Complexidade Clássica e Quantum e Termodinâmica Não-Extensiva, Notas de aula de Engenharia Elétrica

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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

Sabir Umarov Yuzuru Sato

Stanly Steinberg^ Ugur Tirnakli

Miguel Fuentes Silvio M.D. Queiros

Luis Moyano

Paul Rivkin

William Thistleton

John Marsh

Kenric Nelson

NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS^ Nonextensive StatisticalNonextensive StatisticalMechanics and Its ApplicationsMechanics andThermodynamics , SRA Salinasand C Tsallis, eds, Brazilian Journalof Physics 29 , Number 1 (^1999 )

, S Abe and Y Okamoto, eds, LecturesNotes in Physics (Springer, Berlin, 2001 )

Non Extensive Thermodynamicsand Physical Applications

, G Kania- dakis, M Lissia and A Rapisarda,eds, Physica A^^305 , Issue 1/2 (

Classical and Quantum Complexity andNonextensive Thermodynamics 2002 )

, P Grigo- lini, C Tsallis and BJ West, eds, Chaos,Solitons and Fractals^^13

Nonadditive Entropy and NonextensiveStatistical Mechanics, , Issue 3 ( 2002)

M Sugiyama, ed, Continuum Mechanics and Thermo-dynamics^^16 (Springer, Heidelberg,

^2004 )

Nonextensive Entropy -Interdisciplinary Applications

, M Gell- Mann and C Tsallis, eds, (OxfordUniversity Press, New York,

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

News and Expectations inThermostatistics G Kaniadakis and M Lissia, edsPhysica A^^340 , Issue 1/3 (

Trends and Perspectives in Extensiveand Non-Extensive StatisticalMechanics^2004 )

H Herrmann, M Barbosa^

and E Curado, eds, Physica A^^344 , Issue 3/4 (

Complexity, Metastability andNonextensivity 2004 ) , C Beck, G Benedek,A Rapisarda and C Tsallis, eds,(World Scientific, Singapore,^^2005 )

Nonextensive Statistical Mechanics:New Trends, New Perspectives,

JP Boon and C Tsallis, eds, EurophysicsNews (European Physical Society,

Fundamental Problems of ModernStatistical Mechanics, 2005 )

G Kaniadakis, A Carbone and M Lissia, eds,Physica A^^365 , Issue 1 (

Complexity and Nonextensivity: NewTrends in Statistical Mechanics, 2006 )

S Abe, M Sakagami and N Suzuki, eds, Progr.Theoretical Physics Suppl

Introduction to NonextensiveStatistical Mechanics -Approaching a Complex World 162 ( 2006 )

, C. Tsallis (in preparation)

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(2107 MANUSCRIPTS)

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[Updated 31 March 2007]^ SCIENTISTS

Newton Einstein 1915Einstein 1905^ Dirac Quantum gravity? (1/^

1. k = B

1

1 1

1

0;^ 0;^

0;^0 (

:^

B

B

c^ h^

G^ k

The full tetrahedronstatistical mechanics of quantum g

corresponds to the at its centerav t

G

r i y^

c^ h^

k

−^

− −^

−

^ >^

>^ >

=^ =^ =^

-1 c >0h >0G > -1c >0h =0G =

-1c >0h =0G > -1c >0h >0G =

-1c =0h =0G > -1c =0h >0G >0 Schroedinger -1c =0h >0-1c =0G =0h =0G =

C.T.,^ Introduction to Nonextensive Statistical Mechanics-Approaching a Complex World

(in progress -1^1 ) ,^ ,^ ,:

B

The four independent universal constants of c

G^ c ontempor

h ary phys^

s^

k ic^

−

HISTORICAL BACKGROUND AND PHYSICALMOTIVATIONS FOR ATTEMPTING TOGENERALIZE 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

Ludwig BOLTZMANN their mean free path.

Enrico FERMI

Thermodynamics

(Dover, 1936)

The entropy of a system composed of several parts is veryoften equal to the sum of the entropies of all the parts. Thisis true if the energy of the system is the sum of the energiesof all the parts and if the work performed by the systemduring a transformation is equal to the sum of the amountsof^ work

performed

by^ all

the^ parts.

Notice

that^ these

conditions are not quite obvious and that in some casesthey may not be fulfilled. Thus, for example, in the case of asystem composed of two homogeneous substances, it willbe^ possible

to^ express

the^ energy

as^ the

sum^

of^ the

energies of the two substances only if we can neglect thesurface energy of the two substances where they are incontact. The surface energy can generally be neglectedonly 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^

(1942); translated into English by R. Mantegna (2005). This is mainly because entropy is an additive quantity as theother ones. In other words, the entropy of a system composedof several independent parts is equal to the sum of entropy ofeach single part. [...]Therefore one considers all possible internal determinations asequally probable. This is indeed a new hypothesis because theuniverse, which is far from being in the same state indefinitively,is subjected to continuous transformations. We will thereforeadmit as an extremely plausible working hypothesis, whose farconsequences

could^

sometime

not^ be

verified,

that^ all

the

internal states of a system are a priori equally probable inspecific

physical

conditions.

Under

this^ hypothesis,

the

statistical ensemble associated to each macroscopic state Aturns out to be completely defined.