**UFRJ**

# Aulas Tsallis

(Parte **1** de 2)

Constantino Tsallis

Centro Brasileirode PesquisasFisicas Rio de Janeiro -Brasil

Rio de Janeiro, CBPF, Abril2007

and | q -GENERALIZED CENTRAL LIMIT THEOREM: |

EXTENSIVITY OF THE NONADDITIVE ENTROPY Sq

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

L.G. Moyano, C. T. and M. Gell-Mann, EurophysLett73, 813 (2006) |

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

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. Moyanoand C. T, PhysicaA 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. Bergaminiand F. Renzoni, Phys Rev Lett96, 110601 (2006)

Sabir U marov

Yuzuru Sato

Stanly Steinberg

Ugur Tirnakli SilvioM.D. Queiros

Miguel Fuentes

Luis MoyanoPaul RivkinWilliam Thistleton John Marsh KenricNelson

Christian Beck E.G.D. Cohen

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 |

dakis, M Lissiaand A Rapisarda, |

and Physical Applications, G Kaniaeds, PhysicaA 305, Issue 1/2 (2002)

Classical and Quantum Complexity and Nonextensive Thermodynamics, P Grigolini, C Tsallis and BJ West, eds, Chaos, Solitonsand Fractals 13, Issue 3 (2002)

NonadditiveEntropy and Nonextensive Statistical Mechanics, M Sugiyama, ed, Continuum Mechanics and Thermodynamics 16(Springer, Heidelberg, 2004)

Nonextensive Entropy - Interdisciplinary Applications, M Gell- Mann and C Tsallis, eds, (Oxford University Press, New York, 2004)

HL Swinneyand C Tsallis, eds, |

Anomalous Distributions, Nonlinear Dynamics, and Nonextensivity PhysicaD 193, Issue 1-4 (2004)

News and Expectations in Th erm ostat istics G Kaniadakisand M Lissia, eds PhysicaA 340, Issue 1/3 (2004)

Trends and Perspectives in Extensive and Non-Extensive Statistical MechanicsH Herrmann, M Barbosaand E Curado, eds, PhysicaA 344, Issue 3/4 (2004)

A Rapisardaand C Tsallis, eds, |

Complexity, Metastabilityand Nonextensivity, C Beck, G Benedek, (World Scientific, Singapore, 2005)

Nonextensive Statistical Mechanics: | |

New Trends, New Perspectives, JP |

News (European Physical Society, 2005) |

Boon and C Tsallis, eds, Europhysics

Statistical Mechanics, G Kaniadakis, |

Fundamental Problems of Modern A Carboneand M Lissia, eds, PhysicaA 365, Issue 1 (2006)

Complexity and Nonextensivity: New Trends in Statistical Mechanics, S Abe, M Sakagamiand N Suzuki, eds, Progr. Theoretical Physics Suppl162(2006)

Approaching a Complex World, |

Introduction to Nonextensive Statistical Mechanics - 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

> | 957 nominal citations |

(including > 1,407 citationsof the 1988 paper) [31 March 2007]

BRAZIL | 303 |

USA | 217 |

ITALY | 116 |

RUSSIA | 31 |

SWITZERLAND | 17 |

ISRAEL | 13 |

CONTRIBUTORS (2107 MANUSCRIPTS) JAPAN105 FRANCE 92 CHINA 8 ARGENTINA 79 SPAIN 57 GERMANY 5 ENGLAND 40 POLAND 37 TURKEY 27 INDIA 23 CANADA 23 AUSTRIA 21 MEXICO 19 UKRAINE 18 KOREA 12

SINGAPORE | 4 |

BULGARIA | 3 |

NETHERLANDS 1 BELGIUM 1 GREECE 1 AUSTRALIA 8 PORTUGAL 8 SOUTH AFRICA7 CUBA 7 HUNG ARY 7 IRAN 7 VENEZUELA 6 CHILE 6 NORWAY5 SWEDEN 4 TAIWAN 4 URUG UAY 4 ROMENIA 4 EGYPT 4 DENMARK 4 SLOVENIA 4

CROATIA | 3 |

BOLIVIA | 2 |

KAZAKSTAN | 2 |

PHILIPINES | 2 |

PUERTO RICO | 2 |

ARMENIA | 1 |

INDONESIA | 1 |

MALAYSIA | 1 |

SAUDI ARABIA | 1 |

SRI LANKA | 1 |

UZBEKISTAN | 1 |

60 | 1550 |

COUNTRIES | SCIENTISTS |

SLOVAK 3 IRELAND 3 CZECK 2 FINLAND 2 MOLDOVA 2 JORDAN 1 SERBIA 1 [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.

0.6309 | 0.6309 |

TRIADIC CANTOR SET: Hence the interesting mea ln 3 d cm cm==

≅ 10 cm

New ton

Einstein 1905 Einstein 1915Dirac

Quantum gravity?

0; 0; 0; 0 | |

( | 0:) |

ch G ks Thefull tetrahe dron tatistic al mechanic s of quantum g corre sponds to the at its centerav t Griy c h k h >0 G >0 h =0 G =0 h =0 G >0 h >0 G =0 h =0 G >0 h >0 G =0 h >0 G >0 h =0 G =0

Schroedi nger

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

, , , :B Thefour inde pendent univ ersal constants of c Gconte mpor hary phy s s kic

A. Pluchinoand C. T. (2006) A. Pluchinoand C. T. (2006)

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 assumingthat their sphere of action is very smallcompared to their mean free path.

Ludwig BOLTZMANN

Elementary Principles in Statistical Mechanics -Developed with |

J.W. GIBBS 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 finitevalued, 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 involvedin the formula (92).

EnricoFERMI Thermodynamics(Dover, 1936)

The entropy of a system composed of several parts is very oftenequal to the sum of the entropies of all the parts. This is true if the energy of the system is the sum of the energies of all the partsand 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 obviousand 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 Scientia36, 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 independentparts 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 hypothesisbecause 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 ofa system are a priori equally probablein 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 intuitivefeelingas to the proper measure. [...] It is mathematically more suitable. [...].}

This theorem and the assumptions required for its proof, are in no way necessaryfor the present theory. It is given chiefly to lend a certain plausibilityto 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 additivitypostulate

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, additivityand homogeneity can be expected to be reasonable approximationsfor systems containing many particles, provided that the intramolecularforces have a short range character.

Radu BALESCU

Equilibrium and NonequilibriumStatistical 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 studyof the way in which the concepts of mixingand the concept of large numbers of degrees of freedominfluence 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 forcesand 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 StatisticalMechanics (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 remarkablysuccessful at explaining facts. The mathematical investigation of the thermodynamic formalism isin 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 equationis dedicated to define the BGentropy form. It is introduced after the words “we define its entropy”without any kind of justification or physical motivation.]

Nicovan 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 gravitationis 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).

U and N, is homogeneous of degree 1: |

These various quantities are connected with one another through thermodynamic relations which make their extensive or intensive nature obvious, as soon as onepostulates, for instance, for a fluid, that the entropy, considered as a function of the volume Omega and of the constants of motion such as S(xOmega, x U, x N)=x S(Omega, U, N) (Eq. 5.43). [...] Twocounter-exampleswill help us to feel whyextensivity is less trivial than it looks. [...] A complete justificationof the

Laws of thermodynamics, starting from statistical physics, requires a proof of the extensivity(5.43), a property which was postulatedin macroscopic physics. This proof is difficult and appeals to special conditions which must be satisfied by the interactionsbetween the particles.

Celestial Mechanics (John Wiley, New York, 1985)

This means that the total energy of any finite collection of selfgravitating 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 Ruelle1966). Basically it is that simple. One can ignore the factthat 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.

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 propertiesfor subregionsof 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 extensiveproperty, 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 whyis the entropy of a black hole proportional to the square of its radius, and notto 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 GibbsianTheory[J. Stat. Phys. 72, 879-1167 (1993)]

We provide a careful, and, we hope, pedagogical, overview of thetheory of Gibssian measures as well as (the less familiar) non-Gibbsianmeasures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. Toward a Non-GibbsianPoint of View:Let us close with some general remarks on the significance of (non-)Gibbsiannessand (non)quasilocalityin statistical physics. Our first observation is that Gibbsiannesshas heretofore been ubiquitous in equilibrium statistical mechanics because it has been put in by hand: nearly all measures that physicists encounter are Gibbsianbecause physicists have decidedto study Gibbsianmeasures! However, we now know that natural operations on Gibbs measures can sometimes lead out of this class. [...] It is thusof 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 Gibbsiannessis not put in by hand, one should expect non-Gibbsiannessto be ubiquitous. This is probably the case in nonequilibriumstatistical mechanics. Since one cannot expect all measures of interest to be Gibbsian, the question thenarises whether there are weakerconditions 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 renormaliztiongroup, even though this measure is provably non-Gibbsian. One may also inquire whether there is a classification of non- Gibbsianmeasures according to their “degree of non-Gibbsianness”.

(Parte **1** de 2)