Aulas Tsallis, Notas de aula de Engenharia Elétrica
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Aulas Tsallis, Notas de aula de Engenharia Elétrica

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Slide 1

Constantino Tsallis

Centro Brasileiro de Pesquisas Fisicas

Rio de Janeiro - Brasil

Rio de Janeiro, CBPF, Abril 2007

MECANICA ESTATISTICA NAO EXTENSIVA

ASPECTOS TEORICOS, EXPERIMENTAIS,

OBSERVACIONAIS E COMPUTACIONAIS

EXTENSIVITY OF THE NONADDITIVE ENTROPY Sq

and q - GENERALIZED CENTRAL LIMIT THEOREM:

C. T., M. Gell-Mann and Y. Sato, Proc Natl Acad Sci (USA) 102, 15377 (2005)

C. T., M. Gell-Mann and Y. Sato, Europhysics News 36, 186 (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 and 0703533 (2006, 2007)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T, Physica A 372, 183 (2006)

U. Tirnakli, C. Beck and C. T., Phys Rev E /Rapid Comm (2007), in press

C. T. and S.M.D. Queiros, preprint (2007)

W. Thistleton, J.A. Marsh, K. Nelson and C. T., preprint (2007)

EXPERIMENTAL VERIFICATION IN COLD ATOMS:

P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)

Sabir Umarov

Yuzuru Sato

Stanly Steinberg

Ugur Tirnakli Silvio M.D. Queiros

Miguel Fuentes

Luis Moyano Paul Rivkin William Thistleton John Marsh Kenric Nelson

Christian Beck E.G.D. Cohen

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 Nonextensivity HL Swinney and C Tsallis, eds, Physica D 193, Issue 1-4 (2004)

News and Expectations in Thermostatistics G Kaniadakis and M Lissia, eds Physica A 340, Issue 1/3 (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)

Nonextensive Statistical Mechanics: New Trends, New Perspectives, JP Boon and C Tsallis, eds, Europhysics News (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)

Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World, C. Tsallis (in preparation)

Full bibliography (regularly updated):

http://tsallis.cat.cbpf.br/biblio.htm

2,107 articles (done by 1,550 scientists from 60 countries) which led to

> 7,520 citations of papers

(including > 1,407 citations of the 1988 paper)

> 957 nominal citations

[31 March 2007]

CONTRIBUTORS (2107 MANUSCRIPTS)

BRAZIL 303 USA 217 ITALY 116 JAPAN 105 FRANCE 92 CHINA 88 ARGENTINA 79 SPAIN 57 GERMANY 55 ENGLAND 40 POLAND 37 RUSSIA 31 TURKEY 27 INDIA 23 CANADA 23 AUSTRIA 21 MEXICO 19 UKRAINE 18 SWITZERLAND 17 ISRAEL 13 KOREA 12

NETHERLANDS 11 BELGIUM 11 GREECE 11 AUSTRALIA 8 PORTUGAL 8 SOUTH AFRICA 7 CUBA 7 HUNGARY 7 IRAN 7 VENEZUELA 6 CHILE 6 NORWAY 5 SINGAPORE 4 SWEDEN 4 TAIWAN 4 URUGUAY 4 ROMENIA 4 EGYPT 4 DENMARK 4 SLOVENIA 4 BULGARIA 3

SLOVAK 3 CROATIA 3 IRELAND 3 BOLIVIA 2 CZECK 2 FINLAND 2 KAZAKSTAN 2 MOLDOVA 2 PHILIPINES 2 PUERTO RICO 2 ARMENIA 1 INDONESIA 1 JORDAN 1 MALAYSIA 1 SAUDI ARABIA 1 SERBIA 1 SRI LANKA 1 UZBEKISTAN 1 ____________ ___________ 60 1550 COUNTRIES SCIENTISTS

[Updated 31 March 2007]

It is the natural (or artificial or social) system itself which, through its geometrical-dynamical properties, indicates the specific informational tool --- entropy --- to be meaningfully used for the study of its thermostatistical and thermodynamical properties.

TRIADIC CANTOR SET:

0.6309... 0.6309

Hence the interesting mea

ln 2 0.6309... ln 3

(10 ) 4.275

sure is

Fd

cm cm

= =

10 cm

Newton

Einstein 1905 Einstein 1915Dirac

Quantum gravity?

(1/ 0)Bk =

1 1

1 1

0; 0; 0; 0 ( 0: )

B

B

c h G k s The full tetrahedron tatistical mechanics of quantum g

corresponds to the at its centerav t Gri y c h k

− −

− −

> > > >

= = = =

c-1>0 h >0 G >0

c-1>0 h =0 G =0

c-1>0 h =0 G >0

c-1>0 h >0 G =0

c-1=0 h =0 G >0

c-1=0 h >0 G =0

c-1=0 h >0 G >0

c-1=0 h =0 G =0

Schroedinger

C.T., Introduction to Nonextensive Statistical Mechanics-Approaching a Complex World (in progress)

-1 1 , , , : BThe four independent universal constants of c G contempor hary phys s kic

A. Pluchino and C. T. (2006)

HISTORICAL BACKGROUND AND PHYSICAL MOTIVATIONS FOR ATTEMPTING TO

GENERALIZE BOLTZMANN-GIBBS STATISTICAL MECHANICS

ALONG THE LAST 135 YEARS…

Vorlesungen uber Gastheorie (Leipzig, 1896) Lectures on Gas Theory, transl. S. Brush (Univ. California Press, Berkeley, 1964), page 13

The forces that two molecules impose one onto the other during an interaction can be completely arbitrary, only assuming that their sphere of action is very small compared to their mean free path.

Ludwig BOLTZMANN

J.W. GIBBS Elementary Principles in Statistical Mechanics - Developed with Especial Reference to the Rational Foundation of Thermodynamics

C. Scribner’s Sons, New York, 1902; Yale University Press, New Haven, 1981), page 35

In treating of the canonical distribution, we shall always suppose the multiple integral in equation (92) [the partition function, as we call it nowadays] to have a finite valued, as otherwise the coefficient of probability vanishes, and the law of distribution becomes illusory. This will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on thermodynamics. It will exclude, for instance, cases in which the system or parts of it can be distributed in unlimited space […]. It also excludes many cases in which the energy can decrease without limit, as when the system contains material points which attract one another inversely as the squares of their distances. […]. For the purposes of a general discussion, it is sufficient to call attention to the assumption implicitly involved in the formula (92).

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 additive 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.

Claude Elwood SHANNON

The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)

It is practically more useful. [...] It is nearer to our intuitive feeling as to the proper measure. [...] It is mathematically more suitable. [...].}

This theorem and the assumptions required for its proof, 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.

Laszlo TISZA

Generalized Thermodynamics (MIT Press, Cambridge, Massachusetts, 1961)

The situation is different for the additivity postulate Pa2, the validity of which cannot be inferred from general principles. We have to require that the interaction energy between thermodynamic systems be negligible. This assumption is closely related to the homogeneity postulate Pd1. From the molecular point of view, additivity and homogeneity can be expected to be reasonable approximations for systems containing many particles, provided that the intramolecular forces have a short range character.

Radu BALESCU

Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley and Sons, 1975, New York)

It therefore appears from the present discussion that the mixing property of a mechanical system is much more important for the understanding of statistical mechanics than the mere ergodicity. [...] A detailed rigorous study of the way in which the concepts of mixing and the concept of large numbers of degrees of freedom influence the macroscopic laws of motion is still lacking.

Peter LANDSBERG

Thermodynamics and Statistical Mechanics (1978)

The presence of long-range forces causes important amendments to thermodynamics, some of which are not fully investigated as yet.

Is equilibrium always an entropy maximum? J. Stat. Phys. 35, 159 (1984)

[...] in the case of systems with long-range forces and which are therefore nonextensive (in some sense) some thermodynamic results do not hold. [...] The failure of some thermodynamic results, normally taken to be standard for black hole and other nonextensive systems has recently been discussed. [...] If two identical black holes are merged, the presence of long-range forces in the form of gravity leads to a more complicated situation, and the entropy is nonextensive.

David RUELLE Thermodynamical Formalism - The Mathematical Structures of Classical Equilibrium Statistical Mechanics (page 1 of both 1978 and 2004 editions)

The formalism of equilibrium statistical mechanics -- which we shall call thermodynamic formalism -- has been developed since J.W. Gibbs to describe the properties of certain physical systems. [...] While the physical justification of the thermodynamic formalism remains quite insufficient, this formalism has proved remarkably successful at explaining facts. The mathematical investigation of the thermodynamic formalism is in fact not completed: the theory is a young one, with emphasis still more on imagination than on technical difficulties. This situation is reminiscent of pre-classic art forms, where inspiration has not been castrated by the necessity to conform to standard technical patterns.

(page 3) The problem of why the Gibbs ensemble describes thermal equilibrium (at least for “large systems”) when the above physical identifications have been made is deep and incompletely clarified. -----------------------------------------------------------------------------------------------------------------

[The first equation is dedicated to define the BG entropy form. It is introduced after the words “we define its entropy” without any kind of justification or physical motivation.]

Nico van KAMPEN

Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)

Actually an additional stability criterion is needed, see M.E. Fisher, Archives Rat. Mech. Anal. 17, 377 (1964); D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York 1969). A collection of point particles with mutual gravitation is an example where this criterion is not satisfied, and for which therefore no statistical mechanics exists.

Roger BALIAN From Microphysics to Macrophysics (Springer-Verlag, Berlin, 1991), p. 205 and 206; French edition (1982).

These various quantities are connected with one another through thermodynamic relations which make their extensive or intensive nature obvious, as soon as one postulates, for instance, for a fluid, that the entropy, considered as a function of the volume Omega and of the constants of motion such as U and N, is homogeneous of degree 1: S(x Omega, x U, x N)=x S(Omega, U, N) (Eq. 5.43). [...] Two counter-examples will help us to feel why extensivity is less trivial than it looks. [...] A complete justification of the Laws of thermodynamics, starting from statistical physics, requires a proof of the extensivity (5.43), a property which was postulated in macroscopic physics. This proof is difficult and appeals to special conditions which must be satisfied by the interactions between the particles.

L.G. TAFF

Celestial Mechanics (John Wiley, New York, 1985)

This means that the total energy of any finite collection of self- gravitating mass points does not have a finite, extensive (e.g., proportional to the number of particles) lower bound. Without such a property there can be no rigorous basis for the statistical mechanics of such a system (Fisher and Ruelle 1966). Basically it is that simple. One can ignore the fact that one knows that there is no rigorous basis for one's computer manipulations; one can try to improve the situation, or one can look for another job.

W.C. SASLAW

Gravitation Physics of Stellar and Galactic Systems (Cambridge University Press, Cambridge, 1985)

When interactions are important the thermodynamic parameters may lose their simple intensive and extensive properties for subregions of a given system. [...] Gravitational systems, as often mentioned earlier, do not saturate and so do not have an ultimate equilibrium state.

John MADDOX

When entropy does not seem extensive Nature 365, 103 (1993)

Everybody who knows about entropy knows that it is an extensive property, like mass or enthalpy. [...] Of course, there is more than that to entropy, which is also a measure of disorder. Everybody also agrees on that. But how is disorder measured? [...] So why is the entropy of a black hole proportional to the square of its radius, and not to the cube of it? To its surface area rather than to its volume?

A.C.D. van ENTER, R. FERNANDEZ and A.D. SOKAL, Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations: Scope and Limitations of Gibbsian Theory [J. Stat. Phys. 72, 879-1167 (1993)] We provide a careful, and, we hope, pedagogical, overview of the theory of Gibssian measures as well as (the less familiar) non-Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. Toward a Non-Gibbsian Point of View: Let us close with some general remarks on the significance of (non-)Gibbsianness and (non)quasilocality in statistical physics. Our first observation is that Gibbsianness has heretofore been ubiquitous in equilibrium statistical mechanics because it has been put in by hand: nearly all measures that physicists encounter are Gibbsian because physicists have decided to study Gibbsian measures! However, we now know that natural operations on Gibbs measures can sometimes lead out of this class. [...] It is thus of great interest to study which types of operations preserve, or fail to preserve, the Gibbsianness (or quasilocality) of a measure. This study is currently in its infancy. [...] More generally, in areas of physics where Gibbsianness is not put in by hand, one should expect non-Gibbsianness to be ubiquitous. This is probably the case in nonequilibrium statistical mechanics. Since one cannot expect all measures of interest to be Gibbsian, the question then arises whether there are weaker conditions that capture some or most of the “good” physical properties characteristic of Gibbs measures. For example, the stationary measure of the voter model appears to have the critical exponents predicted (under the hypothesis of Gibbsianness) by the Monte Carlo renormaliztion group, even though this measure is provably non-Gibbsian. One may also inquire whether there is a classification of non- Gibbsian measures according to their “degree of non-Gibbsianness”.

/ /

: - :

, ( ) ( ( )

i i

i

i E kT

E kT i

i

The values of p are determined by the following if the energy of the system in the i th state is E and if the temperature of the system is T then

ep where Z T e this l Z

dogma

T

− −= = ∑

1).

.

"

;

i i

i We shall give no justification for t

ast constant is taken so that p

This choice of p is called the Gibbs distribution even a physicist like Ruelle dihi sposes of this question ass dogma

de

=∑

".ep and incompletely clarified

The advantages of the method of postulation are great; they are the same as the advantages of theft over honest toil.

(Bertrand Russell)

ENTROPY

ENTROPIC FORMS

Concave

Extensive

Lesche-stable

Finite entropy production per unit time

Pesin-like identity (with largest entropy production)

Composable

Topsoe-factorizable

BG entropy

(q =1)

Entropy Sq

(q real)

1

1

1

: :

, :

( 1)

ln ( 0) 1

ln ln

q

q xx x

q x

DEFINITION q logarithm

Hence the entropies can be rewritten

equal probabilities general probabilities

BG entrop

x

y q

− − ≡ >

=

=

W

i=1

W

i=1

ln (1/ ) ln

ln ln (1/ )

( ) q

i i

q i q ientr

k p pk W

k W k p popy S

q R

C.T., M. Gell-Mann and Y. Sato Europhysics News 36 (6), 186 (European Physical Society, 2005)

2

(A. Robledo, Mol Phys 103 (2005) 3025)

(F. C

IDEAL GAS

CRITICAL PHENOM

aru

ENA

so and C. T., 2006)

1

2 9 2 3

q

cq c

δ+ =

+ − =

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 (de Moivre-Laplace-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)

ARISTOTLE (384 – 322 BC)

( )x y symmetry

ABOUT ORDINARY DIFFERENTIAL EQUATIONS

-

ABOUT MEAN VALUES

ABOUT BIAS

ABOUT REACTION UNDER BIAS

S. Abe Phys Lett A 224, 326 (1997)

SANTOS THEOREM: RJV Santos, J Math Phys 38, 4104 (1997)

(q -generalization of Shannon 1948 theorem)

({ }) { } ( 1/ , )

( ) ( ) ( ) ( ) ( ) ( )

({ }) ( , ) ({ / }) ({

(1 )

i i

i

A B A B ij i j

i L M L l L M q q

S p continuous function of p S p W i monotonically increases with W S A B S A S B S A S B with p p p

k k k k k S p S p p p S

IF AND

AND

A p p S

q

ND p

+

= ∀ +

= + +

+

− =

= +

1

1

1 ({ }) 1 ({

}) ln

/ }) ( 1

( ) : " ,

)

1

W

i W i

i i i

m M

i i

M L

q

CE SHANNON The Mathematica

p S p k q S p k p

l Theory

THEN AND ONLY THEN

of

p p with

Communication This theorem and th

p

p q

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 necessary

The 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 p S p W i monotonically increases with W S p p p S p p p S A B S A S B A S A S B A

k

AND AND

AND

THEN AND ONLY

k k k

THE

k

N

q

= ∀ =

+ = + + −

1

1

1 ({ }) 1 ({ }) ln

1

(1996, 1999).

W

i W i

i i i i i

q

The possibility of such theorem was conjectured b

p S p k q S p k

y AR Plastino and A Plas n

p p

o

q

ti

=

=

− ⎛ ⎞

= = ⇒ = −⎜ ⎟− ⎝ ⎠

∑ ∑

STABILITY

or CONTINUITY

or EXPERIMENTAL ROBUSTNESS

B. Lesche J Stat Phys 27, 419 (1982)

1

1

1 1

1 ln ( 0) 1

ln ln

[1 (1 ) ] 1 (1 ) 0 0

:

-

:

:

q

q

qx q

DEFINITIONS

q log xx x

q x x

q x i

arithm

q ex

f q

ponent

xe

ial

− ≡ >

=

+ − + − ≥≡

1

x x otherwise

e e

⎧ ⎫⎪ ⎪ ⎨ ⎬ ⎪ ⎪⎩ ⎭

=

0.315 1.35

Total # of papers 230993 119558 18759

25358

(0) (1)

( ) c

N N

N c e

= = =

=

Citations

N

u

m

b

e

r

o

f

p

a

p

e

r

s

January 30, 2007r ,

ISI citations of Brazilian scientists (1945 - …)

[“Brazil” in the address field]

M.P de Albuquerque and D.B. Mussi (2007)

PREDECESSORS

. . , . . . . (1954) [ . (1974,1978)] . , . 4 (1969)

ln :

1 :

th

q i

i

q i

i

M P Schutzenberger Publ Inst Statist Univ Paris according to I Csiszar A Renyi Proc Berkeley Symposium

RENYI ENTROPY p

ENTROPY p

∝ −

∑ . . , 3, 30 (1967)

. , 4, 105 (1968) . , . 16, 36 (1970) . . ,

J Harvda and F Charvat Kybernetica I Vajda Kybernetica Z Daroczy Inf Control J Lindhard and V Nielsen Det Kongelige Danske

- ( ) 38 (9), 1 (197 [

1) . . . . , . . . 10, 28 (1975)

Videnskabernes Selskab Matematisk fysiske Meddelelser Denmark

B D Sharma a unification of bnd D P M othittal J M previouath Sc si en . , . . . 50, 221 (1978)

(1733)

]

:

tropic forms

Gaussian di

A

stribution

Wehrl Rev Mod Phys

Abraham de Moivre q GAUSSIANS

(1774) (1808)

Pierre Simon de Laplace Robert Adrain

(1809) (~ 182 1)

Carl Friedrich Gauss Agustin LoCauchy Lo uis Caucrentz Bre hyit− −

'

(~ 1880)

(1908) Wigner distribution

Stu Hendric Antoon Lorentz William Sealy Godent s t distributio sn set

− −

1

i=1 i=1 1

DISCRETE CLASSICAL STATES (Shannon for 1):

CONTINUOUS CLASSICAL STATES (Boltzmann, Gibbs for 1):

1 1ln ln 1

1

1 [ ( )] 1 ( ) ln 1 (

( )

)

W q i W W W

q qi i q i q i i

ii

q q

q

q

q

S

S

wi p

p p p p k q p

dx p x dx p x

k q p x

th= =

=

=

− = = = − =

− = =

∑ ∑ ∑ ∑

1

1

1

QUANTUM STATES (von Neumann for 1):

1 Particu

[ ( )] ln ( ) ( ) 1

1 ( ln ) (

lar case: ( )

ln ) 1

( ) 1

( )

(

W q iW

q i i i

i

q q

q q q

q q

S

S

q

dx p x p x dx p x

Tr Tr Tr Tr k

with

p p x p x x

k q

q withρ ρ ρ ρ ρ ρ

δ

=

=

=

= − =

− = = = − =

− = − ⇒ =

∑ ∑

∫ ∫ ∫

1 ij ij

1 Particular case:

)1

1

W q i

q i i

S p p

k q ρ δ =

− = ⇒ =

THE VARIOUS FORMS OF THE NONADDITIVE q-ENTROPY Sq:

( , ) qS N t versus t DISSIPATIVE MAPS:

Strongly chaotic (i.e., maximal Lyapunov exponent > 0) Weakly chaotic (i.e., maximal Lyapunov exponent = 0)

CONSERVATIVE MAPS: Strongly chaotic (i.e., maximal Lyapunov exponent > 0) Weakly chaotic (i.e., maximal Lyapunov exponent = 0)

DISSIPATIVE MAPS

LOGISTIC MAP:

2 1 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)

1

1 1

1 1

(0) 0

( )lim

(

( )( ) lim (0)

)

t

t x

We verify

w

Pesin like ide

here

S t

ntity

K t

and x tt e

K

x λ

Δ Δ Δ

λ

ξ

→∞

=

−=

q = 0.1

q = 0.2445

q = 0.5

S (t)q

t

N = W = 2.5 10 6

a = 1.4011552

x = 1 - ax t +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

(0) 0

( ) lim sup

( )

( ) sup lim

(0)

( )

q q t

x

q q

q t q

It can be p q generalized Pesin lik

roved that

where S t

K t

and

x tt x

e identit

w

yK

ith

eΔ λΔ

Δ

λ

ξ

→∞

⎧ ⎫ ≡ ⎨ ⎬

⎩ ⎭

⎧ ⎫ ≡ =⎨ ⎬

⎩ ⎭

−= −

min

1 mi

ma

n max

x

1 1 1 ln ( )1 | | ( 1) 1 ( ) ( ) ( )

1 1 1 ln 1 1 ln

2

2

n

1

l z F

t t

F q

zx a x z q z z

n

z

q a

q d

α α α

α λ α α

+

⎡ ⎤ = − ⇒ = − =

= − = = − −

−⎢ ⎥−⎣ ⎦

EDGE OF CHAOS OF THE LOGISTIC MAP: (Using result in http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt)

q =

0.2444877013412820661987704234046804052344469354900576736703650 986327749672766558665755156226857540706288349640382728306063600 193730331818964551341081277809792194386027083194490052465813521 503174534952074940448165460949087448334056723622466488083333072 142318987145872992681548496774607864821834569063370205946820461 899021675321457546117438305008496860408846969491704367478991506 016646491060217834827889993818382522554582338038113118031805448 236757944990397074395466146340815553168788535030113821491411266 246328940130370152354936571471269917921021622688833029675405780 630706822368810432015790352123740735444602970006055250423142028 089193578811239731977974844235152456040926446709579570304658614 129566479666687743683240492022757393004750895311855179558720483 992696896827555852445024436526825609423780128033094877954403542 524859043379761802711830004573585550738941136758784400629135630 421674541694092135698603207859088199859359007319336801069967496 707904456092418632112054130547393985795544410347612222592136846 219346009360… (1018 meaningful digits)

CONSERVATIVE MAPS

BAKER MAP:

(t=0) (t=1)

q=0.85

q=0.90

q=0.95

q=1 q=1.05

q=1.20

q=0.80

ln W

slope = Lyapunov exponent = ln 2

G. Ruiz-Lopez and C. T. (2006)

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)]

0 20 40 60 80 100n 0

50

100

150

200

S q (n) q=-0.2

q= 0

q=+0.2

(a) 4000 4000 1000

100

[ 0 0.99993]

W cells N initial conditions randomly chosen in one cell Average 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

0 0

( ) lim 1

t

n

Also e S nwith

n

λξ

λ →∞

=

= =

q - generalization of Pesin (- like) theorem

G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)

( , ) qS N t versus N

( N = 0 )

( N = 1 )

( N = 2 )

1 1

1

1 1

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)

1 2

1 p

2 1- p

p 1- p 1

2p κ+ (1 )p p κ− −

(1 )p p κ− − 2(1 )p κ− +

A B

(N=2)

2 2

( 0) ( 1)

1 (

( 2)

1 1 1

1 1- )

[ ] [ (1 ) ] [(1 ) ]2 1 p p

p p p

N

p N N κ κ κ

= × = × ×

= × − −×+ − +×

EQUIVALENTLY:

100

90

80

70

60

50

40

30

20

10

0 1009080706050403020101

N S p q=1.0

q=0.9

q=1.1

(b) 20

10

0 20101

N S p

q=1.0

q=0.9

q=1.1

(c)

100

90

80

70

60

50

40

30

20

10

0 1009080706050403020101

N S p

q=1.0

q=0.9

q=1.1

(a)

1. ., 1 SY

STEM (

S ) ( )i e such that

q S N N N∝ → ∞

=

.0

1/ 2

N Np p

with p

Stretched exponential α

α

⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟= =⎝ ⎠

,0

1/ 2

N

Np p

N independen

with p

t coins

⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎝ ⎠

,0

1

1N

Leibnitz triangle

p N

⎛ ⎞=⎜ ⎟+⎝ ⎠

(All three examples strictly satisfy the Leibnitz rule) C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)

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)

. ., 1 S

YSTEM ( ) ( )

S qi e such that

q S N N N∝ → ∞

11q d

= −

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)

Continental Airlines

1( ) ( )

(

( ) ( ) ( )

( )

,

. ., ,

) ( ) ( )

( ) ( 1)

(

q q q q

BG B

q

A B A B ij 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 B k

S A S B if

p

then

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 B ea

A S B s

S B A S+

+ ≠ +

= +

ADDITIVITY:

additive probabilistically independent Hence, and ( ) are additive, and ( 1) is nonadd

An entropy is if,

for tw ( ) (

o systems and ,

it ) ( )

Renyi BG q q

S A B S A S S

B A

q S

B

S q∀ ∀ ≠ + = +

ive .

EXTENSIVITY:

1 2

1 2 Consider a system made of (not necessarily independent) identical elements or subsystems and , ..., . An entropy is if

... extensive

0 lim

N

N

N

N A A

A A

A AΣ

→∞

≡ + + +

<

Re

Re

The and are the subsystems ar

( ) , . ., ( ) ( )

ad e

(strictly or as

ditive entropies ext

ymptotically) indep

ensive if an

endent; oth

d

erwise, a

only if

nd ar

e no

nyi BG q

nyi BG q

S S

S N i e S N N

S S

N

N

N< ∞ ∝ → ∞

nonadditive entropy extensive for special

nextensive.

The ( 1) is if the

subsystems are specially (globall

valu

y) c

es of

orrel

ated. q q qS

CONSEQUENTLY:

Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon, Oxford, 1970), page 167

O. Penrose,

MEPHISTOPHELES:

Denn eben wo Begriffe fehlen,

Da stellt ein Wort zur rechten Zeit sich ein.

Wolfgang von Goethe

[Faust I, Vers 1995, Schuelerszene (1808)]

For at the point where concepts fail,

At the right time a word is thrust in there.

King Thutmosis III 18th Dynasty c. 1460 B. C.

BG

1 1

( ) ( 0, 1, ) ( ) ln [ ( )] (ln ) S !

( ) [ ( )]

1)

1ln [ ( )]

TYPICAL EXAMPLES FOR THE CASE OF EQUAL PROBABILITI

,

1

1

. .,

S:

E

N

BG

q q q

q

W N A A N S N W N

If

then iN is extensive k S N W N AW N

e

wher k q

eas

μ μ

μ

− −

∼ > > → ∞

=

− = =

− −

∼ (1 )

,

. 1 ( 1).

( )

.,

2)

( 0, 0, ) ( ) ln [ ( )] ln , . .,

N q

q

BG BG

q S is nonextensive for q and neither is for q

W N B N B N S N W N N

constant N

i e

If

then i e S is nonex k

ρ

μ

ρ

ρ

< >

> > → ∞

=

1 1 (1 )

11 , . .,

3)

( ) [ ( )] 1ln [ ( )] ! 1 1

( ) ( 0, 1, 1, , ( 1 1

) )

q q q q

q

N

tensive

S N W N BW N Nwhereas i e

If Hence there is

S is extensive k q q

W N C C N no value of q neither q nor q for which S

ρ

ρ

γ μ μ γ

− − −

− = =

− −

∼ =

> ∞ ≠

> ≠ →

.q is extensive

(Bayes theor( ) ( ) ( | ) ( | ) ( )

( ) ( ) ( | ) ( ) ( | ) (1 )

:

e

)

m

?

ij i j i j

q q q q q

We consider generic subsystems A and B

impli

p A B p A p B A p A B p B

S A B S A S B A S A S B A q

k es

k k k k

HOW IT WORKS

+ = =

+ = + + −

( ) ( | ) ( ) ( | ) (1 )

( ) ( ) ( | ) ( ) ( | )

( ) ( | ) ( )

)

1

(

|1

q q q q

q q q q q

q q q q

S B S A B S B S A B q

k k k k

hence S A B S A S B A S A S B A

S

A special class of correl

B S A B S B S A

ations

B

e

k

x

q

q k

= + + −

+ = +

+ + −

+

=

( | ) ( ) ( | ) ( )

1

1

( | ) ( ) ( | ) (

,

(

,

)

) (

q q q q

q q q q

q q

S A B S

ists such that for a spec

B S A B S A

and

S B A S A S B A S B

henc

ial val q

k

q k

Se A B

ue o q

S

f

+ =

+

+

=

= −

) ( ) (extensivity!)qA S B+

S. Abe, Phys Lett A 271, 74 (2000) and Physica A 368, 430 (2006)

A MANY-BODY HAMILTONIAN ILLUSTRATION OF THE EXTENSIVITY OF Sq FOR

ANOMALOUS VALUES OF q

SPIN ½ XY FERROMAGNET WITH TRANSVERSE MAGNETIC FIELD:

| | 1 0 | | 1

0

Ising ferromagnet anisotropic XY ferromagnet isotropic XY ferrom

transverse magnetic field L length of a block within

ag

a N cha

n t

in

e

γ γ

λ

γ

= → < < → =

≡ → ∞

F. Caruso and C. T., cond-mat/0612032

F. Caruso and C. T., cond-mat/0612032

( ) ( ) ent entq qS L s L L → ∞∼

Ising model

XY model

37 6 entq = − →

F. Caruso and C. T., cond-mat/0612032

1 ( ) ( ) 3.56 0.03 ent ent entq q qS L s L L with sλ = ⇒ → ∞ = ±∼

F. Caruso and C. T., cond-mat/0612032

2

. . , 06002 (2004)

9 3

,

,

ent

Using a Quantum Field Theory result in P Calabrese and J Cardy JSTAT P we obtain at the critical transverse magnetic field

with c

cq c

cent

+ − =

1 2

1

37 6 0.0828

10

3 0.

16

ent

ent

Ising and anisotropic XY ferromagnets c

Isotropic XY fer

in conformal f

romagne

ral charge

q

ield theory

Hence

and

c qt

⇒ = ⇒

⇒ = −⇒

=

=

F. Caruso and C. T., cond-mat/0612032

F. Caruso and C. T., cond-mat/0612032

1

1 1 1 1

(1 ) (1 ) 2 N

x x y y z z z i i i i i i i

i

H γ σ σ γ σ σ Δ σ σ λ σ −

+ + + =

⎡ ⎤= − + + − + +⎣ ⎦∑

29 3( )ent cq c c

+ − =

(| | 1; | | 1; 0)

(1/ 2) 37

( 0; | | 1; 0)

6 0.0828ent Ising a q

nd anisotropic XY ferromagnets

γ λ Δ γ λ Δ

= = = −

= ≠ = =

( 0; | | 1; 0) ( 0; 0;

(1) 10 3 0.1

1)

6

ent

Isotropic XY and He

q

isenberg ferromagnets

γ λ Δ γ λ Δ

= − = ≤ =

= = =

BG

c = 26: P. Ginsparg and G. Moore Lectures on 2D Gravity and 2D String Theory (Cambridge University Press, Cambridge, 1993) [hep-th/9304011, page 65]

2 19 3 ( )

(extensive!)

(nonextensive!)

In

other words,

where

( ) ln

s

a

BG

c c S L L

S L L

⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦

−+ − ∝

REVISITING THE DIFFUSION OF ONE ELECTRON

IN A MANY-BODY QUANTUM HAMILTONIAN

1D ANDERSON MODEL WITH LONG-RANGE CORRELATED DISORDER (METAL-INSULATOR TRANSITION):

F.A.B.F. de Moura and M.L. Lyra, Phys Rev Lett 81, 3735 (1998)

[ ]H

- One electron in a disordered linear chain [the disorder comes from a randomly- correlated potential characterized by a spectral density ( )

| | | 1| | 1|

( :

]

0)

nH n n t n n n

s k k

n

α

ε

α

= >< + >< + + >< −

∝ ≥

H 1

1/ 2/ 2

1

1

2H 1 (H Hurst exp

( 1)

2 cos

/ 2 [0, 2

onent)

0

]

| 2 / |

N

n n

N

n k k

k

t

nkwith k N

where are N independent rand

standard Anderson mo

om phases uniformly distributed i

de

n

Nα α πε

α

π

α

Φ

Φ

π

=

=

− −

=

⎛ ⎞⎡ ⎤= +⎜ ⎟⎣ ⎦ ⎝

= + ≡

= ⇒

∑ ∑

2 ' , '

[ , . ., ), . ., ]

( ) n n n n n

l white noise spectrum i e no correlation from site to site i e crystal no disorder

ε ε ε δ

α

< >=< >

→ ∞ ⇒

F.A.B.F. de Moura and M.L. Lyra, Phys Rev Lett 81, 3735 (1998)

Zero “Lyapunov exponent”

Positive “Lyapunov exponent”

(8000, )BGS t

( , 1)BGS N t >>

B. Santos, L.P. Viana, M.L. Lyra and F.A.B.F. de Moura, Sol State Comm 138, 585 (2006)

0 5000 10000 15000 N

0

5000

10000

15000

20000

S

q

(

N

,

8

0

0

0

)

q=0.0 q=0.025 q=0.05 q=0.1 q=0.2

0 5000 10000 15000 6

7

8

9

10

q=1

α=3

0 2000 4000 6000 8000 t

0

5000

10000

15000

20000

25000

S

q

(

1

6

0

0

0

,

t

)

q=0 q=0.025 q=0.05 q=0.1 q=0.2

0 2000 4000 0

2

4

6

8

q=1

0 0.1 0.2 0.3 0.4 0.5 t/N

0

0.2

0.4

0.6

0.8

1

S

0

(

t

,

N

)

/

N

N=4000 N=8000 N=16000

(q = 0)

F.A.B.F. de Moura and M.L. Lyra (2007)

( 3)α =

REVISITING THE PREVIOUS RESULTS:

( , ) ( , )qS N t versus t N

0 10 20 30 40 TIME

0

100

200

300

400

500

600

E

N

T

R

O

P

Y

0 10 20 30 40 0

2

4

6

0.0 0.5 1.0 0.0

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 <1sen

0 2000 4000 6000 8000 10000

2000

4000

6000

8000

10000

N

S p

(b)

q=1/2-0.1

q=1/2

q=1/2+0.1

q <1sen

100

90

80

70

60

50

40

30

20

10

0 1009080706050403020101

N

S p

q=1

q=0.9

q=1.1

(a)q =1sen

C.T., M. Gell-Mann and Y. Sato, Europhysics News 36 (6), 186 (2005) [European Physical Society]

fixed N’>N

fixed N

S p

q<q sen q=q sen

q>q sen

0 0

t

q=q sen

ε ’< ε

ε’ < ε

ε ’< ε

ε

ε

ε

ε 0ε 0

ε 0

RELAXATION

SENSITIVITY

STATIONARY STATE

ε 0

C.T., M. Gell-Mann and Y. Sato, Europhysics News 36 (6) (Nov-Dec 2005) Special Issue Nonextensive Statistical Mechanics – New Trends, New Perspectives (European Physical Society)

, . ,

) , ( , )

( , ):

sen se

sen

q q

q

For q q N and t play essentially the same role In particular

i Under conditions of graining in phase space S N t K

infinitely fine

A conjecture for S N t

= → ∞ → ∞

( ) 1

( ) 1 1

/ 0

( )

) ,

( 1 , . ., Pesin )

lim ( , )

n

se

j

nt q

j sen

j

N t N t

ii Under con

q K i e like identity for fini

ditions of graining in phas

te N

e space S N t N

finite

λ

λ

→∞

>

= ⇒ =

−∑

(Cl

aus

ius)

C.T., M. Gell-Mann and Y. Sato Europhysics News 36, 186 (2005)

NONEXTENSIVE STATISTICAL MECHANICS AND

THERMODYNAMICS

C. T. Possible generalization of Boltzmann-Gibbs statistics J Stat Phys 52, 479 (1988)

E.M.F. Curado and C. T. Generalized statistical mechanics: connection with thermodynamics J Phys A 24, L69 (1991) [Corrigenda: 24, 3187 (1991) and 25, 1019 (1992)]

C. T., R.S. Mendes and A.R. Plastino The role of constraints within generalized nonextensive statistics Physica A 261, 534 (1998)

NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS (CANONICAL ENSEMBLE):

1

1

1

1

1 [ ]

1

1

W q i

i q i

W q i iW

i i qW

qi i

i

q i

Extremization of the functional

with the constraints an

p S p k

q

p E d

yi

p

el

p U

ds e

p

=

=

=

=

− ≡

= =

=

∑ ∑

( )

1

1

( )

, , q i q W

E U q q qW

q i i

i

q qiE U

q

energy Lagwith andrange parameter e p

β

β

ββ β − − =

=

− −

≡ ≡ ≡ ∑ ∑

Z

Z

'

'

' '

1

'

1 (1 )

1 1( )

1( ) ln

,

ln ln

(

q i

q

W Eq

q q q iq q

q

q

q q q q q q q q q q

i

q

E q

iWe can rewrite

with and

And we can

Z e q U

S i T

T U k

ii F U T

p

S Z Z

rove

with

U

i

h e

Z

w er

e p

β

β

β β

β

β

β β

=

≡ ≡ + −

∂ = ≡

≡ − = − = −

=

Z

2

2

( . .,

) ln

( )

-

!)

q q q

q q q q

i e the Legendre structure of Thermod

ii U Z

S U F iv C T T

T T ynamics is q invariant

T

β ∂

= − ∂

∂ ∂ ∂ ≡ = = −

∂ ∂ ∂

WHY USING ESCORT DISTRIBUTIONS FOR THE CONSTRAINTS? 1) The optimizing probability distribution is automatically invariant with regard to uniform tra zero-point invariancenslation of the energy eigenvalues ( ).

2) The constrain 1 ts and i i i i i

p P E co= =∑ ∑

{ } { }

1/

1/

2

, where and ,

are up to a upper bound for ( . , for ( ) , it must be 3).

3) yields instead of in

finite common

( ) steepest-desthe i i

q q i i

i iq q j j

j j

x q x q x

q q

q e g E x x q

p Pnstant P p p P

de P pe e

dx

<

≡ ≡

=

∑ ∑

calculation

of the stationary-state distribution [ , 33, 87

cent-method

conditional entr

33 (2000)].

4) The naturally appears [ , 271, 74 (2000)] as a

opy

Abe and Rajagopal J Phys A

Abe Phys Lett A -expectation value .

5) The principle of minimal relativ

without involving any optimization principle

consistent as a rule of statistical inference entropy is (1980 Shore-Johnson axioms) only

e if

q

[ , Re 71, 016139 (2005)] we select the -expectation values if we use the entropy .q

Abe and Bagci Phys v E q S

GENERALIZATION OF THE CENTRAL LIMIT THEOREM

ONE OF MANY CONNECTIONS OF THE CENTRAL LIMIT THEOREM WITH BOLTZMANN-GIBBS STATISTIC

- ( ) ln[ ( )]

( )

AL MECHANIC

1

S

Optimization of

S k dx p x p x

with

dx p x

and

=

=

( )

( )

2

(Boltzmann-Gibbs distributio

( ) ( ) ( )

( )

0

n for thermal equilibriu

m

)

Example:

E x

E y

E x dx p x E x constant

yields

ep x dy e

x and x constant yields

β

β

≡ =

=

= =

2

2 ( ) (Gau ssian distr

ibution) x

y ep x

dy e

β

β

− =

D. Prato and C. T., Phys Rev E 60, 2398 (1999)

q-GAUSSIANS: 2

2( / ) 1 -1

1( ) ( 3) 1 ( -1) ( / )

q x

q q

p x q q x

e σ

σ

−∝ ≡ < ⎡ ⎤+⎣ ⎦

q - CENTRAL LIMIT THEOREM:

2( , ) [ ( , )] (0 2; 3) | |

( 1)

qp x t p x tD q xt

α

α

β

β α

β

−∂ ∂ = < ≤ <

∂∂

= independent variables; divergent variance; Levy attractor

independent variables; finite variance; Gaussian attractor

globally correlated variables; finite q-variance; q -Gaussian attractor

M. Bologna, C. T. and P. Grigolini, Phys. Rev. E 62, 2213 (2000)

C. T., Milan J. Math. 73, 145 (2005)

(conjecture)

a

(q,a) - attractor

q - PRODUCT: L. Nivanen, A. Le Mehaute and Q.A. Wang, Rep. Math. Phys. 52, 437 (2003) E.P. Borges, Physica A 340, 95 (2004)

1 1 1 1

1 1 1 1 11

1 1

( , ) ( )

1 1, 0 0, 0

| | |

1 1

| 1

q q q q

q q q

q

q q q

q

x y if x y qFor x and y x y if x y

It can be extended to all x y through x y sign xy

a q

x

nd

y

− − −

− −

⎧ ⎪⎡ ⎤+ − + > ∀⎣ ⎦≥ ≥ ⊗ ≡ ⎨ ⎪

⎛ ⎞ ⎡ ⎤⊗

+ ≤ <⎩

= + −⎜ ⎣ ⎦ ⎝

1

2

1 1 1 1 1

) )

[ ln ( ) ln ln (1 )(ln )(ln )]

)

)

)

ln ( ) ln ln

1 1 1

( ) ( ) 2

:

q q q q

q

q q

q

q q

q q q q q q q q

q

q

q q

q

i ii

whereas x y x y q x y

iii

iv

v

Properti x y x y

x y x y

x y x y

x y y x

x y z x y z x

v

y z

i

es

z x y

− − − −

⊗ = ⊗ = +

= ⊗ ⊗

⊗ = ⊗

⎡ ⎤⊗ ⊗ = ⊗ ⊗

⎟ ⎠

= + + −

= ⊗ ⊗ = + + −⎣ ⎦

1 1 1

1

0 ( 1

)

) 0) ( 1 0 1)

0 = ( 1)

q

q q q

x x

if q and x vii

or if q and x x

x otherwise− −

⊗ =

≥ ≥ < ≤ ≤⎧⎪⊗ ⎨ ⎪ −⎩

C. T. and S.M.D. Queiros (2007)

C. T. and S.M.D. Queiros (2007)

1q =

5q = −

2q =

0q =

q → −∞

q → ∞

C. T. and S.M.D. Queiros (2007)

q - CENTRAL LIMIT THEOREM (q-product and de Moivre-Laplace theorem):

1 1 1 1

1

: ) )

[ ln ( ) ln ln (1

1

)(ln )(ln ) ln ( ) n ln

] l

q

q q q q q

q q q

q q q q

Properties i ii

whereas x y

x y x

x

y

x y x y x y

q x y

y x y

− − −

= +

⎡ ⎤⊗ ≡ + −⎣ ⎦

⊗ =

+

⊗ =

+

The q - product is defined as follows:

The de Moivre-Laplace theorem can be constructed with

,0 1/ 2

N Np p with p

Leibnitz and

rule

= =

0 0.2 0.4 0.6 0.8 1 q

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

q

e

q e =2-1/q

0.5 0.6 0.7 0.8 0.9 1 q

0

0.2

0.4

0.6

0.8

1

q

e

0 0.2 0.4 0.6 0.8 1

x 2

-0.4

-0.3

-0.2

-0.1

0

l

n

-

4

/

3

[

p

(

x

)

/

p

(

0

)

]

N=50 N=80 N=100 N=150 N=200 N=300 N=400 N=500 N=1000

0 0.02 0.04 0.06 0.08 1/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

1 1 1

,0

. .

1 1 1 1 ... ( )

( 1)

,

( 1/ 2)

q q N

q q N

We q generalize the de Moivre Laplace theorem with

i

N terms p 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

L.G. Moyano, C. T. and M. Gell-Mann, Europhys Lett 73, 813 (2006)

0 0.2 0.4 0.6 0.8 1 1.2

x 2

-0.4

-0.3

-0.2

-0.1

0 l

n

-

4

/

3

[

p

(

x

)

/

p

(

0

)

]

-1 -0.5 0 0.5 1 x=(n-N/2)/N/2

0

0.5

1

1.5

2

p

(

x

)

*

N

/

2

(q=3/10; p=1/2)

(N=200)

L.G. Moyano, C. T. and M. Gell-Mann, Europhys Lett 73, 813 (2006)

0 0.2 0.4 0.6 0.8 1

x 2

-3

-2.5

-2

-1.5

-1

-0.5

0 l

n

2

/

3

[

p

(

x

)

/

p

(

0

)

]

N=10 N=26 N=50 N=100 N=300

0 0.05 0.1 1/N

1

1.5

2

2.5

3

β(N)

(q=3/4; p=1/2)

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]

1

1[ ( )] (nonline

-Fourier transform:

-independen

[ ]( ) ( ) = ( )

[ c

( ) ] [ ( e:

)

)

]

ar!

X Y

q ix

f xix q q q qF f f x dx f x dx

Two random variables X with densit

q

y f x and Y with density f y having q

e e ξ

ξξ ∞ ∞

−∞ −∞

− ⊗≡ ∫ ∫

1

1

1

(1 ) /(3 )

-

1[X+Y]( ) = [X]( ) [Y]( ) , 3

. .,

( ) ( ) ( ) ,

q q q

X Y X q q Y

q

q q q iyiz ix

q q q

zero q mean values are said q independent if

qF F F q q

i e if

dz f z dx f x dy f y

wi

e e e ξξ ξ

ξ ξ ξ

∞ ∞ ∞

+ + −−∞ −∞ −∞

⊗⊗ ⊗ ⊗

⎛ ⎞+ ≡⎜ ⎟−⎝ ⎠

⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫ ∫ ( ) ( , ) ( ) ( , )

- 1 , . .,

( , )

( , ) ( ,

. ) ( ) ( )

X Y

X Yth f z dx dy h x y x y z dx h x z x dy h z y y

where h x y is the joint densi q independence means

ty independence if q i e h x y f x f y

δ ∞ ∞ ∞ ∞

+ −∞ −∞ −∞ −∞ = + − = −

= =

= −∫ ∫ ∫ ∫

1 , ( , ) ( ) ( )X Yglobal correlation if q hence h x y f x f y ⎛ ⎞ ⎜ ⎟≠ ≠⎝ ⎠

( )

1 2

1 2

2 12 2(1 ) 2(1 )

1

1

1

1

22

1 3

3 3 ( ) 8 8

12 1

3(3 ) (1 ) 2(

q

q q

q qq q q

q

q

t q q

qwhere q q

q Fouri

q qand K q C C

q qq q

with C

erTransform C

e e βββ ω

β β β β

π Γ

Γ

− −

− − −

−−⎡ ⎤ =⎢ ⎥ ⎢ ⎥⎣ ⎦

+ =

⎡ ⎤− − = ⇔ = ≡⎢ ⎥

⎢ ⎥⎣ ⎦

⎛ ⎞ ⎜ ⎟−⎝ ⎠

− − −

=

1

1 )

1

3 2( 1)

1 3 11

1

if q

q

if q

q q

if q q

q

π

π Γ

Γ

⎧ ⎫ ⎪ ⎪ ⎪ ⎪< ⎪ ⎪⎛ ⎞ ⎪ ⎪⎜ ⎟−⎝ ⎠⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎪ ⎪⎛ ⎞−⎪ ⎪⎜ ⎟⎪ ⎪−⎝ ⎠ < <⎪ ⎪⎛ ⎞⎪ ⎪− ⎜ ⎟⎪ ⎪−⎝ ⎠⎩ ⎭

Heisenberg-like uncertainty

( )

1 2

1 2

2 1 2(1 )

3 ( ) 8

q

q q

q q

q K q C

β β

− −

⎡ ⎤− = ≡⎢ ⎥

⎢ ⎥⎣ ⎦

C. T. and S.M.D. Queiros (2007)

1

1

, | |

-

[ 0, 0, 0 2, ( 1) /(3 )]

( , ) -

( )

q

b q

A random variable X is said to have a

if its q Fourier transform has the form

a

q stable distribut

b q q q

ion L x

a e α α

ξ

α

α −

> > < ≤ ≡ + −

S. Umarov, C. T., M. Gell-Mann and S. Steinberg (2006)

cond-mat/0606038 and cond-mat/0606040

1

1

[ ( )], , , ,

1,2

1,

,2

| |

. .,

( )

)

)

= ( ) =

[ ]( )

( ) ( ) ( )

( ) ( ) (

( ) ( ) (

q ix

L xqix q q q q qq q

qq

b q

i e if

L x dx L x d

stable Levy distribution

q Gau

F

si n

x

s a

L

L x G x Gaussian

L x L x

L x G x

e e a e ξ

α α

ξ α α

α

αα

ξξ

α

−∞ ∞

−∞ −∞

⊗ −≡

∫ ∫

1 0

- - ( ) -

1( ) (- ,

CLOSURE:

ITERATION:

3) 3-

2 (1 )( ) ( ( )) ( 0, 1, 2,...; 2 (1

)

n n n

The q Fourier transform of a q Gaussia qz q q

q n qq z q z

n is a z q Gaussian wit

z q n q n q

h

+ = ∈ ∞

+ − ≡ ≡ = = ± ± =

+ −

1 1

(as in R.S. Mendes and C.T. [Phys Lett A 285, 273 (2001)] calculating marginal probabilities!)

(as in L.G.

)

(i) (1) 1 ( ), ( ) 1 ( ),

1

Moy

a

(i

no

i

) 2

,

,

n

n n

hence

q

q n q q q

q q

±∞

− +

= ∀ = ∀

= −

( ) 2

C.T. and M. Gell-Mann (2005)!) (as in A. Robledo [Physica D 193, 153 (2004)] for pitchfork an

(1 ) (iii) 2 =0, 2, 4, ... 1 (1 )

d tangent bifurcations!

)

m m q m qn m qyields q

m q + −

= ± ± ≡ = + −

(as 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 -triangle for m = 0, 1!)q ±

1 1 2

1 1

1 3 2

( 1.75 )

( 4 )

( 0.5

2,

)

k n stat k n

k n rel k n

k n sen k n

for solar wind

for solar wind

for solar wi

q q q

q q q

q q q

For the following properties

nd

are saα

≡ − − −

≡ − +

≡ − + +

→ ⎯⎯⎯→

→ ⎯⎯⎯→

→ ⎯⎯⎯→ −

=

∼ ∼ ∼

1 2

1 1 (2 ) (2 )

1 2(2

: 1 2

1 2

, , 2,

k n

stat rel

rel sen

q q

tisfied

q q

q q

The scaling of the sum of the random variables is given by

N N

N

hence for

α α

α

− −− −

+ =

+ =

= =

1 1) 2 2

k n k

q q q N N

+ − = =

,

1 1

( 0, 1, 2,...)

n n

q q

n

α

α α = +

− −

= ± ±

S. Umarov, C.T., M. Gell-Mann and S. Steinberg (2006), cond-mat/0606040

ALGEBRA ASSOCIATED WITH q-GENERALIZED CENTRAL LIMIT THEOREMS:

1 [ ]q independent= 1 ( . ., 2 1 1) [ ]q i e Q q globally correlated≠ ≡ − ≠

1

( ) ( )

(

Classic CL

)

,

T

with same of x Gaussian G x

f xσ =F

<

( 2) Qσ

α

=

(0 2)

Qσ α< < → ∞

1/[ (2- )] 1

CENTRAL LIMIT THEOREM

-

-

(

)

( )

q scaled attractor when summing N independent identical random variables

with symmetric distribut

x

ion

N q

f x

α → ∞F

2 1

1 [ ( )] / [ ( )] 2 1, 3

Q Q Q

qwith dx x f x dx f x Q q q q

σ ⎛ ⎞+

≡ ≡ − =⎜ ⎟−⎝ ⎠ ∫ ∫

2

1

| |

( ) | |

( )

Levy-Gn

(1, ) L ( )

( ) / | | | | (1, )

lim (1, ) edenko

( ),

CLT

c

c

c

with same x behavior

G x if x x

x f x C x

x Levy distributio

if x x with x

n L x

α

α α

α

α α

α α→

+

→ ∞

⎧ ⎪ <<⎪ ⎨ ⎪ ⎪

=

>>⎩ = ∞

∼ ∼

F

( ) ( )1 11

1

3 / 1

2/( 1)

( )

( ) | | ( , 2)

( ) / | | | | ( , 2)

( ) ,

lim ( , 2

)

( )

( ) (

c

c

q c

q qq Q

q q q

with same of f x

G x if x x q

f x C x if x x q

with q

x

x

G

G x G x

x

σ+

<<⎧ ⎫⎪ ⎪ ⎨ ⎬

>>⎪

=

⎪⎩ ⎭

∼ ∼

F

S. Umarov, C. T. and S. Steinberg (2006) [

)

con

d-ma

t/060359

3

]

= ∞

( ) ( ) ( ) ( )

( ) ( ) ,

2 1 3 *

2 1 1 ,

)

2

2 1 3

( 1 ,

2 3, 21 ,

, (1 ) /(1

( | |

/ | |

/

(

~

( )

) ,

)

| |

( )

L

q q

q

q

q

q

q q

qq

q

q

q

w ith sx L

interm edi

G x

at

am e x asym ptotic behavior

C

e regim e w i

xx

x

C

th L

G

α α

α

α

α α α

α

α

α

α α α

− − −

− − −

− +

− −

− + +

+ + −

→ ∞

=F

)

S . U m arov, C . T ., M . G ell-M ann and S . S teinberg (2006) [cond-m at/0606038] and [cond-m at/0

( ) distant regim e

α

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

606040]

-1 | |(

)

p x Fourier e αξ−⎡ ⎤∝ ⎣ ⎦

STANDARD AND LEVY-GNEDENKO CENTRAL LIMIT THEOREMS:

1.999

1.5

999 2

1.9

)

1. 9

(

9

Gaussian

α

α

α α

α =

=

=

= =

C. T. and S.M.D. Queiros (2007)

N INDEPENDENT RANDOM VARIABLES

N=1: q-Gaussian with q = 3/2 = 1.5

C. T. and S.M.D. Queiros (2007)

N INDEPENDENT RANDOM VARIABLES

N=1: q-Gaussian with q = 9/5 = 1.8

C. T. and S.M.D. Queiros (2007)

q=5/3, =2 (5/3-Gaussian)α q=5/3, =1.99α q=5/3, =1.9α

q=1.0, =2 (Gaussian)α

log X

P (X)log P (0)

q q q

= -( ) - -( )

-( ) 2 1 3

2 1 αslope

q =

- 2

1 slope slope

q = +

+ -( ) α α

1 1 1

2 2(1 ) (3 ) 1 ( 1) 2(1 ) 1

q q q q q

α α α α

− − − + ≥ >

− − + −

C. T. and S.M.D. Queiros (2007)

q-CENTRAL LIMIT THEOREMS: 1

-1 ,

| |( ) ( ) (

- )

q qp x L x q Fourier be

α

α β ξ−⎡ ⎤= = ⎣ ⎦

2 1 1 12

3 -

attractorcorrelation Gaussian

q q q

tail exponentq qα

+ 2 1

2(1 ) (1 ) 2(1 ) (3 ) ( ) 2(1 ) (3 ) 2(1 )1 2 3 13- ( )

1 1 -

q q q q q intermediate regime q q qq

qq distant regime q

α α α

α α α α

α α α

− − + − − −⎧ ⎫ ⎪ ⎪− − − −+ ⎪ ⎪ ⎨ ⎬− + +⎪ ⎪ ⎪ ⎪+ +⎩ ⎭

Convolution of (5/3)-independent random variables following a 3/2-Gaussian

1 0 1 3

qq q q q

+ = ≡

2 1/( 1)

1 [1 ( 1) ( ) ]

(1) 1

q x

x

q N xβ β

−∝ + − =

C. T. and S.M.D. Queiros (2007)

Convolution of (5/3)-independent random variables following a 3/2-Gaussian

C. T. and S.M.D. Queiros (2007)

Convolution of (5/3)-independent random variables following a 3/2-Gaussian

ln3/2

C. T. and S.M.D. Queiros (2007)

0 50 100 150 200 250 300 0

50

100

150

200

250

300

β−1

N 2

1 21/ ( )

q x N Nβ

−=

xx N=16

N=1,2 N=4

N=8

N=16

N=8

N=4

N=2

N=1

C. T. and S.M.D. Queiros (2007)

3/2-Fourier Transform of PN(x)

C. T. and S.M.D. Queiros (2007)

3/2-Fourier Transform of PN(x)

C. T. and S.M.D. Queiros (2007)

0 2 4 6 8 10 12 14 16 18 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

βk

N

slope = 1

βk for the 3/2-Fourier Transform

2 2( 1)

3( ) ( ) 8[ (1)]k q qx q

qN N q C

β β − −

− = ∀

C. T. and S.M.D. Queiros (2007)

Convolution of (7/3)-independent random variables following a 9/5-Gaussian

C. T. and S.M.D. Queiros (2007)

Convolution of (7/3)-independent random variables following a 9/5-Gaussian

C. T. and S.M.D. Queiros (2007)

0.0 2.0x105 4.0x105 6.0x105 8.0x105 1.0x106 1.2x106 0.0

2.0x105

4.0x105

6.0x105

8.0x105

1.0x106

1.2x106

0 20 40 0

20

40

N 5

β−1

x N=16

N=8

N=2

N=1

N=1,2,4

C. T. and S.M.D. Queiros (2007)

7/3-Fourier Transform of PN(x)

C. T. and S.M.D. Queiros (2007)

7/3-Fourier Transform of PN(x)

C. T. and S.M.D. Queiros (2007)

1 2 1 2

( )

( , ,..., ) ( , ,...,

: ?

N N

N body joint probabili

q INDEPENDENCE IT CORRES

It appears to be no proof available yet

PONDS TO WHA

dx h x x x h x

T

x

ty

=∫ 1

. .,

!

)N

i e scale invariance

x

MORE ON THE NATURE OF q-CORRELATION: W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

1 1

1 1

1

ρρ ρ

ρ ρρ

Z

1 1/ 2 1/ 2

( ) 0

,

Let us consider N correlated random variables if x

f x otherwise

Correlation emerges through the following multivariate Gaussian N N covariance matri

un f r

x

i o m

an

− ≤ ≤⎧ ⎫ = ⎨ ⎬

⎩ ⎭

× ( ) :

( 1 1)

d probability integral transform component by component

ρ− ≤ ≤

0 1

independence full correlation

ρ ρ

= ⇒ = ⇒

-0.5

0

0.5

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

Marginal U1

Bivariate Uniform Distribution, ρ=0

Marginal U2

f

r

e

q

u

e

n

c

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0

0.5

1

Marginals U1, U2

x

f

U

1

(

x

)

,

f

U

2

(

x

)

U1 Histogram

U2 Histogram

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0

0.5

1

Usum=U1+U2, r=0.011257

x

f

U

s

u

m

(

x

)

U1 + U2 Histogram

-0.5

0

0.5

-0.5

0

0.5

0

0.5

1

1.5

2

Marginal U1

Bivariate Uniform Distribution, ρ=0.1

Marginal U2

f

r

e

q

u

e

n

c

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0

0.5

1

Marginals U1, U2

x

f

U

1

(

x

)

,

f

U

2

(

x

)

U1 Histogram

U2 Histogram

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0

0.5

1

Usum=U1+U2, ρ=0.1

x

f

U

s

u

m

(

x

)

U1 + U2 Histogram

( 2; 0)N ρ= =

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

( 2; 0.1)N ρ= =

-0.5

0

0.5

-0.5

0

0.5 0

1

2

3

4

5

6

7

Marginal U1

Bivariate Uniform Distribution, ρNormal=0.5

Marginal U2

f

r

e

q

u

e

n

c

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0

0.5

1

Marginals U1, U2

x

f

U

1

(

x

)

,

f

U

2

(

x

)

U1 Histogram

U2 Histogram

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0

0.5

1

Usum=U1+U2, ρ=0.5

x

f

U

s

u

m

(

x

)

U1 + U2 Histogram

-0.5

0

0.5

-0.5

0

0.5 0

5

10

15

20

25

Marginal U1

Bivariate Uniform Distribution, ρ=0.9

Marginal U2

f

r

e

q

u

e

n

c

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0

0.5

1

Marginals U1, U2

x f

U

1

(

x

)

,

f

U

2

(

x

)

U1 Histogram

U2 Histogram

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0

0.5

1

Usum=U1+U2, ρ=0.9

x

f

U

s

u

m

(

x

)

U1 + U2 Histogram

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

( 2; 0.5)N ρ= =

( 2; 0.9)N ρ= =

-1.5 -1 -0.5 0 0.5 1 1.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=2 q

∞ =-0.27112 β∞=0.675

Empirical Fitted

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=3 q

∞ =-0.042292 β∞=0.39472

Empirical Fitted

-3 -2 -1 0 1 2 3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=5 q

∞ =0.15505 β∞=0.18051

Empirical Fitted

-4 -3 -2 -1 0 1 2 3 4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=6 q

∞ =0.19471 β∞=0.13266

Empirical Fitted

-15 -10 -5 0 5 10 15 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=20 q

∞ =0.31718 β∞=0.014581

Empirical Fitted

-15 -10 -5 0 5 10 15 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF system size=24 q

∞ =0.32112 β∞=0.010238

Empirical Fitted

(N=2 ) (N=3 )

(N=5 )

(N=6 )

(N=20 ) (N=24)

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

PDF of Umean: Analytic Representation and Fitted qGaussian

System Size=2000 ρ=0.5 Fitted q=0.354122

fitted analytical

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

PDF of Umean: Analytic Representation and Fitted qGaussian

System Size=1000 ρ=0.5 Fitted q=0.353723

fitted analytical

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

10-4

10-3

10-2

10-1

100

101

PDF of Umean: Analytic Representation and Fitted qGaussian

System Size=3000 ρ=0.5 Fitted q=0.354256

fitted analytical

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

10-4

10-3

10-2

10 -1

100

101

PDF of Umean: Analytic Representation and Fitted qGaussian

System Size=5000 ρ=0.5 Fitted q=0.354363

fitted analytical

(N=1000) (N=2000)

(N=3000) (N=5000)

( 0.5) ρ =

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

-40 -30 -20 -10 0 10 20 30 40 10-6

10-5

10-4

10-3

10-2

10-1

Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF: UNIFORM system size=100 ρ=0.2 q∞=0.8347 β∞=0.0024114

Empirical Fitted

(N=100 )

(q = 0.8347)

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

( ) ( )( , ) ( )

(0.5) 0.3545 . ., (0.5) 0.5338

(0.5) 1.9535

Aq N q N

q e g A

δ ρ ρρ ρ

δ

= −

⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -12

-10

-8

-6

-4

-2

0

2

ρ

F

i

t

t

e

d

q

q Vs ρ, System Size=1000

Fitted q y=(1-φ⋅ρ)/(1-ρ)

Fitted q Value as a function of correlation coefficient, ρ . The ansatz appearing in the legend is the golden ratio, ( )/251+=ϕ .

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

1

1 1 (5/ 3)( ) 1 1

q ρ ρρ ρ ρ∞

− Φ − =

− −

0 0.2

0.4 0.6

0.8 1

0

0.5

1

1.5 -6

-4

-2

0

2

ρ

Fitted q as a function of Normal Correlation (ρ) and Scaling Exponent (α) System Size = 35

α

1 (2) (3) ... ( -1) ( ) (2) 1 (2) ... ( - 2) ( -1) (3) (2)

Multivariate Gaussian N N covariance matrix given by

N N N N

ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

×

1 ... ( -3) ( - 2) ... ... ... ... ... ...

( -1) ( - 2) ... (2) 1 (2) ( )

N N

N N N

ρ ρ

ρ ρ ρ ρ ρ ( -1) ... (3) (2) 1

( 1 1; 0; 2, 3,..., )

( )

N

with

r N

r r

ρ ρ ρ

ρ α

α ρρ

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠

− ≤ ≤ ≥ =

=

q

W. Thistleton, J.A. Marsh, K. Nelson and C. T. (2006)

INFLUENCE OF THE RANGE OF CORRELATIONS DECAYING FAR FROM THE DIAGONAL OF THE COVARIANCE MATRIX:

(N=35 )

1 1

1 (5 / 3) 1

q Φ ρ ρ

ρ ρ

− −

− −

1

LOGICAL CONSEQUENCES OF THE q-CENTRAL LIMIT THEOREM, OR JUST A COINCIDENCE? – AN OPEN QUESTION

1

1- The dynamical attractor at the edge of chaos [ ( )] of the one-dimensional dissipative

map -Ga us 1 ( ) | | is, , well fitted by a [U. Tirn

sian with 1.75 akli

c

t c t z

a

z

z

qx a z x q+ = ∀− , C. Beck and C. T., Phys Rev E / RC (in press), cond-mat/0701622]

2- The distribution associated with the fluctuating magnetic field within the solar wind is, as observed in the data from the spacecraft Voyager 1, well fitted by a [L.F. Burlaga and A. F.-Vinas, Physica A 356, 375 (20

-Gaussian 05)]

3- Th

with 1

e histo

.75 0.06

gram of Br

q q = ±

azilian stock market index changes is, for a considerable range of time delays, well fitted by a [A.A

-Gaussian wit .G. Cortines

h 1 and

.7

5 R. R

q q iera, Physica A 377, 181 (2007)]

4- The probability distribution of energy differences of subsequent earthquakes in the World Catalog and in Northern California is well fit -Gausted b sian y h a witq [F. Caruso, A. Pluchino, V. Latora, S. Vinciguerra and A. Rapisarda, cond-ma

1 t/0606118]

.75 0.15q ±

1

1- The dynamical attractor of the sum at the edge

1

of

|

chaos

| ( 2)

t t zx a x

z + = − =

U. Tirnakli, C. Beck and C. T., Phys Rev E / RC (2007), cond-mat/0701622

(a = 1.7, 1.8, 1.9, hence positive Lyapunov exponent)

q = 1

U. Tirnakli, C. Beck and C. T., Phys Rev E / RC (2007), cond-mat/0701622

(a = a (z), hence vanishing Lyapunov exponent)

q = 1.75

c

2- The distribution associated with the fluctuating magnetic field

L.F. Burlaga and A.F. Vinas, Physica A 356, 375 (2005)

3- The histogram of Brazilian stock market index changes

A.A.G. Cortines and R. Riera, Physica A 377, 181 (2007)

Bovespa index (November 2002 - June 2004)

A.A.G. Cortines and R. Riera, Physica A 377, 181 (2007)

q = 1.75

4- The probability distribution of energy differences of subsequent earthquakes

F. Caruso, A. Pluchino, V.Latora, S. Vinciguerra and A. Rapisarda, cond-mat/0606118

q = 1.75

THERMODYNAMICAL CONSEQUENCES OF LONG-RANGE INTERACTIONS IN MANY-BODY CLASSICAL SYSTEMS

2

1

( )

CLASSICAL SYSTEM:

( ) 2

( )

, , , ,

( 0; 0

)

N i

ij i i j

where V r has no singularity at the origin or an integrable singularity and for r

The characteristic potent

pH K V V r m

AV

ial energy would be

r A

ty rα

α

= ≠

→ ∞

= + = +

> ≥

∑ ∑

1

*

1

*

1/

1

1

1 /

/

( ) / 1

0 / 1

( )

,

1 ln 1

/

pot

pot d

d

N d

d

pically given by

Consequently a mo

U N if d A dr r r

if dN

U N AA dr r r N N d

NN d

re appropriate estimation is given by

with

α

α

α

α

α α

α

∞ −

< ∞ >⎧ ∝ − ⎨→ ∞ ≤ ≤⎩

∝ − = −

− ≡ =

1 /

1 / 1 / 1

ln / 1

0 / 1 1 /

d d

if d d

N N if d

N if d d

α

α α

α

α α

⎧ >⎪ −⎪⎪ =⎨

⎪ ⎪ ≤ < ⎪ −⎩

short-range interaction

long-range interaction

Me an

Fi eld

st an

da rd

sc ali

ng

/ 1 leads to the traditional classification of thermodynamical variables

-intensive extensivedα >

/ / 1

d d

α α −

( , , , ) ( , , , ) ( , , , ) ( , , , ) ( SHORT-RANG

, , , )

( , , , ) ( , , , ) ( , , , )lim lim lim

E INTERACTIONS ( . ., / 1 ):

N N N

G N T p H U N T p H T S N T p H p V N T p H H M N T p H hence

G N T

i e d f

p H U N T p H S N T p

or clasical systems

HT N N N

α

→∞ →∞ →∞

= − + >

= −

( , , , ) ( , , , ) lim lim

. ., ( , , ) ( , , ) ( , , ) ( , ,

LONG-RANGE INTERACTIONS ( . ., 0 / 1

) ( , , )

( , , , ) ( , , , ) ( )

, :

N N V N T p H M N T p Hp H

N N i e g T p H u T p H T s T p H p v T p H H m T p H

G N i e d

T p H U N T p H T S N T for clasical systemsα

→∞ →∞+ −

= − + −

= − ≤ ≤

* * *

* *

* * * * *

, , ) ( , , , ) ( , , , )

( , , , ) ( , , , ) ( , , , )lim lim lim

( , , , ) ( , , , ) lim lim

. ., ( , , ) ( ,

N N N

N N

p H p V N T p H H M N T p H hence

G N T p H U N T p H T S N T p H NN NN N N

p V N T p H H M N T p H N N N N

i e g T p H u T p

→∞ →∞ →∞

→∞ →∞

+ −

= −

+ −

= * * * * * * * * * * * * *, ) ( , , ) ( , , ) ( , , )H T s T p H p v T p H H m T p H− + −

Intensive variables : , , (they do not scale with si SHORT-RANGE INTERACTIONS ( / 1

z e Extensive variables: , , , , ,

) (the y scale with

):

d for clasical system T p H N G U S V

s

M N

α >

i.e., the equations of states are expressed in the variables ( ,

BOTH SHORT-

)

AND LONG-RANGE INTERACTIONS

, , , ) versus

(

, , )

N g N u N s N v N m T p H

N

*

*

Pseudo-intensive variables : , , (they scale ( / 0

Pseudo-exte

with ) (they scale nsive variables: , th

):

wi NT p H

d for clasical systems

U NG N

α ≥

* * * *

i.e., the equations of states are expressed in th

) (th

e va ey scale

riables Extensive variables:

(

with

, , , , )

, , ) ,

versus , ,

T pNN g NN u N s N v N m N

S M

N

N NV

*

- THE THERMODYNAMICAL IN ALL CASES! - THE VARIABLES DEFINING "THERMAL EQUILIBRIUM" ( , , ) D

ENTROPY IS EXTENSIVE

NOT NECESSARILY COINCIDO WITH THOSE DEFINING EE FINIT U TE Q A ION A B A B A BT T p

H

p H H

N

=

⎝ ⎠

=

=

S OF STATES!

0 1 2 3 4 5 0

1

2

3

4

5

α

d

EXTENSIVE SYSTEMS

NONEXTENSIVE SYSTEMS

dipole-dipole

Newtonian gravitation

dip ole

-m on

op ole

(ti de

s)

d-d im

en sio

na l g

rav ita

tio n

( )

/ 1 ( - ) - 0 / 1

( )

( 0, 0)

( - ) integrable if d short ranged

non integrable if d

AV r

long

r

r

A

ange

r

d

α

α α

α

− ∞

> > ≤

ILLUSTRATIONS OF THE FINITENESS OF THE EQUATIONS OF STATES

Ferrofluid-like model:

P.Jund, S.G. Kim and C. T., Phys Rev B 52, 50 (1995)

Lennard-Jones-like fluids:

R. Grigera, Phys Lett A 217, 47 (1996)

Magnetic systems:

L.C. Sampaio, M.P. de Albuquerque and F.S. de Menezes,

Phys Rev B 55, 5611 (1997)

C. Anteneodo and C. T., Phys Rev Lett 80, 5313 (1998)

R.F.S. Andrade and S.T.R. Pinho, Phys Rev E 71, 026126 (2005)

Percolation:

H.H.A. Rego, L.S. Lucena, L.R. da Silva and C. T., Physica A 266, 42 (1999)

U.L. Fulco, L.R. da Silva, F.D. Nobre, H.H.A. Rego and L.S. Lucena,

Phys. Lett. A 312, 331 (2003)

EXTENSIVITY OF THE NONADDITIVE ENTROPY Sq

and

N>>1 ATTRACTORS IN THE SENSE OF THE CENTRAL LIMIT THEOREM

Attractor Entropy

independent variables

1 ( ( ) ) 1 ( ( ) )

( . ., 1)

( . .,

?

Leibnitz triangle ( ) ?1MTG mode1)

2

l

entent BG ent q

att

att

att

att

corr att

q S N N q S N N

Gaussian attractor i e q

q Gaussian attractor i e q

q

q q

→ ∞

⎛ ⎞ =

= ∝ < ∝

=

⎝ ≠ −⎜ ⎟

⎠ TGS1 model TGS2, TGS3 (stretched exponential) MFMT1 ( 0) MFMT2 ( 0) model MFMT2 ( 0)

MFM

( . ., ( , ) )

T3

other attractors e g q stable

distributions

α α α

α ≠

> < −

N DISTINGUISHABLE BINARY RANDOM VARIABLES

C. T., M. Gell-Mann and Y. Sato (2005) Central Limit Theorem L.G. Moyano, C. T. and M. Gell-Mann (2006) Levy-GnedenkoTheorem

TGS MTG M J.A.

CLT LG

Marsh, M.AFMT . F e te

u n

→ → →

→ →

s, L.G. Moyano S. Umarov, C. T. and S. Steinberg (2006)

and C. T. UTS UT (2006) S. Umaro v and C. T. (2 007

)

→ →

S. Um arov, C. T., M. Gell-Mann

UT

GS

→ and S. Steinberg (2006)

CLT

UTS

UT

LG

UTGS

BG NONEXTENSIVE

(Leibnitz rule)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

INDEPENDENT VARIABLES

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

MTG MODEL

12 -att corr

q q

=

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

MTG MODEL

TGS1 MODEL

(STRETCHED

EXPONENTIAL)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

TGS2 MODEL

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

TGS2 MODEL

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

TGS3 MODEL

(Leibnitz rule is asymptotically valid)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

MFMT1 MODEL

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

MFMT1 MODEL

(Q ~ 5.0)

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

MFMT2 MODEL

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

MFMT2 MODEL

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T. Physica A 372, 183 (2006)

(Leibnitz rule is not valid)

MFMT3 MODEL

11- 0

1~ | |

ent

if dq

if

α

α α

⎧= ≥⎪⎪ ⎨ ⎪ → −∞ ⎪⎩

J.A. Marsh, M.A. Fuentes, L.G. Moyano and C. T., Physica A 372, 183 (2006)

balistic

localized

CONNECTION WITH

(ASYMPTOTICALLY) SCALE-FREE NETWORKS

R. Albert and A.-L. Barabasi, Phys. Rev. Lett. 85, 5234 (2000)

AN ANALYTICALLY SOLVABLE MODEL:

, ,

1 ,

- -

( )

k q

At each time step new links are added with probability

or existing links are rewired with probability or a new node with links is added with probabi

m p m r

m p r

degree distribution p

ty

k

li

e−≡ ∝ /

2 (2 - ) 1 - - (3- 2 ) 1- -

m r p rq m r

with p r

κ

+ =

+

0

1 ( ) ( )

written in the form p k by the authors k k γ

⎡ ⎤ ∝⎢ ⎥+⎣ ⎦

(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 G r distance to the baricenter of the pre existing cluster

P r α α+

≡ −

∝ ≥

4) Repeat

A

( )

( )

/ ( 0) Ai i A i

i

k links already attached to site i

r distance to site i

k rp α α ≡

∝ ≥

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)

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) / ]

k q

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∀

( 0)

( )

1

ij

ij ij

Merging probability

d shortest path chemical distance connecting nodes i and j on the network

p d α

α ≥

(Kim, Trusina, Minnhagen and Sneppen, . . . 43 (2005) 369)

0 and recover th random neighbore and the schemes respectively Eur 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= = =

S. Thurner, Europhys News 36, 218 (2005)

2000 4000 6000 8000 10000 12000 14000 0

10

20

30

40

50

60

70

80

time

k

m

a

x

;

k

i

k i

k max

[ ] [ ] 1

( )

( 1.84

1 ( ) ln ( )

1 )

q

q q

c

P k Z k

optimal

P k q

q

=

> − ≡ > ≡

( ;α → ∞

( ; 8)rα → ∞ < >=

S. Thurner and C. T., Europhys Lett 72, 197 (2005)

- ( -2)/( ) ( 2, 3, 4,...)c k

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)

GENERATIVE MODEL FOR FEEDBACK NETWORKS: D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

0 ( )

routing parameter u l number of neighbors of node l that have not yet been visited

γ > ≡

1; 1, 2, 3...distance decay parameter dβ > =

MODEL:

0attachment parameter α ≥

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

D.R. White, N. Kejzar, C. T., D. Farmer and S. White, Phys Rev E 73, 016119 (2006)

The quality of the fittings has been shown to be satisfactory through the

nonparametric statistical Kolmogorov-Smirnov and the Wilkoxon rank sum tests

HOW COME THE DEGREE DISTRIBUTION

COINCIDES WITH THAT MAXIMIZING Sq ?

" " , ( 1, 2,..., )

Therefore the degree distribution coincides with the energy distributio

2

n!

/ .i

If we associate with each bond an energy we may associate with each node i N the energy k

ε ε=

TRAIN DELAYS ON THE BRITISH RAILWAY NETWORK: K. Briggs and C. Beck, Physica A 378, 498 (2007)

Bath Spa to London Paddington Swindon to London Paddington

K. Briggs and C. Beck, Physica A 378, 498 (2007)

Reading to London Paddington

K. Briggs and C. Beck, Physica A 378, 498 (2007)

ANOMALOUS DIFFUSION, ESCAPE TIME

AND GENERALIZED ARRHENIUS LAW

E.K. Lenzi, C. Anteneodo and L. Borland, Phys Rev E 63, 051109 (2001)

E.K. Lenzi, C. Anteneodo and L. Borland, Phys Rev E 63, 051109 (2001)

THEORETICAL PREDICTIONS

AND

EXPERIMENTAL OBSERVATIONAL COMPUTATIONAL VERIFICATIONS

− −

2 2 /(3 )

2 2

2

1/(1 )2 2 /(3 ) / ( )

2 2

( , ) [ ( , )] [ ( ,0) (0)] ( 3)

( , ) 1 (1 ) / ( ) ( )

( . .,

)

q

q

qq x t q

The solution of p x t p x tD p x q

t x is 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 , . ., ) q

e g q i e normal diffusionγ

γ

= ⇒ =

= −

PREDICTION OF A SCALING RELATION:

C.T. and D.J. Bukman, Phys Rev E 54, R2197 (1996) (see also AR Plastino and A Plastino, Physica A 222, 347 (1995))

Hydra viridissima: A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada Physica A 293, 549 (2001)

q=1.5

1.24 0.1 2

3

slope

hence is satisfied q

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 with q

γ γ≈ ≈ = −

Silo drainage: R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)

(intermediate regime)

q=3/2 q=1

(fully developed regime)

4 / 3 2

3

slope

hence is satisfied q

γ

γ

=

= −

R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)

(outlet size 3.8 d)

2

1 ,

1 /

1

1 cos( )1 ( 0, 0) 2

0 / 1 ln / 1

/ 1

N i j

i i i j ij

d N

ij j

JH K V L I J I r

N if d with r N if d

constant if d

and periodic boundary

α

α

α

ϑ ϑ

α α α

=

=

− − = + = + > >

⎧ ⎫≤ < ⎪ ⎪≡ ∝ =⎨ ⎬ ⎪ ⎪>⎩ ⎭

∑ ∑

A

A

.

[ / 0]The HMF model corresponds t

condit

o

io s

d

n

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)

A. Campa, A. Giansanti, D. Moroni and C. T., Phys Lett A 286, 251 (2001)

B.J.C. Cabral and C. T., Phys Rev E 66, 065101 (2002)

V. Latora, A. Rapisarda and C. T., Phys Rev E 64, 056134 (2001)

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)

N = 10000 τ = 0.01 TQSS = 10000 Coupling = 0.1 Init = CATANIA

HOST = NCSA Program Version = 1.0.a # Processors = 10

τ .

SS li .

I it I

r r r i . . r r

Contact

U1 = 0.69 U2 = 0.72

. .

Charm++ NExtComp Molecular Dynamics – 2 Systems Interactions

Time

T

e

m

p

e

r

a

t

u

r

e

BOLTZMANN-GIBBS STATISTICAL MECHANICS (Maxwell 1860, Boltzmann 1872, Gibbs ≤ 1902)

Entropy

Internal energy

Equilibrium distribution

Paradigmatic differential equation

1

W

B G i i i

U p E =

= ∑ 1

ln W

B G i i i

S k p p =

= − ∑

/iEi BGp e Z β−=

1

j W

E BG

j Z e β−

=

⎛ ⎞ ≡⎜ ⎟

⎝ ⎠ ∑

( 0 ) 1

d y a y d x y

⎫= ⎪ ⇒⎬ ⎪= ⎭

x a y(x) Equilibrium distribution Ei -β Z p(Ei) Sensitivity to initial conditions t λ Typical relaxation of observable Ο t -1/τ

SBG→ extensive, concave, Lesche-stable, finite entropy production

( 0 ) 0

( )lim (0 )

t

x

x t e x

λξ Δ →

Δ ≡ =

Δ

/( ) ( ) (0) ( )

tO t O e O 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 state distribution

Paradigmatic differential equation

x a y(x) Stationary state distribution

Ei Sensitivity to initial conditions t Typical relaxation of observable Ο t

1

1 /( 1) W

q q i

i

S k p q =

⎛ ⎞= − −⎜ ⎟ ⎝ ⎠

( ) /q i qE Ui q qp e Z β− −=

( )

1

E Uq j q W

q q j

Z e β− −

=

⎛ ⎞ ≡⎜ ⎟

⎝ ⎠ ∑

(0) 1

qdy a y dx y

⎫ ⎪ ⎬ ⎪⎭

= ⇒

=

Sq→ extensive, concave, Lesche-stable, finite entropy production

qsen

sen

t qe λξ =

/ qrel rel

t qe

τ− Ω =

[ ] 1

11 (1 ) qq a x

q a xy e −+ −= ≡

1 1 /

W W q q

q i i j i j

U p E p = =

= ∑ ∑

(typically 1)senq

1 / r e lq

τ−

senq λ

(typically 1)relq

statq β− ( )statq iZ p E

C. T., Physica A 340,1 (2004)

(typically 1)statq

Prediction of the q - 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 = ±

IHY 2007: VOYAGER 1: Fundamental Physics The atmosphere of the Sun beyond a few solar radii, known as HELIOSPHERE, is fully ionized plasma expanding at supersonic speeds, carrying solar magnetic fields with it. This solar wind is a driven non-linear non-equilibrium system. The Sun injects matter, momentum, energy, and magnetic fields into the heliosphere in a highly variable way. Voyager 1 observed magnetic field strength variations in the solar wind near 40 AU during 1989 and near 85 AU during 2002. Tsallis’ non-extensive statistical mechanics, a generalization of Boltzmann-Gibbs statistical mechanics, allows a physical explanation of these magnetic field strength variations in terms of departure from thermodynamic equilibrium in an unique way:

( 2 ) ( 1/ )

1 2

(

)

rel sen

Playing with additive duality and with multiplicative duality and using numerical results related to the q generalized central limit theorem

we conject

q q q q

q q

ure

→ − →

+

=

!

1 2

1 1 3 2

( )

stat rel

stat sen

stat

stat

and

hence

Burlaga and Vinas NASA most precise value of the q tripl

hence onl

et

y one independe

q q

qq q

t

is

n

q

+ =

− − =

( 0.6 0.2 !) (

1.75 7 / 4 0.5 1/ 2

4 3 .8 0.3 ) ! sen

r

sen

rel el

hence a

consistent with q consistent wind th q

q q

= − ± = ±

= = = − = − =

C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)

1

1

Generic pitchfork bifurcations:

Generic tangent bifu

( 1; 0)

( 1

The fixed point map is a q-exponential with

rcatio ; 0)

and the s

n

e

s:

( ) | |

| |

z t t tt

z t tt

z

q z

b

z

x x b sign x x

x x b x b

+

+ > >

=

> >= +

= +

1

Example:

nsitivity

to the initial c

( 1; 0 2; 1)

3

onditions is a -exponential with

1

2

1 | |

sen

sen

tt

The logistic family of maps a

has

z for pitchfor

q

q

k

q

x a x ς ς

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)

ELEMENTARY 1D CELLULAR AUTOMATA

(SHORT AND LONG MEMORY)

T. Rohlf and C. T., Physica A (2007), in press

3

-1 1

1

1

2

1( ) ( 1), ( ) , ( 1) 2

where

is one of the Wolfram 2 256 two-state 3-neighborhood 1D elementary CA rules ( )

( ) lim [0,1] ( 0) 1

Heaviside s

i i i i

T i

i T T

t f t t t

f t

t α

τ

α τ

σ σ Θ Ξ σ

σ τ τΞ α

τ Θ

+

= →∞

=

⎡ ⎤⎛ ⎞= − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

=

≡ ∈ ≥

∑ tep function

<

α

α

→ ∞

CA WITH SHORT AND LONG MEMORY:

T

.

R

o

h

l

f

a

n

d

C

.

T

.

,

P

h

y

s

i

c

a

A

(

2

0

0

7

)

,

i

n

p

r

e

s

s

Difference patterns with initial configurations differing in one randomly chosen bit

T

.

R

o

h

l

f

a

n

d

C

.

T

.

,

P

h

y

s

i

c

a

A

(

2

0

0

7

)

,

i

n

p

r

e

s

s

Rule 61

Rule 111

N

=

1

0

0

0

;

T

=

3

2

0

2

0

0

d

i

f

f

e

r

e

n

t

i

n

i

t

i

a

l

c

o

n

d

i

t

i

o

n

s

T. Rohlf and C. T., Physica A (2007), in press

( ) ; 1/(1 ) H t t qγ γ∝ ≡ −

(T=320)

(Rule 61; T=320)

slo pe

=1

slo pe

=1 /2

( , )

( ,

1 0

1 1

1.4 1

11.4 ) ( )

WHY?

2

Rule 61

tH t N N

H t N t N

q

q

γ γ

γ γ

α γ

α γ

⎧ < ⇒ ⇒⎪⎪

⎨ ⎪ > ⇒ ⇒

⎡ ⎤∝⎢ ⎥⎣ ⎦

⎡ ⎤∝ ∀

− =

⎪ − ⎣

− = ⎦⎩

A TYPICAL OPEN QUESTION: For 1<< t < N

T. Rohlf and C. T., Physica A (2007), in press

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

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 Gauss Boltzmann machine Gaussian

Boltzmann machine Boltzmann

Visiting algo

weight

rithm

Acceptance a ian

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 machine t

Typ

Cool

ica

ne t

ing algori

l values and q

thm

q

→ =

→ −

− → =

+ −

< < <

+

C. T. and D.A. Stariolo, Physica A 233, 395 (1996)

4 4 2 2

1 2 3 4 1 1

( , , , ) ( 8) 5

(15 )

: i i i 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)

Genetic algorithm

Present with q=2.7slo

pe 4

.7

slo pe

2. 9

N

u

m

b

e

r

o

f

f

u

n

c

t

i

o

n

c

a

l

l

s

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)

P. Serra, A.F. Stanton, S. Kais and R.E. Bleil J. Chem. Phys 106, 7170 (1997)

HYBRID LEARNING OF NEURAL NETWORKS

A.D. Anastasiadis and G.D. Magoulas, Physica A 344, 372 (2004)

A.D. Anastasiadis, Proc. Int. Summer School Complex Systems (June 2005, Santa Fe Institute, NM)

HLS (Hybrid Learning Scheme; q > 1)

Rprop CHLS (Cooling Hybrid Learning Scheme; q > 1)

Rprop

Sarprop

HLS

Anastasiadis and Magoulas (2004)

DIABETES CANCER

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)

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

q = 1.5

q = 1

(Jensen- Shannon)

Canny edge detector

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

Canny edge detector

q = 1.5

q = 1

(Jensen- Shannon)

Magnetic Resonance and Computed Tomography Images

alpha=q

Image Fusion

q=1 q=1.85

INFRARED VISIBLE INFRARED VISIBLE

A. Plastino and O.A. Rosso Europhysics News 36 (6), 224 (2005) [European Physical Society]

ELECTROENCEPHALOGRAMS (tonic-clonic transition in epilepsy):

starts

ends

TONIC CLONIC

(

I

N

C

L

U

D

I

N

G

M

U

S

C

U

L

A

R

A

C

T

I

V

I

T

Y

C

O

N

T

R

I

B

U

T

I

O

N

)

EARTHQUAKES

n / nw 1.05

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 / n w

1.05 0 2 4 6 8 10 12 14 16

l

n

q

[

D

(

n

+

n

w

,

n

w

)

]

-12

-10

-8

-6

-4

-2

0

S. Abe, U. Tirnakli and P.A. Varotsos Europhysics 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

INTERNET (PING): S. Abe and Suzuki, Phys Rev E 67, 016106 (2003)

(q=1.7) (q=1.12)

(q=1.7) (q=1.12)

S. Abe and Suzuki, Phys Rev E 67, 016106 (2003)

(q=1.16)

(q=0.73)

(q=0.73)

(q=1.16)

S. Abe and Suzuki, Phys Rev E 67, 016106 (2003)

ASTROPHYSICS

FLUX OF COSMIC RAYS:

C. T, J. Anjos and E.P. Borges Phys. Lett. A 310, 372 (2003)

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 = ±

Using the release of 3-year data:

A. Bernui, C. T. and T. Villela, Europhys Lett 78, 19001 (2007)

1.04 0.01q = ±

max

min

1.43 1.50

q q

SOLAR FLARES:

M. Baiesi, M. Paczuski and A. Stella, Phys Rev Lett 96, 051103 (2006)

All

min

max

1.46 1.66 1.35

q q q

CHEMICAL RE-ASSOCIATION

C.T., G. Bemski and R.S. Mendes Phys. Lett. A 257, 93 (1999)

C.T., G. Bemski and R.S. Mendes Phys. Lett. A 257, 93 (1999)

C.T., G. Bemski and R.S. Mendes Phys. Lett. A 257, 93 (1999)

FINGERING

P. Grosfils and J.P. Boon, 2005

P. Grosfils and J.P. Boon, 2005

512 1024 x

-6

-3

0

3

6

c

0,01 0,1 1 c

10 -5

10 -4

10 -3

10 -2

10 -1

P(c)

-6 -4 -2 0 2 4 6 c

0,0001

0,001

0,01

0,1

1

P(c)

0.78 0.82

P. Grosfils and J.P. Boon, 2005

MANGANITES

Reis et al, Phys Rev B (2003)

ECONOMICS

J de Souza, LG Moyano and SMD Queiros, Eur Phys J B 50, 165 (2006)

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, Quantitative Finance 4, 499 (2004) L Borland, Europhys News 36, 228 (2005) See also H Sakaguchi, J Phys Soc Jpn 70, 3247 (2001)

C Anteneodo and C T, J Math Phys 44, 5194 (2003)

3 1

[ :

] n n

REMARK Student t distributions are the particular case

of q Gauss q with n intians whe rn ege+ +

=

LONDON STOCK EXCHANGE (Block market):

Data: I.I. Zovko; Fitting: E.P. Borges (2005)

VODAPHONE stocks (31 May 2000 to 31 December 2002)

Daily net exchange of shares (between all pairs of two institutions)

4

6 '

3.28 ; 1.1 10

' 1.45 ; 1.1 10 q

q

q

q

β

β

= = ×

= = ×

C

u

m

u

l

a

t

i

v

e

d

i

s

t

r

i

b

u

t

i

o

n

WEALTH DISTRIBUTION: E.P. Borges, Physica A 334, 255 (2004)

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 RR Eq

U E

U

= ≡

+

Experimental and computational verifications by P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)

(Computational verification: quantum Monte Carlo simulations) (Experimental verification)

0

441 REq U

= +

THERMODYNAMICS

MECHANICS (classical, quantum, …)

LANGEVIN EQUATION

FOKKER-PLANCK EQUATION

MASTER EQUATION

STATISTICAL MECHANICSLIOUVILLE EQUATION

VON NEUMANN EQUATION

VLASOV EQUATION

BOLTZMANN KINETIC EQUATION B

r

a

u

n

a

n

d

H

e

p

p

t

h

e

o

r

e

m

Sofia e la scoperta delle fragole (Marco Bersanelli)

A Gutenberg, tra le verdissime colline austriache, una mattina saliamo per il sentiero che attraversa il bosco scuro e profumato alle spalle del paese. Dopo mezz'ora di cammino troviamo sulla destra una sorgente presso una radura e ci fermiamo a bere. Con una grande espressione di felicità ad un tratto Sofia, la piccola di tre anni, esclama: «Mamma, mamma!! una fragola!!». Gli altri due accorrono e, constatato che la sorellina ha prontamente raccolto e inghiottito il frutto della sua scoperta, si mettono a cercare, presto seguiti dai genitori. «Un'altra!» e dopo un po': «Guarda qui, ce ne sono altre tre, quattro...». La caccia è aperta. Cercando in quel prato abbiamo presto riempito un bicchiere di fragole di bosco. Poi al ritorno, con mia sincera sorpresa, ripercorrendo lo stesso sentiero dalla sorgente in giù ne abbiamo trovate altrettante! Zero fragole all'andata, forse un centinaio al ritorno: un effetto statisticamente schiacciante. Cos'era cambiato?

Sofia e la scoperta delle fragole (Marco Bersanelli)

A Gutenberg, tra le verdissime colline austriache, una mattina saliamo per il sentiero che attraversa il bosco scuro e profumato alle spalle del paese. Dopo mezz'ora di cammino troviamo sulla destra una sorgente presso una radura e ci fermiamo a bere. Con una grande espressione di felicità ad un tratto Sofia, la piccola di tre anni, esclama: «Mamma, mamma!! una fragola!!». Gli altri due accorrono e, constatato che la sorellina ha prontamente raccolto e inghiottito il frutto della sua scoperta, si mettono a cercare, presto seguiti dai genitori. «Un'altra!» e dopo un po': «Guarda qui, ce ne sono altre tre, quattro...». La caccia è aperta. Cercando in quel prato abbiamo presto riempito un bicchiere di fragole di bosco. Poi al ritorno, con mia sincera sorpresa, ripercorrendo lo stesso sentiero dalla sorgente in giù ne abbiamo trovate altrettante! Zero fragole all'andata, forse un centinaio al ritorno: un effetto statisticamente schiacciante. Cos'era cambiato? Eravamo cambiati noi.

HOKKAIDO UNIVERSITY – SAPPORO

WILLIAM S. CLARK (1826-1886)

1877: BOYS, BE AMBITIOUS!

qthan

Aulas do Prof. Andrea Rapisarda:

www.ct.infn.it/rapis/rio-lectures

Aulas do Prof. Constantino Tsallis:

http://www.cbpf.br/NextCurso2007/AulasTsallis.pdf

Aulas do Prof. Alberto Robledo:

Foi distribuido em cada pasta de participante

O conjunto permanecera disponivel (a partir de uma semana) em

http://tsallis.cat.cbpf.br/NextCurso2007

Logos Quotes 26-JAN-2007, Every day a new quotation translated into many languages.

Quotation of the day: Author - Mark Twain

English - the man with a new idea is a crank until the idea succeeds Albanian - njeriu që ka një ide të re është një i çmendur deri sa ajo ide të ketë sukses Basque - ideia berri bat duen gizakia ero bat da, ideiak arrakasta lortzen duen arte Bolognese - un òmen con un'idê nôva l é un mât, infénna che cl'idê la n à suzès Brazilian Portuguese - um homem com uma ideia nova é um louco até que a ideia tenha sucesso Breton - ken na zeu e vennozh da vat, un den ideet eo an hini en deus ur mennozh nevez Calabrese - 'n uomu cu 'n'idea nuova è nu pacciu finu a quannu l'dea nun teni successu Catalan - un home amb una idea nova és un boig fins que la idea no triomfa Croatian - čovjek s novom idejom je čudak sve dok ideja ne uspije Danish - en mand med en ny ide er skør, indtil ideen lykkes Dutch - iemand met een nieuw idee is een dwaas, totdat het slaagt English - the man with a new idea is a crank until the idea succeeds Esperanto - homo kun nova ideo estas frenezulo ĝis kiam la ideo sukcesas Estonian - inimest, kellel on uued ideed, peetaske hulluks senikaua, kuni tema ideid kroonib edu Finnish - ihminen jolla on idea on hullu, kunnes ideasta tulee menestys French - un homme avec une nouvelle idée est un fou tant que l'idée n'a pas de succès Furlan - un omp con une gnove idee al è un mat fintremai che chê idee no à sucess Galician - un home cunha idea nova é un tolo ata que a idea teña éxito German - ein Mensch mit einer neuen Idee ist so lange ein Spinner, bis die Idee zum Erfolg wird Griko Salentino - nan àntrepo me mian idea nea (cinùria) ene na ppàccio sara ka cin idea en echi successo Hungarian - az új ötlettel rendelkező embert mindaddig hülyének nézik, amíg az ötlet nem lesz sikeres Italian - un uomo con un'idea nuova è un matto finché quell'idea non ha successo Judeo Spanish - tun ombre kon una idea mueva es un loko asta ke la idea triunfe Latin - homo quidam cum nova cogitatione stultus est antequam cogitatio firmetur Latvian - cilvēks ar jaunu ideju ir jucis, kamēr ideja nav realizēta Leonese - un home cun una idega nueva ye un home alloriáu fasta que la idega triunfa Mudnés - un àmm ch'àl ghà n'idèa nôva l'è un mât fintânt che c'l'idèa l'àn ghà sucês Neapolitan - n'ommo cu na penzata nova è nu schirchio nfin'a quanno chella penzata nun trionfa Papiamentu - un persona ku un idea nobo ta un loko te ora e idea logra Portuguese - um homem com uma ideia nova é um louco até que a ideia tenha sucesso Roman - ´n´ omo co´ ´n´ idea nova è solo un matto, fino a che l´ idea nun ciabbia successo Spanish - un hombre con una idea nueva es un loco hasta que la idea triunfa Umbro-Sabino - n'omo co' n'idea nòa è sciurnu finaquanno ell'idea nun c'à succiessu Venetian - l'omo co na idèa el xe un móna fin che l'idèa no ła ga suceso Wallon - l' sakî k'a ène noûve idêye dimère on sot djusk' à c'k' èle fuchisse riconèxheuwe come boune Welsh - hyd nes bod ei syniad yn llwyddo, dyn â chwilen yn ei ben yw'r un â syniad newydd Zeneize - un òmmo con unn'idea neuva o l'é un sciòllo scin che quell'idea a no l'à successo

The man with a new idea is a crank until the idea succeeds.

(Phenomenological model for collisions in a diluted gas with probability r of forming clusters of q correlated particles)

Monte Carlo

Single-parameter fitting

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