entropia nÃo aditiva, mecÂnica estatÍstica...

Constantino TsallisCentro Brasileiro

de Pesquisas

Físicas

e Instituto

Nacional

de Ciência

e Tecnologia

de Sistemas

Complexos

Rio de Janeiro

e

Santa Fe Institute, New Mexico, USA

EconoRio, Rio de Janeiro, Novembro

2010

ENTROPIA NÃO ADITIVA, MECÂNICA ESTATÍSTICA NÃO EXTENSIVA,

E ECONOFÍSICA:

VALORES EXTREMOS EM FINANÇAS

THERMODYNAMICS

MECHANICS (classical, quantum, relativistic ... )

LANGEVIN EQUATION

FOKKER-PLANCK EQUATION

STATISTICAL MECHANICSLIOUVILLE EQUATION

VON NEUMANN EQUATION

VLASOV EQUATION

BOLTZMANN KINETIC EQUATION

BBGKY HIERARCHY

Brau

n an

d H

epp

theo

rem

MASTER EQUATION

H theorem

N

THEORY OF PROBABILITIES

ENTROPY FUNCTIONALENERGY

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)

POSTULATE FOR THE ENTROPIC FUNCTIONAL

nonadditive (if 1)q

additive

1

1

1ln ( 0)1

ln ln

( ) :

,

:

q

qDEFINITION q logarithm

Hence the entropies can be rewritten

equa

xx xq

x

l probabili

x

W

i=1

W

i=1

( 1)

1 l

( )

n ln

1 ln ln

ii

q i qqi

ties generic probabilities

k pk Wp

k W

BG entropyq

entropy S

q R

k pp

TYPICAL SIMPLE SYSTEMS: Short-range space-time correlations

Markovian processes (short memory), Additive noise

Strong chaos (positive maximal Lyapunov exponent), Ergodic, Euclidean geometry

Short-range many-body interactions, weakly quantum-entangled subsystems

Linear/homogeneous Fokker-Planck equations, Gausssians

Boltzmann-Gibbs entropy (additive)

Exponential dependences (Boltzmann-Gibbs weight, ...)

TYPICAL COMPLEX SYSTEMS:Long-range space-time correlations

Non-Markovian processes (long memory), Additive and multiplicative noises

Weak chaos (zero maximal Lyapunov exponent), Nonergodic, Multifractal geometry

Long-range many-body interactions, strongly quantum-entangled sybsystems

Nonlinear/inhomogeneous Fokker-Planck equations, q-Gaussians

Entropy Sq (nonadditive)

q-exponential dependences (asymptotic power-laws)

e.g., ( ) ( 1)NW N

e.g., ( ) ( 0)W N N

- Additive versus Extensive

- Central Limit Theorem

- Predictions, verifications and applications

ADDITIVITY:

additive probabilistically independent

( ) ( )Therefo

An entropy is if, for any two systems an

( )

re

( ) ( ) ( ) (1, since

d ,

q q q

S A B S A S B

S A B S A

A B

S B

and ( ) are additive, an

) ( )

d

( 1) is nonadditiv

(

,

e .

)Renyi

BG q q

q q

S S q S

q S A S B

q

EXTENSIVITY:

1 2

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

...

extensive

trop

y is

0 l

m

i

i

f

N

N

N

NA A A

A A A

( ) , . ., ( ) ( ) S N i e S N N NN

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

O. Penrose,

SPIN ½

XY FERROMAGNET WITH TRANSVERSE MAGNETIC FIELD:

| | 1 0 | | 1

0

Ising ferromagnetanisotropic XY ferromagnetisotropic XY ferrom

transverse magnetic fieldL length of a block within

ag

a N cha

n t

in

e

F. Caruso and C. T., Phys Rev E 78, 021101 (2008)

2

. . , 06002 (2004)

9 3

,

,

ent

Using a Quantum Field Theory resultin P Calabrese and J Cardy JSTAT Pwe obtain at the critical transverse magnetic field

with c

cqc

cent

1 2

37 6 0.08

1

28

10 3 0.1623

ent

ent

Ising and anisotropic XY ferromagnets c

Isotropic XY ferr

in conformal f

omagnet c

ral char ield theoryge

q

q

Hence

and


29 3cqc

1.67 1.671 1ln (2 1)

qc S

(d = 1; T = 0)

(pure magnet with critical transverse field)

(random magnet with no field)

BG

Summarizing, for a wide class of quantum systems or subsystems with elements, we know that

ln ln for 1 quantum chains

for 2

(

boson

)BG

N

L N N d

L N

S N

N d

2 2/3

1 ( 1)/

ic systems for 3 for -dimensional bosonic systems

black

holed d d

L N N dL N N d

1

2

area law1 ln ( 1) (NONEXTENSIVE!)

1

( 1; )

For the same class of quantum systems, we expe

c

t

dd

d

enq

L L L Nd

d

d

S

(EXTENSIVE!( ) ( 1; 1)

(analytically and/or computationally shown for 1

)

,2)

dentt

N L N d q

d


SYSTEMS ENTROPY SBG

(additive)ENTROPY Sq (q<1)

(nonadditive)

Short-range interactions,weakly entangled blocks, etc

EXTENSIVE NONEXTENSIVE

Long-range interactions (QSS),strongly entangled blocks, etc

NONEXTENSIVE EXTENSIVE

quarks-gluons, plasma, curved space ...?

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

q-GAUSSIANS:2

2( / )1-1

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

qx

qq

p x qq x

e

q -

GENERALIZED CENTRAL LIMIT THEOREM:

S. Umarov, C.T. and S. Steinberg, Milan J Math 76, 307 (2008)

1[ ( )]

[ ]( ) ( ) = ( )

-Fourier trans

( )

1

form:

ix ix f xq q q q

qF f f x dx f x dx

q

q

e e

(nonline ar!)

For q<1

see K.P. Nelson and S. Umarov, Physica A 389, 2157 (2010)

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

q

qqf y F f x y y d q

1 1( ) [ ( )]( , ) [ ( )]( ) 2 2

i yf y F f x y y d F f x de

Particular case q = 1:

ON THE INVERSE q-FOURIER TRANSFORM:

M. Jauregui and C. T. (2010)

3/2-Gaussian


Hilhorst function

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

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

( 2)/( 1)

| |

| | 1 ( 1) | |

( ) if 0 | | 0

qq q

qq qq q qq

q qA

x A

C x q x A

f x A x

( 2)/( 1) if 0 | |

with ( ) 1

q q

A

x A

dx f x

0 2 1/( 1)

Particular case: 01 ( )

[1 ( 1) ] qq

A

f xC q x

Hilhorst function (q=5/4; A=1)


1 [ ]q independent 1 ( . ., 2 1 1) [ ]q i e Q q globally correlated

1

( ) ( )

(

Classic CL

)

,

T

with same ofx Gaussian G x

f x

<

( 2)Q

(0 2)

Q

1/[ (2- )]

CENTRAL LIMIT THEOREM

-

-

( )

( )

q scaled attractor when summing N independent identical random variables

with symmetric distribut

x

ion w

N q

f x

21

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

Q QQ

qith dx x f x dx f x Q q qq

2

1

| |

( )

L

evy

| | (1, )

L ( )( ) / | |

| |

(

(

-Gned

1, ) lim (

enk

) ( )

o CLT 1 )

,

,

c

c

c

with same x behavior

G xif x x

xf x C x

if x xwith x

x Levy distribution L x

1 11

1

3 / 1

2/( 1)

( )

( ) | | ( , 2)

( ) / | | | | ( , 2)

( ) ,

lim

( )

( ) ( )

c

c

q

q q

c

q

Q

q

q

q

with same of f x

G x if x x q

f x C x if x x q

with

x

x

G x

G x G x

S. Umarov, C. T. and S. Steinberg, Milan J Math 76, 307 (2008)

( , 2)

q

2 1 3* 2 (1 )

2 1

,

1,

2 1 3,

,2 3, 21

,(1 )/(1 )

| |

( ) / | |

( ) ,

( ~

( ) / | |

)

L

q qq

q qq q

q

qq

q

qq

x L

intermediate regime

with same x asymptotic behavior

G x C x

L

G x C x

S. U m arov, C . T ., M . G ell-M ann and S. SteinbergJ M ath Phys 51, 0335

02

(2010)

( )

distant regime

LASER COOLING:

21

1

(0)( )

1 ( 1)q

TT xxq

Devoe, Phys Rev Lett

102

(2009) 063001

SPIN RELAXATION IN SPIN GLASSES (NEUTRON SPIN ECHO):

Pickup, Cywinski, Pappas, Farago, Fouquet, Phys Rev Lett

102

(2009) 097202

SPIN RELAXATION IN SPIN GLASSES (NEUTRON SPIN ECHO):

21

11

1( ) 12

1

1 ( 1)

1 2

rel

rel

rel

qq

q

q ttq

tq

qq

Pickup, Cywinski, Pappas, Farago, Fouquet, Phys Rev Lett

102

(2009) 097202

hence2 2=1+ 1 2( 1) 1 2( 1)ret retsize sizeq q q q

A. Celikoglu, U. Tirnakli and S.M.D. Queiros, Phys Rev E 82, 021124 (2010)

Gutenberg-Richter- like law

avalanche size avalanche return

0 0

. .,

1) New representation of Dir

( ) ( ) ( )

Question: For what cla

ac delta:

2

ss

( )

of functions ( ) is this so?

(1 2)2

ikxq

i e

dx x

qx

x f x f x

f x

dk qe

M. Jauregui and C. T., J Math Phys 51, 063304 (2010)

q-PLANE WAVES:


( ) | | (| | ; )f x A x x R

A. Chevreuil, A. Plastino and C. Vignat J Math Phys 51, 093502 (2010)

standard Dirac delta

max1 1q

2) New representation of :


Archimedes

(c. 287 BC – c. 212 BC)

2 22

2

2

2( ) satisfies

3) -plane waves ar

with

( ) (cos sin ) with | ( ) | 1

1( 1

e square integrable (

) 1

3| | 2(

0 3):( , ) ( , ) ( , )

q qi xq

i kx tq

q qx t x tc

x N N x i x dx x

q

q

ckt x

qN

x t

e

e

1 2

1)q


q=1.1q=1

22

2

0 0

(quantum non-relativistic spinless free particle)4) generalized Schroedinger equation

, ,1

is giv

( )2

e by

2

n

Its exact soluti

on

q

q

x t x ti i c q R

t q m

x

0 0

2

. .

with

,

(Newtonian re

2

lation !)

k x ti p x Etq q

q

it

pEm

e e

F.D. Nobre, M.A. Rego-Monteiro and C. T. (2010)

2( 1)2 2 2

22 2 2

0

(quantum relativistic spinless free particle: e.g.5) -generalized Klein-Gordon equation:

, ,1, ,

, mesons )

is giv

( )

Its exact solution

q

q

x t x tm cx t q x t q Rc t

0 0

2 2 2 2 4

. .

en by

with (Ei

Particular c

nstein relat

0 -plane waves

,

ion

a

s

!

)

e:

k x ti p x Et

q qi

x t

E p c m c q

m q

e e


(quantum relativistic spin 1 2 matter and anti-matter free particles:

5) -generalized Dirac equation:

e.g., electron and positro

n)

q

i

2 ( )

1

( ) (1)

with

(4 4

,. , , , ( )

0 1 0 matrice ;

0 -1 0

,

s

, ,

)

q

q

jqi j ij i j ij

j

x ti c x t mc A x t x t q R

t

x tA x t A x t

a

(4 4 matrix)

where are complex consta

nt

.

sja


11 1

2 2 2

3 33

4 44

1

2

3

4

. .

is given by

with

,

,

,

, ,

Its exact solutio

n

k x ti p x Etq q

x ta a

x t ia ax t

a ax ta a

x t

aaaa

e e

2 2 2 2 4 (Einstein relation

(

being the same

hence !) )

q

E p c m c q R


LHC (Large

Hadron

Collider)CMS (Compact

Muon

Solenoid) detector

~ 2500 scientists/engineers

from

183 institutions

of

38 countries

http://physicsact.files.wordpress.com/2007/10/lhc1.jpg

q=1.15 T=0.145

PHENIX @ RHIC

BOOKS ON NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS

1999 2001 2002 2002 2004

2004 2004 2004 2005 2005 2006

2006 2007 2009

2004

2009 2009 2010

Full bibliography (regularly updated):

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

3246 articles by 5175 scientists from 72 countries

[25 November 2010]

CONTRIBUTORS

(3246 MANUSCRIPTS)

USA 1296BRAZIL 476ITALY 396CHINA

293

FRANCE 280GERMANY

264

JAPAN 264RUSSIA 192SWITZERLAND 180UNIT. KINGDOM 181SPAIN

160

INDIA

143TURKEY

97

ARGENTINA

89POLAND

78

SOUTH KOREA 77GREECE

64

HUNGARY 63CANADA

55

AUSTRIA

51TAIWAN

40

PORTUGAL 38ISRAEL 32MEXICO 30UKRAINE 29AUSTRALIA 26CZECK 24FINLAND 24BULGARIA 19IRAN

19

BELGIUM 18CROATIA 15NETHERLANDS 15DENMARK 12CUBA

11

SOUTH AFRICA

11VENEZUELA

11

CYPRUS 10PUERTO RICO 10COLOMBIA 8LITHUANIA 8CHILE

7

ESTONIA 7PAKISTAN 7ROMENIA 7NEW ZEALAND 6NORWAY 6SLOVAK 6SWEDEN 6IRELAND 5SERBIA 5SINGAPORE 5ARMENIA 4BOLIVIA 4EGYPT 4MALAYSIA 4SLOVENIA 4URUGUAY 4ALGERIA 3BELARUS 3KAZAKSTAN 2MOLDOVA 2PHILIPINES 2

[Updated 25 November 2010]

ECUADOR 1GEORGIA 1INDONESIA 1JORDAN 1PERU 1SAUDI ARABIA 1SRI LANKA 1THAILAND 1UZBEKISTAN 1

72 COUNTRIES 5175 SCIENTISTS

entropia nÃo aditiva, mecÂnica estatÍstica...

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