2
3
4
5
α
d
EXTENSIVE
NONEXTENSIVE
dipole-dipole
Newtonian gravitation
dipole
-mon
opole
(tide
s)
d-dim
ensio
nal g
ravita
tion
( )
/ 1 ( - ) - 0 / 1
( )
( 0, 0)
( - ) integrable if d short ranged
non integrable if d
AV r
long
r
r
A
ange
r
d
α
αα
α
− ∞
>>≤
→
≤
≥
∼
-
[ ]
{ }1
( 1) ( ) sin 2 ( )2sin 2 ( ) ( )
(mod1)2
i i i
i j
ijj
N
ap t p t t
t tbN rα
πθπ
π θ θ
π=
+ = + +
⎡ ⎤−⎣ ⎦∑( 1) ( ) ( 1) (mod1)i i it t p tθ θ+ = + +
1 / 1 ; 0 ; 0 ; 0 ; 11 /
dNN a b dd
α
αα
− −≡ ≥ ≥ ≥ =
−
with
MANY COUPLED STANDARD MAPS:L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)
-
L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)
-
L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)
-
2
1 ,
1 /
1
1 cos( )1 ( 0, 0)2
0 / 1 ln / 1
/ 1
Ni j
ii i j ij
dN
ijj
JH K V L I JI r
N if dwith r N if d
constant if d
and periodic boundary
α
α
αϑ ϑ
ααα
=
−
−
=
− −= + = + > >
⎧ ⎫≤ <⎪ ⎪≡ ∝ =⎨ ⎬⎪ ⎪>⎩ ⎭
∑ ∑
∑
A
A
[
.
0, ]The HMF model correspond
conditi
s to d
ons
α = ∀
d-DIMENSIONAL CLASSICAL INERTIAL XY FERROMAGNET:
(We illustrate with the XY (i.e., n=2) model; the argument holds however true for any n>1 and any d-dimensional Bravais lattice)
C. Anteneodo and C. T., Phys Rev Lett 80, 5313 (1998)
M. Antoni and S. Ruffo, Phys Rev E 52, 2361 (1995)
-
C. Anteneodo and C. T. Phys Rev Lett 80, 5313 (1998)
A. Campa. A. Giansanti, D. Moroni and C. T. Phys Lett A 286, 251 (2001)
-
V. Latora, A. Rapisarda and C. T., Phys Rev E 64, 056134 (2001)
-
C. T., A. Rapisarda, V. Latora and F. Baldovin, Lecture Notes in Physics 602 (Springer, Berlin, 2002)
-
V. Latora, A. Rapisarda and C. T., Phys Rev E 64, 056134 (2001)
-
F. Baldovin and E. OrlandiniPhys Rev Lett 96, 240602 (2006)
F. Baldovin and E. Orlandini, cond-mat/0603659
In contact with a thermostat (canonical ensemble):In contact with a thermostat (canonical ensemble):
-
8Wt =
2.35 ? 1.5 ?
0 :rel
stat
qq
modelα =
F.A. Tamarit and C. AnteneodoEurophysics News 36 (6), 194 (2005) [European Physical Society]
(q=2.35)
t
-
HMF MODEL:
L.G. Moyano, F. Baldovin and C. T., cond-mat/0305091
-
ECONOMICS
-
J de Souza, SD Queiros and LG Moyano, physics/0510112 (2005)
STOCK VOLUMES:
-
q-GENERALIZED BLACK-SCHOLES EQUATION:L Borland, Phys Rev Lett 89, 098701 (2002), and Quantitative Finance 2, 415 (2002)L Borland and J-P Bouchaud, cond-mat/0403022 (2004)L Borland, Europhys News 36, 228 (2005) See also H Sakaguchi, J Phys Soc Jpn 70, 3247 (2001)
C Anteneodo and CT, J Math Phys 44, 5194 (2003)
31
[ :
] nn
REMARK Student t distributions are the particular case
of q Gauss q with n intians whe rn ege++
=
−
−
-
EARTHQUAKES
-
n / nw1.05
10-4 10-3 10-2 10-1 100 101 102
D(n
+n w
, n w
)
0.1
1.0 nw=250
nw=1000
nw=500
nw=2000
nw=5000
n / nw
1.050 2 4 6 8 10 12 14 16
lnq[
D(n
+n w
, n w
)]
-12
-10
-8
-6
-4
-2
0
S. Abe, U. Tirnakli and P.A. VarotsosEurophysics News 36 (6), 206 (2005) [European Physical Society]
MODEL FOR EARTHQUAKES (OMORI REGIME):
(q=2.98)
-
U. Tirnakli, in Complexity, Metastability and Nonextensivity, eds. C. Beck, G. Benedek, A. Rapisarda and C. T. (World Scientific, Singapore, 2005), page 350
-
GENERALIZED SIMULATED ANNEALING AND RELATED ALGORITHMS
-
q-GENERALIZED SIMULATED ANNEALING (GSA):
C.T. and D.A. Stariolo, Notas de Fisica / CBPF (1994); Physica A 233, 395 (1996)
:
:
VGeneralized machine q GaussBoltzmann machine Gaussian
Boltzmann machine Boltzmann
Visiting algo
weight
rithm
Acceptance aian
Generalized mac
lgorithm
hi
→ −→
→
1
1
T(t) ln 2 T(1) ln(1 )
T(t) 2 1 T(1)
[ :
(1 ) 1
1 3
:
] 1
A
V A
V
V
q
q
ne q exponential weight
Gen
Bo
er
ltz
alized machi
mann machinet
Typ
Cool
ica
net
ing algori
l values and q
thm
q
−
−
→ =
→ −
−→ =
+ −
< < <
+
-
C. T. and D.A. Stariolo, Physica A 233, 395 (1996)
-
4 42 2
1 2 3 41 1
( , , , ) ( 8) 5
(15 )
: i ii i
E x x x x x x
local minima and one global minimum
Illustration= =
= − +∑ ∑
q-GENERALIZED SIMULATED ANNEALING (GSA):
( 1 50000)Vq mean convergence time= ⇒ ≈
-
q-GENERALIZED PIVOT METHOD:
P. Serra, A.F. Stanton and S. Kais, Phys Rev E 55, 1162 (1997)
(Branin function) (Lennard-Jones clusters)
Num
ber o
f fun
ctio
n ca
lls Genetic algorithm
Present with q=2.7slo
pe 4
.7
slope
2.9
Recently: M.A. Moret, P.G. Pascutti, P.M. Bisch, M.S.P. Mundim and K.C. MundimClassical and quantum conformational analysis using Generalized Genetic AlgorithmPhysica A 363, 260 (2006) (presumably better than both!)
-
IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]
Original image
q = 1.5
Canny edge detector
q = 1
(Jensen-Shannon)
-
IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]
Original image
q = 1.5
Canny edge detector
q = 1
(Jensen-Shannon)
-
M.P. de Albuquerque, I.A. Esquef, A.R.G. Mello and M.P. de Albuquerque Pattern Recognition Letters 25, 1059 (2004)
-
IMAGE THRESHOLDING:
M.P. de Albuquerque, I.A. Esquef, A.R.G. Mello and M.P. de Albuquerque Pattern Recognition Letters 25, 1059 (2004)
-
qthan
-
0
1 1 1 1 1 7 3 3 4 36
36 12 3 2 2 180 2 1 37
3 !!!
7
?
number of independent physical universal constants in conte
HOW MANY PHYSICAL UNIVERSAL CONST
mporary physics
ANT
he
S
nce
ν
ν
ν
ν π
≡
⎡ ⎤= + + =⎢ ⎥∞⎣ ⎦
⎛ ⎞= = + + =
=
+ = + = + =⎜ ⎟⎝ ⎠
Nino Constantino Gerardus
Euclid Gerardus
NINO WAS RIGHT!!!
-
( 0)
( )
1
ij
ijij
Merging probability
d shortest path chemical distance connecting nodes i and j on the network
pd α
α ≥
≡
∝
(Kim, Trusina, Minnhagen and Sneppen, . . . 43 (2005) 369)
0 and recover th random neighbore and the schemes respectivelyEur Phys J B
α α= →∞
GAS-LIKE (NODE COLLAPSING) NETWORK:
S. Thurner and C. T., Europhys Lett 72, 197 (2005)Number N of nodes fixed (chemostat); i=1, 2, …, N
Degree of the most connected node Degree of a randomly chosen node
7( 2 ; 0; 2)N rα= = =
-
2000 4000 6000 8000 10000 12000 140000
10
20
30
40
50
60
70
80
time
k max
; k i
k
ik
max
-
S. Thurner, Europhys News 36, 218 (2005)
-
[ ] [ ]1
( )
( 1.84
1( ) ln ( )
1)
q
q q
c
P kZ k
optimal
P kq
q
−
=
> −≡ > ≡
−
( ;α →∞
( ; 8)rα →∞ < >=
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
-
- ( -2)/( ) ( 2, 3, 4,...)ck
qP k ke κ≥ = =[0.999901,0.999976]linear correlation∈
9( 2 ; 2)N r= =
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
-
9( 2 )N =
( 2)r =
[ ]( ) ( ) (0) ( ) c c c cq q q q e αα −= ∞ + − ∞
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
-
SANTOS THEOREM: RJV Santos, J Math Phys 38, 4104 (1997)
(q - generalization of Shannon 1948 theorem)
({ }) { } ( 1/ , )
( ) ( ) ( ) ( ) ( ) ( )
({ }) ( , ) ({ / }) ({
(1 )
i i
i
A B A Bij i j
i L M L l L Mq q
S p continuous function of pS p W i monotonically increases with WS A B S A S B S A S B with p p p
k k k k kS p S p p p S
IFAND
AND
A p p S
q
ND p
+
= ∀+
= + +
+
− =
= +
1
1
1 ({ }) 1 ({
}) ln
/ }) ( 1
( ) :" ,
)
1
W
i Wi
i i i
m M
ii
M L
q
CE SHANNON The Mathematica
pS p k q S p k p
l Theory
THEN AND ONLY THEN
of
p p with
CommunicationThis theorem and th
p
pq
p
=
=
−⎛ ⎞= = ⇒ = −⎜ ⎟− ⎝ ⎠
+ =
∑∑
,
. .
e assumptions required for its proof for the
present theory It is given chiefly to lend a certain plausibility to some of our later definitions
are in no way necessaryThe
real justification of the , , .se definitions however will reside in their implications
-
ABE THEOREM: S Abe, Phys Lett A 271, 74 (2000)
(q - generalization of Khinchin 1953 theorem)
1, 2 1, 2
({ }) { } ( 1/ , )
( ,..., ,0) ( ,..., )( ) ( ) ( | )
( ) ( |(
) ) 1
i i
i
W W
IF S p continuous function of pS p W i monotonically increases with WS p p p S p p pS A B S A S B A S A S B A
k
ANDAND
AND
THEN AND ONLY
k k k
THE
k
N
q
= ∀=
+= + + −
1
1
1 ({ }) 1 ({ }) ln
1
(1996, 1999).
W
i Wi
i i i ii
q
The possibility of such theorem was conjecturedb
pS p k q S p k
y AR Plastino and A Plas n
p p
o
q
ti
=
=
−⎛ ⎞= = ⇒ = −⎜ ⎟− ⎝ ⎠
∑∑