Constantino Tsallis

Transcription

Constantino Tsallis

COMPLEXITY AND NONEXTENSIVE STATISTICAL MECHANICS
THEORY, EXPERIMENTS,
OBSERVATIONS AND COMPUTER SIMULATIONS
Constantino Tsallis
Centro 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
ORDINARY DIFFERENTIAL EQUATIONS
FURTHER APPLICATIONS
(Physics, Astrophysics, Geophysics,
Economics, Biology, Chemistry,
Cognitive psychology, Engineering,
Computer sciences, Quantum
information, Medicine, Linguistics …)
ENTROPY Sq
(Nonextensive statistical mechanics)
PARTIAL DIFFERENTIAL EQUATIONS
(Fokker-Planck, fractional derivatives,
nonlinear, anomalous diffusion, Arrhenius)
IMAGE PROCESSING
SIGNAL PROCESSING
(ARCH, GARCH)
GLOBAL OPTIMIZATION
(Simulated annealing)
CENTRAL LIMIT THEOREMS
(Gauss, Levy-Gnedenko)
UBIQUITOUS LAWS IN
COMPLEX SYSTEMS
STOCHASTIC DIFFERENTIAL EQUATIONS
(Langevin, multiplicative noise)
NONLINEAR DYNAMICS
(Chaos, intermittency, entropy production, Pesin,
quantum chaos, self-organized criticality)
SUPERSTATISTICS
(Other generalizations)
THERMODYNAMICS
q-TRIPLET
AGING (metastability, glass, spin-glass)
q-ALGEBRA
LONG-RANGE INTERACTIONS
(Hamiltonians, coupled maps)
CORRELATIONS IN PHASE SPACE
GEOMETRY
(Scale-free networks)
Enrico FERMI
Thermodynamics (Dover, 1936)
The entropy of a system composed of several parts is very often
equal 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.
Ettore MAJORANA
The 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 A
turns out to be completely defined.
The values of pi are determined by the following dogma :
if the energy of the system in the i - th state is Ei and if the
temperature of the system is T then :
e − Ei / kT
pi =
, where Z (T ) = ∑ e − Ei / kT
Z (T )
i
(this last constant is taken so that
∑p
i
= 1).
i
This choice of pi is called the Gibbs distribution. We shall give
no justification for this dogma ; even a physicist like Ruelle
disposes of this question as " deep and incompletely clarified ".
ENTROPIC FORMS
Concave
Extensive
Lesche-stable
Finite entropy production
per unit time
Pesin-like identity (with
largest entropy production)
Composable
Topsoe-factorizable
C.T., M. Gell-Mann and Y. Sato
Europhysics News 36 (6), 186 (2005)
[European Physical Society]
S q ( N , t ) versus t
LOGISTIC MAP:
xt +1 = 1 − a xt
2
(0 ≤ a ≤ 2; −1 ≤ xt ≤ 1; t = 0,1,2,...)
(strong chaos, i.e., positive Lyapunov exponent)
V. Latora, M. Baranger, A. Rapisarda and C. T., Phys. Lett. A 273, 97 (2000)
We verify
K1 = λ1 ( Pesin − like identity )
where
K1 ≡ limt →∞
S1 (t )
t
and
Δ x(t )
= eλ
ξ (t ) ≡ lim Δ x (0)→0
Δ x(0)
1
t
(weak chaos, i.e., zero Lyapunov exponent)
50
q = 0.1
x t +1= 1 - a x t2
Sq (t)
a = 1.4011552
40
6
N = W = 2.5 10
#
realizations
q = 0.2445
= 15115
30
20
q = 0.5
10
0
0
20
40
60
t 80
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
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)
[G. Casati, C.T. and F. Baldovin, Europhys Lett 72, 355 (2005)]
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)
(a)
200
W = 4000 × 4000 cells
N = 1000 initial conditions randomly chosen in one cell
Average done over 100 initial cells
q=-0.2
Sq(n)
150
[q = 0 → linear correlation = 0.99993]
q= 0
100
Also
with
50
q=+0.2
0
0
20
40
60
80
n 100
ξ = e0
λ0 t
λ0 = lim n→∞
S 0 ( n)
=1
n
q - generalization of
Pesin (- like) theorem
G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)
S q ( N , t ) versus N
HYBRID PASCAL - LEIBNITZ TRIANGLE
1×
(N = 0 )
1×
(N = 1 )
1×
(N = 2 )
1×
(N = 3 )
1×
(N = 4 )
(N = 5 )
1×
1
6
1
5
5×
1
30
1×
2 ×
3×
4 ×
1
2
1
3
1
4
1
1
1
6
1
12
1
20
10 ×
1
60
Blaise Pascal (1623-1662)
Gottfried Wilhelm Leibnitz (1646-1716)
Daniel Bernoulli (1700-1782)
1×
3×
6 ×
1
2
1
3
1
12
1
30
1×
4×
10 ×
1
60
1
4
1
20
1×
5×
1
30
Σ =1
1
5
1×
1
6
(∀ N )
(N=2)
A B
1
2
1
2
p2 + κ
p (1 − p ) − κ
p (1 − p ) − κ (1 − p ) 2 + κ
p
p
1- p
1- p
1
EQUIVALENTLY:
( N = 0)
( N = 1)
( N = 2)
1× 1
1× p
1× [ p 2 + κ ]
1× (1- p)
2 × [ p(1 − p ) − κ ]
1× [(1 − p )2 + κ ]
q = 1 SYSTEMS
i.e., such that S1 ( N ) ∝ N ( N → ∞)
100
100
(a)
90
80
(b)
90
q=0.9
80
q=0.9
70
70
60
60
Sp 50
Sp 50
(c)
20
q=0.9
40
Sp 10
q=1.0
40
q=1.0
30
q=1.0
30
20
20
10
q=1.1
q=1.1
q=1.1
10
10
20
30
40
50
60
70
80
N
Leibnitz triangle
1 ⎞
⎛
p
=
⎜ N ,0
⎟
N +1 ⎠
⎝
90
100
0
1
10
20
30
40
50
60
70
80
90
N
1
100
0
1
10
20
N
0
N independent coins
Stretched exponential
⎛ pN ,0 = p N ⎞
⎜
⎟
⎜
⎟
⎜ with p = 1/ 2 ⎟
⎝
⎠
α
⎛ pN .0 = p N
⎞
⎜
⎟
⎜
⎟
⎜⎜ with p = α = 1/ 2 ⎟⎟
⎝
⎠
(All three examples strictly satisfy the Leibnitz rule)
Proc Natl Acad Sc USA 102, 15377 (2005)
Asymptotically scale-invariant (d=2)
d+1
(It asymptotically satisfies the Leibnitz rule)
q ≠ 1 SYSTEMS
i.e., such that S q ( N ) ∝ N ( N → ∞)
(d =1)
(d = 2)
10000
(d = 3)
10000
10000
(a)
q=-0.1
8000
(c)
(b)
8000
q=0.0
Sp
8000
q=1/2-0.1
Sp
Sp
6000
q=2/3-0.1
6000
6000
q=1/2
4000
4000
q=+0.1
2000
4000
2000
q=2/3
2000
q=1/2+0.1
q=2/3+0.1
4000
6000
8000
10000
0
2000
4000
6000
8000
10000
N
2000
0
2000
4000
6000
8000
10000
N
0
N
1
q = 1−
d
(All three examples asymptotically satisfy the Leibnitz rule)
Continental Airlines
10000
100
(a)
qsen =1
90
80
qsen <1
(b)
8000
q=0.9
70
q=1/2-0.1
60
Sp
6000
Sp
50
q=1/2
40
4000
q=1
30
20
2000
q=1/2+0.1
q=1.1
10
0
10
20
30
40
50
60
70
80
90
100
0
2000
4000
6000
10000
8000
N
1
N
600
(c)
qsen =1
a=1.40115519
qsen <1
(d)
1.0
500
q=0.05
R
0.5
ENTROPY
400
(b)
Sp
Sp
q=0.8
0.0
0.0
0.5 q
300
1.0
q=0.05
6
4
200
q=1
q=0.2445
q=0.2445
q=1
2
(a)
q=1.2
0
100
0
0
0
10
20
30
40
10
t
Europhysics News 36 (6), 186 (2005) [European Physical Society]
q=0.5q=0.5
20
TIME t
30
40
If A and B are independent ,
i.e., if pij
A+ B
= pi p j ,
A
B
then
S BG ( A + B) = S BG ( A) + S BG ( B )
whereas
S q ( A + B) = S q ( A) + S q ( B) +
≠ S q ( A) + S q ( B)
1− q
S q ( A) S q ( B)
kB
(if q ≠ 1)
But if A and B are especially ( globally ) correlated ,
then
S q ( A + B) = S q ( A) + S q ( B)
whereas
S BG ( A + B) ≠ S BG ( A) + S BG ( B)
NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS
(CANONICAL ENSEMBLE):
W
S q [ pi ] ≡ k
Extremization of the functional
1 − ∑ piq
i =1
q −1
W
W
with the constraints
∑p
i =1
i
=1
q
p
∑ i Ei
and
i =1
W
q
p
∑ i
= Uq
i =1
eq− βq ( Ei −U q )
pi =
Zq
yields
with
βq ≡
β
W
W
∑p
i =1
q
i
, β ≡ energy Lagrange parameter , and Z q ≡ ∑ eq
i =1
− β q ( Ei −U q )
pi =
We can rewrite
with
β ≡
'
q
Z q'
βq
1 + (1 − q ) β qU q
And we can prove
1 ∂S q
=
with
(i )
T ∂U q
(ii )
− βq' Ei
eq
Fq ≡ U q − TS q = −
W
, and Z ≡ ∑ e
'
q
i =1
− β q' Ei
q
1
T≡
kβ
1
β
ln q Z q
where
ln q Z q = ln q Z q − βU q
∂
(iii ) U q = −
ln q Z q
∂β
(iv)
Cq ≡ T
∂S q
=
∂U q
= −T
∂ 2 Fq
∂T
∂T
∂T 2
(i.e., the Legendre structure of Thermodynamics is q - invariant !)
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 Kaniadakis, M Lissia and A Rapisarda,
eds, Physica A 305, Issue 1/2 (2002)
Classical and Quantum Complexity and
Nonextensive Thermodynamics, P Grigolini, C Tsallis and BJ West, eds, Chaos,
Solitons and Fractals 13, Issue 3 (2002)
Nonextensive Entropy - Interdisciplinary
Applications, M Gell-Mann and C Tsallis,
eds, (Oxford University Press, New York,
2004)
Anomalous Distributions, Nonlinear
Dynamics, and Nonextensivity
HL Swinney and C Tsallis, eds,
Physica D 193, Issue 1-4 (2004)
Nonextensive Statistical Mechanics:
New Trends, New Perspectives, JP
Boon and C Tsallis, eds, Europhysics
News (European Physical Society, 2005)
Fundamental Problems of Modern Complexity and Nonextensivity: New
Statistical Mechanics, G Kaniadakis, Trends in Statistical Mechanics, S Abe,
M Sakagami and N Suzuki, eds, Progr.
A Carbone and M Lissia, eds,
Theoretical Physics Suppl 162 (2006)
Physica A 365, Issue 1 (2006)
News and Expectations in
Thermostatistics
G Kaniadakis and M Lissia, eds
Physica A 340, Issue 1/3 (2004)
Nonadditive Entropy and Nonextensive
Statistical Mechanics, M Sugiyama, ed,
Continuum Mechanics and Thermodynamics 16 (Springer, Heidelberg, 2004)
Trends and Perspectives in Extensive
and Non-Extensive Statistical Mechanics
H 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)
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
It can be proved that
K q = λq (q − generalized Pesin − like identity )
where
⎧ Sq (t ) ⎫
K q ≡ limt →∞ sup ⎨
⎬
⎩ t ⎭
and
λq t
⎧
Δ x(t ) ⎫
ξ (t ) ≡ sup ⎨lim Δ x (0)→0
⎬ = eq
Δ x(0) ⎭
⎩
with
1
1
1
ln α F
=
−
=
1 − q α min α max
ln 2
and
1
λq =
1− q
⎡
1
1
1
ln α F ( z ) ⎤
z
=
−
= ( z − 1)
⎢ xt +1 = 1 − a | xt | ⇒
⎥
1
(
)
(
)
(
)
l
n
2
−
q
z
z
z
α
α
min
max
⎣
⎦
Generic pitchfork bifurcations:
xt +1 = xt + b sign( xt ) | xt |z ( z > 1; b > 0)
Generic tangent bifurcations:
xt +1 = xt + b | xt |z ( z > 1; b > 0)
The fixed point map is a q - exponential with
q=z
and the sensitivity to the initial conditions is a qsen - exponential with
1
q
Example : The ς − logistic family of maps
qsen = 2 −
xt +1 = 1 − a | xt |ς (ς > 1; 0 ≤ a ≤ 2; ς > 1)
has
5
;
3
3
= .
2
z = 3 for pitchfork bifurcations (∀ς ), hence q = 3 and qsen =
z = 2 for tangent bifurcations (∀ς ), hence q = 2 and qsen
A. Robledo, Physica D 193, 153 (2004)
DEFINITIONS :
q − logarithm :
1− q
x −1
ln q x ≡
(ln1 x = ln x)
1− q
q - exponential :
1
1− q
e ≡ [1 + (1 − q ) x]
x
q
(e = e )
x
1
x
(if 1 + (1 − q ) x > 0; vanishes otherwise)
q-GAUSSIANS:
pq ( x) ∝
− ( x / σ )2
eq
1
≡
⎡⎣1 + (q -1)
1
2 q -1
(x / σ ) ⎤
(q < 3)
⎦
D. Prato and C. T, Phys Rev E 60, 2398 (1999)
q - CENTRAL LIMIT THEOREM
THEOREM: (conjecture)
∂ p ( x, t )
∂ γ [ p ( x, t )]2− q
=D
(0 < γ ≤ 2; q < 3)
γ
∂t
∂| x|
globally correlated variables;
finite q-variance;
q-Gaussian attractor
independent variables;
finite variance;
Gaussian attractor
independent variables;
divergent variance;
Levy attractor
C.T., Milan J. Math. 73, 145 (2005)
q - CENTRAL LIMIT THEOREM (q-product and de Moivre-Laplace theorem):
The q- product is defined as follows:
x ⊗q y ≡ ⎡⎣ x
1− q
+y
1− q
− 1⎤⎦
1
1− q
Properties :
i ) x ⊗1 y = x y
ii ) ln q ( x ⊗q y ) = ln q x + ln q y
[ whereas ln q ( x y ) = ln q x + ln q y + (1 − q )(ln q x)(ln q y )]
[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
pN ,0 = p N with
and
Leibnitz rule
p = 1/ 2
q - CENTRAL LIMIT THEOREM: (numerical indications)
We q − generalize the de Moivre − Laplace theorem with
⎛1⎞ ⎛1⎞
⎛1⎞
1
= ⎜ ⎟ ⊗q ⎜ ⎟ ⊗q ... ⎜ ⎟ ( N terms )
pN ,0 ⎝ p ⎠ ⎝ p ⎠
⎝ p⎠
i.e.,
pN ,0 = ⎡⎣ N p q −1 − ( N − 1) ⎤⎦
1
q −1
( with p = 1/ 2)
de Moivre - Laplace
1
0.44
(q = 3/10)
-0.1
0
0.43
-1
β(N) 0.42
-2
qe=2-1/q
1
0.8
-3
-0.2
-0.3
-0.4
0
0.2
0
0.02
0.04
0.06
0.08
1/N
-4
0.6
qe
0.41
N=50
N=80
N=100
N=150
N=200
N=300
N=400
N=500
N=1000
qe
ln-4/3[p(x)/p(0)]
0
-5
0.4
-6
-7
0.2
-8
0
-9
0.4
0.6
2
x
0.8
1
-10
0
0.2
0.4
0.5
q
0.6
0.6
0.7
q
0.8
0.8
0.9
1
1
[Hence q Æ 2 – q (additive duality) and q Æ1/q (multiplicative duality) are involved]
L.G. Moyano, C. T. and M. Gell-Mann, Europhys. Lett. 73, 813 (2006)
q - GENERALIZED CENTRAL LIMIT THEOREM: (mathematical proof)
S. Umarov, C.T. and S. Steinberg [cond-mat/0603593]
q-Fourier transform:
Fq [ f ](ξ ) ≡
∞
ixξ
e
∫ q ⊗q f ( x) dx =
−∞
∞
∫
ixξ
[ f ( x )]1−q
eq
f ( x) dx
(nonlinear!)
−∞
q-correlation:
Two random variables X [ with density f X ( x)] and Y [ with density fY ( y )]
are said q - correlated if
Fq [X+Y](ξ ) =Fq [X](ξ ) ⊗q Fq [Y](ξ ) ,
i.e., if
izξ
⎡ ∞ dx eixξ ⊗ f ( x) ⎤ ⊗ ⎡ ∞ dy eiyξ ⊗ f ( y ) ⎤ ,
=
e
⊗
dz
f
(
z
)
q
q
q X
q Y
∫−∞ q q X +Y
⎢⎣ ∫−∞
⎥⎦ q ⎢⎣ ∫−∞
⎥⎦
∞
∞
∞
∞
∞
−∞
−∞
−∞
−∞
with f X +Y ( z ) = ∫ dx ∫ dy h( x, y ) δ ( x + y − z ) = ∫ dx h( x, z − x) = ∫ dy h( z − y, y )
where h( x, y ) is the joint density.
⎛ q - correlation means
⎜
⎝
independence
global correlation
if q = 1 , i.e.,
h ( x, y ) = f X ( x ) fY ( y ) ⎞
⎟
if q ≠ 1 , hence h( x, y ) ≠ f X ( x) fY ( y ) ⎠
⎡ β −β t2 ⎤
− β1 ω 2
q − FourierTransform ⎢
eq ⎥ = eq1
⎢⎣ Cq
⎥⎦
1+ q
where
q1 =
3− q
3− q
β = 2− q 2(1− q )
and
1
8 β Cq
with
⎧
⎛ 1 ⎞
2 πΓ ⎜
⎪
⎟
−
1
q
⎝
⎠
⎪
⎪
⎛ 3− q ⎞
⎪ (3 − q ) (1 − q )Γ ⎜
⎟
−
2(1
)
q
⎝
⎠
⎪
⎪
Cq = ⎨ π
⎪
⎪ π Γ ⎛⎜ 3 − q ⎞⎟
⎪
⎝ 2(q − 1) ⎠
⎪
⎪ q − 1Γ ⎛⎜ 1 ⎞⎟
⎪
⎝ q −1 ⎠
⎩
⎫
⎪
if q < 1 ⎪
⎪
⎪
⎪
⎪
if q = 1 ⎬
⎪
⎪
⎪
if 1 < q < 3⎪
⎪
⎪
⎭
Closure:
The q - Fourier transform of a q - Gaussian is a z ( q) - Gaussian with
1+ q
z (q) =
∈ (-∞,3)
3- q
Iteration:
2q + n(1 − q )
(n = 0, ± 1, ± 2,...; q0 = q )
2 + n(1 − q )
(the same as in R.S. Mendes and C.T. [Phys Lett A 285, 273 (2001)] when calculating marginal probabilities!)
qn ≡ zn (q ) ≡ z ( zn −1 (q )) =
hence
(i) qn (1) = 1 (∀n),
(ii) qn −1 = 2 −
q±∞ (q ) = 1 (∀q ),
1
,
qn +1
(the same as in L.G. Moyano, C.T. and M. Gell-Mann (2005)!)
(the same as in A. Robledo [Physica D 193, 153 (2004)] for pitchfork and tangent bifurcations!)
q + m(1 − q )
(iii) n = 2m =0, ± 2, ± 4,... yields q( m ) ≡ q2 m =
1 + m(1 − q )
(the same obtained in C.T., M. Gell-Mann and Y. Sato [Proc Natl Acad Sci (USA) 102, 15377 (2005)],
by combining only additive and multiplicative dualities, and which was conjectured
to be a possible explanation for the NASA-detected q-triangle for m = 0, ± 1!)
ALGEBRA ASSOCIATED WITH q-GENERALIZED CENTRAL LIMIT THEOREMS:
α
1 − qα ,n
=
α
1− q
+n
(n = 0, ± 1, ± 2,...)
S. Umarov, C.T., M. Gell-Mann and S. Steinberg (2006), cond-mat/0606040
A random variable X is said to have a ( q, α ) - stable distribution Lq,α ( x)
−b |ξ |α
if its q - Fourier transform has the form a e
( a > 0, b > 0, 0 < α ≤ 2)
q
i.e., if
Fq [ Lq ,α ](ξ ) ≡
L1,2 ( x) ≡ G ( x)
∞
∞
−∞
−∞
ixξ
e
∫ q ⊗q Lq,α ( x) dx =
ixξ
[ Lq ,α ( x )]1−q
∫ eq
−b |ξ |α
Lq ,α ( x) dx =a eq
(Gaussian)
L1,α ( x) ≡ Lα ( x) (α − stable Levy distribution)
Lq,2 ( x) ≡ Gq ( x) (q − Gaussian)
S. Umarov, C. T., M. Gell-Mann and S. Steinberg (2006)
cond-mat/0606038
cond-mat/0606040
CENTRAL LIMIT THEOREMS: N 1/[α (2-q )] - SCALED ATTRACTOR F( x) WHEN SUMMING N → ∞
q - CORRELATED IDENTICAL RANDOM VARIABLES WITH SYMMETRIC DISTRIBUTION f ( x)
q =1
[independent ]
q ≠ 1 (i.e., Q ≡ 2q − 1 ≠ 1) [globally correlated ]
F( x) = G3q −1 ( x) ≡
3q − 1
q +1
F( x) = Gaussian G ( x),
σQ < ∞
with same σ 1 of f ( x)
(α = 2)
Classic CLT
F( x) = Levy distribution Lα ( x),
σQ → ∞
(0 < α < 2)
with same | x | → ∞
asymptotic behavior
⎧ G ( x)
⎫
⎪
if | x |<< xc (1, α ) ⎪⎪
⎪
Lα ( x) ⎨
1+α ⎬
∼
∼
f
x
C
x
(
)
/
|
|
⎪
⎪
α
⎪
if | x |>> xc (1, α ) ⎪⎭
⎩
with limα → 2 xc (1, α ) = ∞
Levy-Gnedenko CLT
q +1
- Gaussian,
with same σ Q ⎡ ≡ ∫ dx x 2 [ f ( x)]Q / ∫ dx [ f ( x)]Q ⎤ of f ( x)
⎣
⎦
if | x |<< xc (q, 2) ⎫⎪
⎧⎪ G ( x)
G3q −1 ( x) ⎨
⎬
( q +1)/( q −1)
(
)
/
|
|
|
|
(
>>
∼
∼
,
2)
f
x
C
x
if
x
x
q
q +1
⎪⎩
q
c
⎭⎪
with lim q → 1 xc (q, 2) = ∞
S. Umarov, C. T. and S. Steinberg (2006) [cond-mat/0603593]
F ( x ) = L 2α
w ith L 2α
q −α + 3
,α
α +1
q −α + 3
,α
α +1
or
F ( x ) = L α q + q −1
,α
s ta b le d is tr ib u tio n ,
~ f (x) ∼ C
(L)
q ,α
/ | x | (1 + α ) / (1 + α q − α )
s ta b le d is tr ib u tio n ,
α + q −1
w ith L α q + q −1 ~ f ( x ) ∼ C
,α
α + q −1
(* )
q ,α
/ | x |2 (α + q −1 ) / α ( q − 1 )
S . U m a r o v , C . T . , M . G e ll- M a n n
a n d S . S te in b e rg (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 ]
WHAT IS IT q − CORRELATION ? :
It appears to be (no proof available yet )
N − body joint probability
∫ dx
N
h( x1 , x2 ,..., xN ) =h( x1 , x2 ,..., xN −1 )
i.e., scale invariance !
BOLTZMANN-GIBBS STATISTICAL MECHANICS
(Maxwell 1860, Boltzmann 1872, Gibbs ≤ 1902)
Entropy
Internal energy
Equilibrium
distribution
Paradigmatic
differential
equation
Equilibrium distribution
Sensitivity to
initial conditions
Typical relaxation of
observable Ο
W
S B G = − k ∑ p i ln p i
U
=
BG
W
∑
i =1
pi = e
− β Ei
i =1
pi Ei
/ Z BG
W
⎛
−β Ej ⎞
Z
e
≡
⎜ BG ∑
⎟
j =1
⎝
⎠
dy
⎫
= ay ⎪
dx
⎬ ⇒
y ( 0 ) = 1 ⎪⎭
y = e ax
x
a
y(x)
Ei
-β
Z p(Ei)
t
λ
t
-1/τ
ξ ≡ lim
Δx (0 )→ 0
Ω≡
Δ x (t )
= e λt
Δ x (0 )
O (t ) − O ( ∞ )
= e − t /τ
O (0) − O ( ∞ )
SBG → extensive, concave, Lesche-stable, finite entropy production
NONEXTENSIVE STATISTICAL MECHANICS
(C. T. 1988, E.M.F. Curado and C. T. 1991, C. T., R.S. Mendes and A.R. Plastino 1998)
W
⎛
⎞
S q = k ⎜1 − ∑ piq ⎟ /(q − 1)
⎝ i =1 ⎠
Entropy
Internal energy
W
U q = ∑ p Ei / ∑ p qj
i =1
Stationary state
distribution
−βq ( Ei −Uq )
q
pi = e
dy
⎫
= a y q⎪
dx
⎬⇒
y (0) = 1 ⎪⎭
Paradigmatic
differential
equation
Stationary state
distribution
x
Ei
Sensitivity to
initial conditions
t
Typical relaxation of
observable Ο
W
q
i
t
W
− β q ( E j −U q ) ⎞
⎛
⎜ Z q ≡ ∑ eq
⎟
=
1
j
⎝
⎠
j =1
/ Zq
ax
y = eq
≡ [1 +
a
Z qstat p ( Ei )
λqsen t
ξ = eq
sen
− 1 / τ
(1 − q ) a x ]
y(x)
− β qstat
λq
1
1− q
q
r e l
Ω
=e
sen
− t / τ qrel
qrel
(typically q stat ≥ 1)
(typically qsen ≤ 1)
(typically qrel ≥ 1)
Sq → extensive, concave, Lesche-stable, finite entropy production
C. T., Physica A 340,1 (2004)
Prediction of the q - triplet:
C. T., Physica A 340,1 (2004)
SOLAR WIND: Magnetic Field Strength
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]
q sen = − 0.6 ± 0.2
qrel = 3.8 ± 0.3
qstat = 1.75 ± 0.06
Playing with additive duality
(q → 2 − q)
and with multiplicative duality (q → 1/ q )
(and using numerical results related to the q − generalized central limit theorem)
we conjecture
qrel +
1
=2
qsen
hence
and
1 − qsen
qstat +
1
=2
qrel
1 − q stat
=
3 − 2 qstat
hence only one independent !
Burlaga and Vinas ( NASA) most precise value of the q − triplet is
qstat = 1.75 = 7 / 4
hence
qsen = − 0.5 = − 1/ 2
(consistent with qsen = − 0.6 ± 0.2 !)
and
qrel = 4
(consistent with qrel =
3.8 ± 0.3 !)
PREDICTION:
The solution of
∂ p ( x, t )
∂ 2 [ p ( x, t )]2− q
=D
∂t
∂x 2
is given by
p ( x, t ) ∝ ⎡⎣1 + (1 − q ) x / (Γ t )
2
[p ( x, 0) = δ (0)]
2 /(3− q ) 1/(1− q )
⎤⎦
(q < 3)
− x 2 / ( Γ t )2 /( 3−q )
q
≡e
(Γ ∝ D )
hence
x 2 scales like t γ (e.g.,
x2 ∝ t γ )
with
2
γ=
3− q
(e.g ., q = 1 ⇒ γ = 1 , i.e., normal diffusion)
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
slope γ = 1.24 ± 0.1
2
hence γ =
is satisfied
3− q
Defect turbulence:
K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
2
q ≈ 1.5 and γ ≈ 4 / 3 are consistent with γ =
3− q
K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
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)
q=3/2
q=1
(fully developed
regime)
R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)
(outlet size 3.8 d)
slope γ = 4 / 3
2
hence γ =
is satisfied
3− q
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) q = 1 +
44 ER
U0
where
ER ≡ recoil energy
U 0 ≡ potential depth
Experimental and computational verifications
by P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)
q = 1+
44 ER
U0
(Computational verification:
quantum Monte Carlo simulations)
(Experimental verification)
HADRONIC JETS FROM ELECTRON-POSITRON ANNIHILATION:
I. Bediaga, E.M.F. Curado and J.M. de Miranda, Physica A 286 (2000) 156
Beck (2000): q=11/9
Hagedorn
(Phenomenological model for collisions in a diluted gas with probability r
of forming clusters of q correlated particles)
Monte Carlo
Single-parameter
fitting
Connections with
asymptotically scale − free networks
GEOGRAPHIC PREFERENTIAL ATTACHMENT GROWING NETWORK:
THE NATAL MODEL
D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva, Europhys Lett 70, 70 (2005)
(1) Locate site i=1 at the origin of say a plane
(2) Then locate the next site with
PG ∝ 1/ r 2+αG (αG ≥ 0)
( r ≡ distance to the baricenter of the pre − existing cluster )
(3) Then link it to only one of the previous sites using
p A ∝ k i / ri
αA
(α A ≥ 0)
( k i ≡ links already attached to site i )
( ri ≡ distance to site i )
4) Repeat
(α G = 1; α A = 1; N = 250)
D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva
Europhys Lett 70, 70 (2005)
P (k )/P (0 )= eq − k /κ
≡ 1 / [1 + ( q − 1) k / κ ]1 / ( q − 1 )
κ = 0 .0 8 3 + 0 .0 9 2 α A
Barabasi-Albert
universality class
(∀ α G )
q=1+(1/3) e −0.526 α A
(∀ α G )
ASTROPHYSICS
(Band Q: 22.8 GHz)
(Data after using Kp0 mask)
(Band V: 60.8 GHz)
q = 1.045 ± 0.005
(Band W: 93.5 GHz)
(99 % confidence level)
A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)
q = 1.045 ± 0.005 (99 % confidence level)
A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)
Connections with conservative
and Hamiltonian systems
A
(r → ∞ )
V (r ) ∼ − α
r
( A > 0, α ≥ 0)
integrable if
α /d >1
non -integrable if 0 ≤ α / d ≤ 1
α
5
( short - ranged )
( long -r ange d )
)
s
e on
EXTENSIVE
d
(ti tati
e vi
l
o ra
p
o lg
n
o na
dipole-dipole
m io
le n s
o
e
p
i
d
im
d
NONEXTENSIVE
d-
4
3
2
1
Newtonian gravitation
0
0
1
2
3
4
d
5
MANY COUPLED STANDARD MAPS:
L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)
a
pi (t + 1) = pi (t ) + sin [ 2πθi (t )] +
2π
b
2π N
N
∑
j =1
{
θi (t + 1) = θi (t ) + pi (t + 1)
with
1−α / d
}
sin 2π ⎡⎣θi (t ) − θ j (t )⎤⎦
α
rij
(mod1)
(mod1)
N
−1
N≡
; a ≥ 0 ; b ≥ 0 ; α ≥ 0 ; d =1
1−α / d
d-DIMENSIONAL CLASSICAL INERTIAL XY FERROMAGNET:
C. Anteneodo and C. T., Phys Rev Lett 80, 5313 (1998)
(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)
1
H = K +V =
2I
N
with A ≡ ∑ rij−α
j =1
1 − cos(ϑi − ϑ j )
J
L + ∑
∑
α
r
A
i =1
i, j
ij
N
2
i
⎧ N 1−α / d
⎪
∝ ⎨ln N
⎪constant
⎩
if
if
if
( I > 0, J > 0)
0 ≤α / d <1 ⎫
⎪
α / d =1 ⎬
α / d > 1 ⎪⎭
and periodic boundary conditions.
[The HMF model corresponds to α = 0, ∀d ]
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)
In contact with a thermostat (canonical ensemble):
F. Baldovin and E. Orlandini
Phys Rev Lett 96, 240602 (2006)
F. Baldovin and E. Orlandini, cond-mat/0603659
(q=2.35)
tW = 8
t
α = 0 model :
qrel
2.35 ?
qstat
1.5 ?
F.A. Tamarit and C. Anteneodo
HMF MODEL:
L.G. Moyano, F. Baldovin and C. T., cond-mat/0305091
ECONOMICS
STOCK VOLUMES:
J de Souza, SD Queiros and LG Moyano, physics/0510112 (2005)
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)
[ REMARK : Student t − distributions are the particular case
of q − Gaussians when
q=
n+3
n +1
with n integer ]
EARTHQUAKES
MODEL FOR EARTHQUAKES (OMORI REGIME):
D(n+nw , nw)
1.0
nw=250
nw=500
nw=1000
nw=2000
nw=5000
0
ln [D(n+n , n )]
q
w w
-2
-4
-6
(q=2.98)
-8
-10
-12
0
2
4
6
8
n/n
0.1
10-4
10-3
10-2
10
1.05
12
14
16
w
10-1
100
101
102
n / nw1.05
S. Abe, U. Tirnakli and P.A. Varotsos
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)
Visiting algorithm :
Boltzmann machine → Gaussian
Generalized machine → qV − Gaussian
Acceptance algorithm :
Boltzmann machine → Boltzmann weight
Generalized machine → q A − exponential weight
Cooling algorithm :
Boltzmann machine
→
T(t)
ln 2
=
T(1) ln(1 + t )
q −1
T(t)
2 V −1
=
Generalized machine →
T(1) (1 + t ) qV −1 − 1
[Typical values : 1 < qV < 3 and q A < 1]
C. T. and D.A. Stariolo, Physica A 233, 395 (1996)
q-GENERALIZED SIMULATED ANNEALING (GSA):
Illustration :
4
4
E ( x1 , x2 , x3 , x4 ) = ∑ ( xi − 8) + 5∑ xi
2
2
i =1
i =1
(15 local minima and one global minimum)
(qV = 1 ⇒ mean convergence time ≈ 50000)
q-GENERALIZED PIVOT METHOD:
P. Serra, A.F. Stanton and S. Kais, Phys Rev E 55, 1162 (1997)
(Lennard-Jones clusters)
e4
.7
Genetic
algorithm
slo
p
Number of function calls
(Branin function)
sl
e
p
o
9
2.
Present
with q=2.7
Recently: M.A. Moret, P.G. Pascutti, P.M. Bisch, M.S.P. Mundim and K.C. Mundim
Classical and quantum conformational analysis using Generalized Genetic Algorithm
Physica A 363, 260 (2006) (presumably better than both!)
IMAGE EDGE DETECTION
Original
image
Canny
edge
detector
[A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]
q = 1.5
q=1
(JensenShannon)
IMAGE EDGE DETECTION
Original
image
Canny
edge
detector
[A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]
q = 1.5
q=1
(JensenShannon)
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)
thanq
HOW MANY PHYSICAL UNIVERSAL CONSTANTS ?
ν ≡ number of independent physical universal constants in contemporary physics
1
ν
=
1 ⎡1 1 1 ⎤ 7
+ + ⎥=
⎢
3 ⎣ 3 4 ∞ ⎦ 36
Nino Constantino Gerardus
hence
Euclid
ν=
Gerardus
36
1⎞
⎛
= 2 + ⎜ 3 + ⎟ = 2 + π = 2 + 1800 = 2 + 1 = 3
7
7⎠
⎝
ν = 3 !!!
NINO WAS RIGHT!!!
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
Merging probability pij ∝
1
dijα
(α ≥ 0)
dij ≡ shortest path (chemical distance) connecting nodes i and j on the network
α = 0 and α → ∞ recover the random and the neighbor schemes respectively
(Kim, Trusina, Minnhagen and Sneppen, Eur. Phys. J . B 43 (2005) 369)
( N = 27 ; α = 0; r = 2)
Degree of the most connected node
Degree of a randomly chosen node
80
ki
kmax
70
kmax ; ki
60
50
40
30
20
10
0
2000
4000
6000
8000 10000 12000 14000
time
S. Thurner, Europhys News 36, 218 (2005)
(α → ∞ ; < r >= 8)
(α → ∞;
Z q (k ) ≡ ln q [ P(> k )]
P(> k )]
[
≡
1− q
−1
1− q
(optimal qc = 1.84)
( N = 29 ; r = 2)
P (≥ k ) = eqc
- ( k -2)/ κ
(k = 2, 3, 4,...)
linear correlation ∈ [0.999901, 0.999976]
(r = 2)
qc (α ) = qc (∞) + [ qc (0) − qc (∞) ] e−α
( N = 29 )
SANTOS THEOREM: RJV Santos, J Math Phys 38, 4104 (1997)
(q - generalization of Shannon 1948 theorem)
IF
S ({ pi }) continuous function of { pi }
AND S ( pi = 1/ W , ∀i ) monotonically increases with W
S ( A + B) S ( A) S ( B)
S ( A) S ( B)
A+ B
A
B
( with p ij = p i p j )
=
+
+ (1 − q )
k
k
k
k
k
q
q
AND S ({ pi }) = S ( pL , pM ) + pL S ({ pl / pL }) + pM S ({ pm / pM }) ( with pL + pM = 1)
AND
THEN AND ONLY THEN
W
S ({ pi }) = k
1 − ∑ pi
i =1
q −1
q
W
⎛
⎞
p
q
S
p
k
p
=
1
⇒
({
})
=
−
ln
∑
i
i
i⎟
⎜
i =1
⎝
⎠
CE SHANNON (The Mathematical Theory of Communication) :
are in no way necessary for the
present theory. It is given chiefly to lend a certain plausibility to some of our later definitions. The
real justification of these definitions, however, will reside in their implications.
"This theorem, and the assumptions required for its proof ,
ABE THEOREM: S Abe, Phys Lett A 271, 74 (2000)
(q - generalization of Khinchin 1953 theorem)
IF
S ({ pi }) continuous function of { pi }
AND S ( pi = 1/ W , ∀i ) monotonically increases with W
AND S ( p1, p2 ,..., pW , 0) = S ( p1, p2 ,..., pW )
AND
S ( A + B) S ( A) S ( B | A)
S ( A) S ( B | A)
=
+
+ (1 − q )
k
k
k
k
k
THEN AND ONLY THEN
W
S ({ pi }) = k
1 − ∑ pi
i =1
q −1
q
W
⎛
⎞
⎜ q = 1 ⇒ S ({ pi }) = − k ∑ p i ln pi ⎟
i =1
⎝
⎠
The possibility of such theorem was conjectured
by AR Plastino and A Plastino (1996, 1999).

Constantino Tsallis

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