Generalized Statistical Mechanics for The

Home , Vlasov equation

CBPF-NF-043/95

GENERALIZED STATISTICAL MECHANICS FOR THE

N-BODY QUANTUM PROBLEM { IDEAL GASES

Sergio Curilef

Centro Brasileiro de Pesquisas Fsicas

Rua Dr. Xavier Sigaud 150, 22290-180{ Rio de Janeiro, RJ { Brazil

Abstract

Within Tsallis generalized thermostatistics, the grand canonical ensemble is derived for

quantum systems. In particular, the generalized Fermi-Dirac, Bose-Einstein and Maxwell-

Boltzmann statistics are de ned. The b ehavior of the chemical p otential is depicted as a

function of the temp erature. Some thermo dynamic quantities at high and low temp erature

are studied as well.

Key-words: Generalized Statistics; Non-extensive Systems; Ideal Gas.

PACS index: 05.30-d; 02.50+s; 05.70Ce; 05.90+m

CBPF-NF-043/95 1

I. INTRODUCTION

If the appropriate distribution function for a system is known, we can compute the

exp ectation values of all thermo dynamic quantities, such as energy and numb er of par-

ticles, as well as sp eci c heat, magnetization, etc. However, the entropy still eludes us.

Boltzmann, followed by Gibbs, intro duced the one which yields the correct results for the

thermo dynamic prop erties of standard systems. This is

X X

(N ) (N )

S = k p ln p :

i i

N =0

An imp ortant prop erty of this entropy is the extensivity (additivity).

Now, non-extensivity (or non-additivity) is an imp ortant concept in some areas of

physics, byway of reference to some interesting generalizations of traditional concepts.

A generalization of the Boltzmann-Gibbs statistics has b een recently prop osed by Tsallis

[1{ 3] for non-extensive systems. This generalization relies on a new form for the entropy,

namely

i h

P P

(N )

p 1

N =0

S k ;

1 q

where q 2<; k is a p ositive constantand S recovers its standard form, in the limit q ! 1.

Various prop erties of the usual entropyhave b een proved to hold for the generalized

one: p ositivity, equiprobability, concavity and irreversibility [4]; its connection with ther-

mo dynamics is now established and suitably generalizes the standard additivity(itis

non-extensiveifq 6=1)aswell as the Shannon theorem [5].

The thermal dep endence of the sp eci c heat has b een studied for some physical sys-

tems, among them, wehave the d = 1 Ising ferromagnet [6]; a con ned free particle

(square well) [7]; two-level system and harmonic oscillator [8] and an anisotropic rigid

rotator [9].

It is imp ortant to remark that, this formalism has already received some physical

and mathematical applications. Among them, let us mention: Self-gravitating systems,

CBPF-NF-043/95 2

Stellar p olytrop es, Vlasov equation [10{ 13]; Levy-like anomalous di usion [14{ 17]; Cor-

related anomalous difusion [18,19]; d = 2 Euler turbulence [10]; Self-organizing biological

systems [20]; Simulated Annealing (optimization techniques in Genetics, Traveling sales-

man problem, Data tting curves, quantum chemistry) [21{ 26]; Neural networks [27].

This generalized statistics has b een shown to satisfy appropriate forms of the Ehrenfest

theorem [28]; von Neumann equation [29]; Langevin and Fokker-Planck equations [19,30];

Callen's identity (used to approximatively calculate the critical temp erature of the Ising

ferromagnet) [31]; Fluctuation-dissipation and Onsager recipro city theorems [32]; its con-

nection with Quantum Groups [33], quantum uncertainty [34,35], fractals [36,37], quantum

correlated many-b o dy problems [38], nite systems [2,39], etc. has b een established. In

additon to this, some asp ects of the generalized statistical mechanics in relation to the

N-b o dy classical problem were discussed [40,41], in order to treat more general situations

than the collisionless one.

There exists an attempt to generalize the quantum (Fermi-Dirac and Bose-Einstein)

statistics [42], but it was not taken into account the diculty asso ciated with the concomi-

tant partition function owing to the factorization pro cess shown in [40]. Consequently,

the quantum ideal gas has not yet b een adequatly discussed within generalized statistics.

The micro-canonical and canonical formulations have b een quite well studied up to

now. In the present pap er, the formalism in the grand-canonical ensemble is generalized.

In Section I I, the grand-partition function is obtained. In Section I I I, the extensions

of the Hilhorst transformations to the grand canonical ensemble are shown. Along the

same lines, the distribution function is generalized as well. In Section IV, the generalized

chemical p otential is depicted as a function of the temp erature for the Fermi-Dirac, and

Bose-Einstein gases. Approaches at high and low temp eratures are derived.

CBPF-NF-043/95 3

II. OPEN SYSTEMS: GENERALIZED GRAND-PARTITION FUNCTION

In general, op en systems can exchange heat and matter with its surroundings; there-

fore, the energy and the particle numb er will uctuate. However, for systems in equilib-

rium we can require that b oth the average energy and the average particle number be

xed. To nd the probability distribution, we need to get an extremum of the entropy

which satis es the ab ovementioned conditions. We pro ceed by the metho d of Lagrange

multipliers with three constraints.

In this problem, we can require that the generalized probability distribution b e nor-

malized over all p ossible numb er of particles and all states of the system. Thus, the

normalization condition takes the form

X X

(N )

p =1 ; (1)

N =0

the generalized average energy is de ned

h i

X X

(N ) (N )

p E = U ; (2)

j j

N =0

it is also called q-expectation value [3] of the energy. The generalized average particle

numb er is de ned

h i

X X

(N )

= N : (3) N p

N =0

or q-expectation value of N .

To obtain the equilibrium generalized probability distribution, wemust nd an ex-

tremum of the Tsallis entropy sub ject to the ab ove constraints. This gives us

" #

h i

q 1

(N ) (N )

+ q E + N p =0; (4)

o E N

j j

q 1

(N )

and where , and are the Lagrange multilpiers. Let us multiply Eq.(4) by p

o E N

sum. It is found

i h

P P

(N )

1 p

N =0

o E N

= U + N : (5)

q q

qk q 1 k 1 q k

CBPF-NF-043/95 4

If we compare the Eq.(5) with the grand p otential = U TS N , taking = 1=T

1=(q 1)

and = =T and de ning ( ; )=[(q 1) =q k ] , the obtained probability

N q o

distribution for the grand-canonical ensemble is the following

h i

(N ) (N )

p = 1 (1 q )(E N ) = ( ; ); (6)

j j

(N )

where =1=k T ,thenumb er of particles N =0; 1; 2:::,and E represents the N-particle

energy sp ectrum (characterized bythequantum numb er or set of quantum numb ers j ).

It is convenient to remark that in general

2 3

1=q

h i

X X

(N ) (N )

(N )

4 5

p p 6= p p (7)

j j

(N )

where p is the probabilityofhaving N particles (no matter the energy value) and p

h i

(N )

is the quantity which enables us re-writting Eq.(3) as N p = N ; unless q =1,

N =0

(N )

p generically di ers from p ( for instance, p = 1 always, whereas in general

N N

N =o

(N )

p 6=1 ).

N =0

The generalized grand partition function is obtained from Eq.(6) with the aid of Eq.(1)

h i

X X

(N )

( ; )= 1 (1 q )(E N ) : (8)

N =0 j

On the other hand, we can also obtain the fundamental equation for op en systems, this

takes the following form

; (9) = kT

1 q

and it is similar to the fundamental equation for closed systems (canonical ensemble [5]).

The average particle number is given by,

@ @ 1

q q

N = = kT : (10)

@ ( ) @

For a quantum system the expresion for the probability density op erator b eing given by

[1 (1 q )(H N)]

= ; (11)

CBPF-NF-043/95 5

where H refers to the Hamiltonian of the system and N to the particle numb er op erator.

The grand-partition function is given by

= Tr [1 (1 q )(H N)] : (12)

The trace in Eq.(12) can b e evaluated regarding some convenient set of basis states.

I I I. HILHORST INTEGRAL TRANSFORMATIONS AND GENERALIZED

DISTRIBUTION FUNCTION

The so called Hilhorst integral transformations [2] and the extension for q<1 shown by

Prato [41] are imp ortant b ecause they connect a thermo dynamic or statistical generalized

quantity to its resp ective standard quantity. Therefore, an extension of the Hilhorst inte-

gral to the grand-canonical ensemble is derived. From the representation of the Gamma

function wehave

= d e : (13)

( )

Using this expresion in the generalized grand partition function (8) with the identi cations

(N )

N ), it is obtained =1=(q 1) and =1+ (q 1)(E

X X

(N )

q 1

( ; )= d exp [1 + (q 1)(E N )] : (14)

( )

N =0

q 1

Whenever

Z Z

1 1

X X X X

d = d ; (15)

0 0

N =0 j N =0 j

Eq.(14) b ecomes

h i

X X

(N )

q 1

exp 1+ (q 1)(E N ) : (16) d ( ; )=

) (

N =0 j

q 1

Finally, it is obtained

q 1

( ; )= e ( (q 1); ); (17) d

q 1

( )

q 1

CBPF-NF-043/95 6

for q>1; and (see [41])

2 q i

e ( (1 q ); ); (18) ( ; )= ( ) d ( )

1 q

1 q 2

for q<1. The contour C in the complex plane is depicted in FIG. 1. The connection

between Eq.(17) and Eq.(18) is shown in App endix A.

Now, we write similar transformations for the q -exp ectation value of the energy.Itis

obtained

q 1

U = e ( (q 1); )U ( (q 1) ) (19) d

q 1 1

[ ( )] ( )

l q 1

for q>1 (for the canonical ensemble it is shown in [9]); and

( )

U = d ( ) e ( (1 q ); )U ( (1 q ) ) (20)

q 1 1

[ ( )] 2

for q<1.

Similar expresions are obtained for the q -exp ectation value of the particle number

q 1

e ( (q 1); )N ( (q 1) ) (21) d N =

1 1 q

[ ( )] ( )

q 1

for q>1; and

( )

N = d ( ) e ( (1 q ); )N ( (1 q ) ) (22)

q 1 1

[ ( )] 2

for q<1.

Finally, let us also write the generalized distribution function in the Hilhorst manner.

We remark that a double sum over all p ossible numb er of particles and all states of the

systems app ears at eachquantity, in particular, the average particle numb er (3). The

double sum can b e transformed to one only sum over all states of a single particle by

standard metho ds.

Now, let us remember that

N = n ; (23)

1 1l l

CBPF-NF-043/95 7

where n is known as the distribution function and it is very well de ned in the Maxwell-

Boltzmann, Bose-Einstein and Fermi-Dirac statistics. By replacing Eq.(23) into Eq.(21)

and Eq.(22), we obtain the generalized distribution functions. We de ne

q 1

n = d e ( (q 1) )n ( (q 1) ) (24)

ql 1 1l

) [ ( )] (

q 1

for q>1; and

( )

n = d ( ) e ( (1 q ) )n ( (1 q ) ) (25)

ql 1 1l

[ ( )] 2

for q<1.

Therefore, wehave de ned the generalized distribution functions in connection with

the standard distribution and partition functions through Eq.(24) and Eq.(25). In addi-

tion, wehave

N = n (26)

q ql

which is the generalization of the Eq.(23).

IV. APPLICATIONS TO QUANTUM IDEAL GASES

The statistics of N-b o dy quantum systems plays a crucial role in determining the

thermo dynamic b ehavior at very low temp erature. It is known however that, in the stan-

dard framework, there is no di erence b etween Bose-Einstein and Fermi-Dirac statistics

at high temp erature. Maxwell-Boltzmann statistics is the name given to the statistics

which describ es the b ehavior of the systems at high temp erature.

(N )

of the system is sp eci ed by the one-particle states. The The quantum state E

total energy is given by

(N )

= + + ::: + = n + n + ::: + n E

1l l l 1 1 2 2 1 1

l N

Where is the energy of the state and in is the o ccupation number, wehave also

i i

n + n + :::+ n = N . The generalized grand partition function for the Maxwell-Boltz-

1 2 1 mann statistics can b e written as

CBPF-NF-043/95 8

X X X X

M B

= ::: [1 (1 q )( + + ::: + N)] ; (27)

1l l l

q 2

N !

1l l l N =0

where wehave inserted the factor 1=N ! in the same wayasinthe q = 1 statistics, b ecause

it gives us the prop er form of the grand partition function for indistinguishable particles

at high temp erature.

The generalized grand partition function in Fermi-Dirac statistics, acordding to Pauli

exclusion principle, is given by

1 1 1 1

X X X X

F D

[1 (1 q )( + + ::: N)] ::: = : (28)

1l l l

2 q

N =0

l =l +1 l =l +1 1l

2 1

N N 1

Each di erent set of o ccupation numb er corresp onds to one p ossible state. Sometimes, it

is convenient to write the partition function by the equivalent form

" #

1 1 1

X X X X

F D

= ::: ::: 1 (1 q ) n ( ) : (29)

l l

n =0 n =0 n =0

o 1

The exclusion principle restricts the o ccupation number (n ) of each state to 0 or 1.

The generalized grand partition function in Bose-Einstein statistics is given by

1 1 1 1

X X X X

B E

[1 (1 q )( + + ::: + N)] ::: = : (30)

1l l l

2 q

N =0

l =l l =l 1l

2 1

N N 1

Here, there is no restriction on the numb er of particles that can o ccupy a given momentum

state. Another form for this partition function is

" #

1 1 1

X X X X

B E

: (31) = ::: ::: 1 (1 q ) n ( )

l l

n =0 n =0 n =0

o 1

The o ccupation number (n )ofeach momentum state can b e 0,1,2,...

A. Particles with Perio dic Boundary Conditions

We consider a gas of non-interacting particles of mass m with the condition exp(ik `)=

1, so k ` =2l and l =0; 1; 2; 3;:::

The sp ectrum for a single particle is given by

2 2

h k h 2l

= = ( )

2m 2m `

whereh = h=2 (h is the Planckconstant).

CBPF-NF-043/95 9

1. High-Temperature Approach

M B

In order to nd in connection with ( Eq.(17) and Eq.(18)), we write

M B

conveniently.Thus, the grand partition function within q = 1 statistics is given by

" #

X X

M B

e = : (32)

N !

l N =0

If the volume is large enough, the particle energies will b e closely spaced and wecan

replace the sum over l byanintegral over a continuous variable k .Thus,

D=2

` 2 2

D 1 h k =2m

! e dk k e ; (33)

2 (D=2)

l=1

M B

b ecomes where D is the dimension. Hence,

ND=2

1 m`

M B N

; (34) = e

N !

2 h

N =0

Replacing Eq.(34) into Eq.(18), we obtain:

ND 1

ND=2

1q 2

X ( )[1 + (1 q )N ]

M B

= ; (35)

2q 2

DN=2

N !(1 q ) ( + ) 2 h

N =0

1q 2

for q<1.

2. Fermi-Dirac Gas at Zero-temperature Limit

The average particle number at low temp erature is given by (see App endix B)

# ! "

D=2

2 D=2

m` 2

+ g (; D )(k T ) : (36) N =

n B 1

(D=2) D

2 h

n=1

We solve Eq.(22) with the aid of Eq.(36) for q<1

D=2

N i

d ( ) = e + (37)

(D )

D 2

= ( ) ( )A

e : d ( ) + g (D; )( (1 q ))

1 n

n=1

(D )

is de ned in Eq.(B4). Finally where A F

CBPF-NF-043/95 10

D=2

= 1 + (38)

(D )

) g (D; )(

+ [kT (1 q )(H N)] :

+2n) (1 q ) (

n=1

Here 1 is the q-exp ectation values of 1 and it is a function of T ; in general 1 6=1,unless

q q

q = 1. It is easy to verify that in the limit q ! 1, Eq.(38) is reduced to Eq.(36). Now,

the generalized Fermi level ob eys the following expresion,

2 3

D=2 1

D=2

g (D; )( )

n F

2 m`

4 5

lim 1 + lim N = (H N) (39)

q q F

2 1

T !0 T !0

(D=2) D

( +2n)

2 h

n=1

Now, we can verify that the sum vanishes when q ! 1, b ecause

) (

=lim(1 q ) =0 lim

q !1 q !1

( +2n)

thus, Eq.(39) is reduced to the known result for the Fermi level in Boltzmann-Gibbs

statistics when q ! 1.

B. Particles in a Box

We consider a gas of non-interacting particles into a b ox. The sp ectrum for a single

particle is given by

h l

= ( )

2m `

where l =1; 2; 3; ::: and ` is the side of the b ox.

The chemical p otential as function of the temp erature is depicted in FIG. 2 for q =1,

typical values of N and D = 1; the thermo dynamic limit is easily computed. See the

Fermi-Dirac case in FIG. 2.(a) and the Bose-Einstein case in FIG. 2.(b).

In the Boltzmann-Gibbs statistics the exp onential form of the probability distribu-

tion allows for the explicit integration, in evaluating the partition function ,ofthe

momentum-dep endent part (kinetic energy) of exp( H). This fact, reduces the work

involved in computing of the evaluation over just the con guration variables of the

one-b o dy con guration space. It is clear, this interesting prop ertyislostforq 6=1.

CBPF-NF-043/95 11

The b ehavior at high temp erature of the generalized partition function for q<1is

given by

1 m

m=2

1 DN

DN m 1q 2 2

X )[1 + (1 q )N ] X (

1 () (DN )! 2m`

M B

= ; (40)

2q 2

m=2

2 (DN m)!m!

N !(1 q ) ( + ) h

m=0

N =0

1q 2

for particles into a b oxinDdimensions.

The computation is very slowwhenq di ers from the unity. FIG. 3 depicts the chemical

p otential versus temp erature for q =0:8 (see Fermi-Dirac case in FIG. 3(a) and Bose-

Einstein case in FIG. 3(b)). The thermo dynamic limit in (a) is found by extrap olating

(to the origin) the trend of the chemical p otential with 1=N for xed temp erature T .

It was not reliable to do the same in (b) b ecause all generalized quantities converge very

slowly, which made numerically inaccessible the region N > 6.

CONCLUSIONS

It is well established the connection b etween the generalized statistical mechanics in

the grand canonical ensemble with thermo dynamics through the relation given by Eq.(9).

Following along the lines of the Hilhorst integral transformations for the grand parti-

tion function ,wehave obtained the analogous expressions for the appropriate averages

of the particle numb er and the energy in the grand-canonical ensemble. In the same style,

the generalized distribution functions are de ned as well.

It is clear that the statistical and thermo dynamic quantities transform to their stan-

dard forms in the q ! 1 limit, as it has b een shown in some cases.

ACKNOWLEDGMENTS

The author is very indebted to C. Tsallis and D. Prato for valuable discussions and

to C. Tsallis, S.A. Cannas, D.A. Stariolo, A. M. de Souza and M. Muniz for their helpful

comments on the draft version of this pap er. This work has b ene tted from partial sup-

CBPF-NF-043/95 12

p ort from the Fundacion Andes/Vitae/Antorchas, grant 12021-10. Part of the numerical

calculation was carried out on a Cray Y-MP2E of the sup ercomputing center at Ufrgs.

APPENDIX A

By taking z as variable of integration, F (z; )= e ( (1 q )z; ) and =

1=(1 q ); the integral in Eq.(18) can b e written as

I Z Z Z

1 1

dz (z ) F (z; )= + + dz (z ) F (z; ); (A1)

C ab bcd de

where ab, bcd and de are lines of C shown in FIG. 1. If we use z = for the integral along

i 2i

the line ab, z = e along the line bcd and z = e along the line de,wehave

I Z

1 i 1

dz (z ) F (z; )= e d e ( (1 q ); ) (A2)

C 1

i i i 1 e i

e d(e )(e ) e ( (1 q )e ;)

1 i

d e ( (1 q ); ): e

Now, putting q>1and ! 0we can see that the second integral vanishes. Thus,

I Z

q 1

dz (z ) F (z; )=2i sin( e ( (q 1); ) (A3) ) d

q 1

C 0

On the other hand, wehave the following prop erty of the function

: (A4) (p)(1 p)=

sin p

Using the Eq.(A3) and Eq.(A4) into Eq.(18), we obtain

q 1

e ( (q 1); ): (A5) ( ; )= d

1 q

( )

q 1

Summarizing, wehave recovered Eq.(17) from Eq.(18).

APPENDIX B

The average particle number N for a fermion system in the large enough volume

approach is given by

CBPF-NF-043/95 13

` 1

N = d : (B1) k

2 exp( ( )) + 1

Thus

Z D=21

D=2

2 2m 1 `

: (B2) N = d

1 k

2 (D=2) 2 exp( ( ))+1

Let us transform this integral with the following change of variable = kT z,

D=2

D=2 D=21

` kT 2 2m ( + kT z)

N = : (B3) dz

2 (D=2) 2 e +1

h =k T

De ning

! !

D D=2

D=2

D=2 2

` 2m 2 m` 2

(D )

= A ; (B4) =

2 2

2 (D=2) (D=2)

h 2 h

the integral can b e written as:

Z Z

D=21 D=21

1 =k T

N ( + kT z) ( kT z)

+ dz : (B5) = dz

(D )

z z

e +1 e +1

0 0

kT A =2

Using the simple transformation

1 1

=1 (B6)

z z

e +1 e +1

in the rst integral, then

=k T

D=21

dz ( kT z) (B7) =

(D )

kT A =2

Z Z

D=21 D=21

=k T 1

( kT z) ( + kT z)

dz + : dz

z z

e +1 e +1

0 0

The second integral converges very fast; therefore, if T vanishes the upp er limit can b e

replaced by 1,so

D=21 D=21 D=2

( + kT z) ( kT z) N

dz = + : (B8)

(D )

(D=2)kT e +1

kT A =2

Finally

! " #

D=2

2 D=2

m` 2

g (; D )(k T ) + ; (B9) N =

n B 1

(D=2) D

2 h

n=1

CBPF-NF-043/95 14

where

12n D=22n

g (; D )=(D=2 1)(D=2 2) (D=2 2n + 1)(1 2 ) (2n); (B10)

and (2n) is the Riemann function:

( )= :

m=1

In the same approach, the average energy U is given by

; (B11) U = d k

2 exp( ( )) + 1

then, we obtain

! " #

D=2

2 D=2+1

2 m`

U = + h (; D )(k T ) ; (B12)

1 n B

(D=2) D +2

2 h

n=1

where

12n D=22n+1

h (; D )=(D=2)(D=2 1) (D=2 2n + 2)(1 2 ) (2n): (B13) n

CBPF-NF-043/95 15

FIGURES

FIG. 1.

Contour C in the complex plane.

FIG. 2.

Chemical p otential for (a)Fermi-Dirac and (b) Bose-Einstein cases for D = 1 and

typical values of N within Boltzmann-Gibbs statistics (q = 1). The thermo dynamic

limit is shown in each case.

FIG. 3.

Chemical p otential for the (a)Fermi-Dirac, (b) Bose-Einstein cases for D =1,typical

values of N and q =0:8 within generalized statistics. The thermo dynamic limit is

extrap olated in (a)by standard metho d.

CBPF-NF-043/95 16

CBPF-NF-043/95 17

CBPF-NF-043/95 18

CBPF-NF-043/95 19

CBPF-NF-043/95 20

CBPF-NF-043/95 21

REFERENCES

[1] C. Tsallis, J. Stat. Phys. 52, 479 (1988).

[2] C. Tsallis, in \New Trends in Magnetism, Magnetic Materials and their Applications",

ed. J. L. Moran-Lop ez and J. M. Sanchez, 451 (Plenum Press, New York, 1994).

[3] C. Tsallis, Chaos, Solitons and Fractals 6, 539 (1995).

[4] A. M. Mariz, Phys. Lett. A 165, 409 (1992); J. D. Ramshaw, Phys. Lett. A 175,

169 (1993); J. D. Ramshaw, Phys. Lett. A 175, 171 (1993).

[5] E. M. F. Curado and C. Tsallis, J. Phys. A: Math. Gen 24, L69 (1991); Corrigenda:

J. Phys. A24, 3187 (1991) and 25, 1019 (1992).

[6] R. F. S. Andrade, Physica A 175, 185 (1991) and Physica A 203, 486 (1994).

[7] E. da Silva, C. Tsallis and E. Curado, Physica A 199, 137 (1993) Erratum: Physica

A203, 160 (1994).

[8] N. Ito and C. Tsallis, N. Cim. D11, 907 (1989).

[9] S. Curilef and C. Tsallis, Physica A 215, 542 (1995).

[10] B.M. Boghosian, \Thermodynamic description of the relaxation of two-dimensional

Euler turbulence using Tsal lis statistics", preprint (1995). E-mail to chao-

[email protected] with \get chao-dyn/9505012" on sub ject line.

[11] A. R. Plastino and A. Plastino, Phys. Lett. A 174, 384 (1993).

[12] J. J. Aly in \N-Body Problems and Gravitational Dynamics", \Pro ceedings of the

Meeting held at Aussois-France", eds. F. Comb es and E. Athanassoula (Publications

de l'Observatoire de Paris), 19 (21-25 March, 1993).

[13] A.R. Plastino and A. Plastino, Phys. Lett. A 193, 251 (1994).

[14] P. A. Alemany and D. H. Zanette, Phys. Rev. E49, 956 (1994).

CBPF-NF-043/95 22

[15] C. Tsallis, A.M.C. de Souza and R. Maynard, in \Levy ights and related topics

in Physics", eds. M.F. Shlesinger, U. Frisch and G.M. Zaslavsky (Springer, Berlin,

1995), in press.

[16] D.H. Zanette and P.A. Alemany,Phys. Rev. Lett (1995), in press.

[17] C. Tsallis, S.V.F Levy, A.M.C. de Souza and R. Maynard \ On the statistical-

mechanical foundation of the ubiquity of Levy distributions in Nature", preprint

(1995).

[18] C. Tsallis, \Nonextensive thermostatistics{Some new results", preprint (1995).

[19] A.R. Plastino and A. Plastino, \Non-extensive statistical mechanics and generalized

Fokker-Planck equation", preprint (1995).

[20] P. T. Landsb erg, in \On Self-Organization", Synergetics 61, 157 (Springer, 1994).

[21] C. Miron, Optimisation par recuit simulegeneralise,DEAdePhysique Statistique et

Phenomenes Non Lineaires, Ecole Normale Superieure de Lyon-France (1994).

[22] T.J.P.Penna, Phys. Rev. E 51, R1 (1995).

[23] D.A. Stariolo and C. Tsallis, Ann. Rev. Comp. Phys., vol. I I, ed. D. Stau er (World

Scienti c, Singap ore, 1995), p. 343.

[24] T.J.P.Penna, Computers in Physics 9 (May/June 1995), in press.

[25] C. Tsallis and D.A. Stariolo, \ Generalized simulatedannealing", preprint (1995);

app eared (in English) as Notas de Fisica/CBPF 026 (June 1994).

[26] K.C. Mundim and C. Tsallis, \Geometry optimization and conformational analysis

through generalized simulatedannealing", preprint (1995).

[27] S.A. Cannas, D. Stariolo and F.A. Tamarit, \Learning dynamics of simple perceptrons

with non-extensive cost functions", preprint(1995).

CBPF-NF-043/95 23

[28] A. R. Plastino and A. Plastino, Phys. Lett. A 177, 177 (1993).

[29] A. R. Plastino and A. Plastino, Physica A 202, 438 (1993).

[30] D. A. Stariolo, Phys. Lett. A 185, 262 (1993).

[31] E. F. Sarmento \Generalization of Single-Site Cal len's Identity Within Tsal lis Sta-

tistics", preprint (1995).

[32] A. Chame and E. V. L. de Mello, J. Phys. A27, 3663 (1994); A. Chame and E. V.

L. de Mello, \The Onsager Reciprocity Relations within Tsal lis Statistics", preprint

(1994); M. O. Caceres, Physica A (1995), in press.

[33] C. Tsallis, Phys. Lett. A195, 329 (1994).

[34] M. Portesi and A. Plastino, \Entropic formulation of uncertainty for quantum mea-

surements with Tsal lis entropy", preprint (1995).

[35] A.K. Ra jagopal, \The Sobolev inequality and the Tsal lis entropic uncertainty rela-

tion", preprint (1995).

[36] C. Tsallis, Fractals 3 (Septemb er 1995), in press.

[37] D.H. Zanette, \Generalized Kolmogorov entropy in the dynamics of multifractal ge-

neration", preprint (1995).

[38] P. Ziesche, in Information: New questions to a multidisciplinary concept, eds. K.

Kornwachs and K. Jacoby (Akademie Verlag Berlin, 1995), in press.

[39] A.K. Ra jagopal, Physica B (July/August 1995), in press.

[40] A.R.Plastino, A.Plastino and C. Tsallis, J. Phys. A: Math. Gen 27, 1 (1994).

[41] D. Prato, Phys. Lett. A 203, 165 (1994).

[42] F. Buy ukkili c and D. Demirhan, Phys. Lett. A181, 24 (1993); F. Buy ukkili c, D.

Demirhan and A. Gule c, Phys. Lett. A197, 209 (1995).