The Classical Gases in the Tsallis Statistics Using the Generalized Riemann Zeta Functions

HAL Id: jpa-00247368

https://hal.archives-ouvertes.fr/jpa-00247368

Submitted on 1 Jan 1997

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The Classical Gases in the Tsallis Statistics Using the Generalized Riemann Zeta Functions

Sorinel Adrian Oprisan

To cite this version:

Sorinel Adrian Oprisan. The Classical Gases in the Tsallis Statistics Using the Generalized Riemann Zeta Functions. Journal de Physique I, EDP Sciences, 1997, 7 (7), pp.853-862. �10.1051/jp1:1997201�.

�jpa-00247368�

The Classical Gases in the Tsallis Statistics Using the Generalized Riemann Zeta Fanctions

Sorinel Adrian Oprisan (*)

Department

of Theoretical

Physics,

"Al. I. Cuza"

University,

Bd.

Capon,

No. ii, 6600 Iasi, Romania

(Received

8

August

1996, revised 22

January

1997,

accepted

24 March

1997)

PACS.05.20.-y

Statistical mechanics

Abstract. In the last few _{years an}

increasing

interest has been

paid

to fractal

inspired

statistics. Our aim is to describe _{some new}

insight

obtained

using

Tsallis statistics. In the framework of the

generalized

statistics _we described _some

properties

of the Maxwell-Boltzmann

gases. The behavior of the

occupation

numbers with respect to the temperature indicates

similarities with Fermi gases.

Using

the Nernst theorem _we also determine the fractal index of statistics.

1. Introduction

In the last few years a number of papers

developed

_{a new area} of interest both for

experimental

and theoretical

physics.

Fractal

inspired

statistics becomes _an

important

tool in

studying analytically

and

numerically

the behavior of

complex systems.

^{A number of research} groups used the Tsallis

Iii

formalism of

generalized

statistics to find _{some new}

analytical

results and

more efficient numerical

algorithms.

Mariz [2]

approaches

the

problem

of the time evaluation of Tsallis entropy. ^In the

meantime,

Ramshaw

[3,4]

^obtained

important

results

concerning

Tsallis

statistic

using

the master

equation

and clarified the

meaning

of

thermodynamic stability

in the fractal

inspired

statistics.

Bfiy6kkili~

et al. [5] established the

general

form of the Maxwell-

Boltzmann,

Fermi-Dirac and Bose-Einstein

generalized (fractal)

distributions.

In the

present

^article we describe _some unusual

properties

of the Maxwell-Boltzmann gen- eralized distribution. Our

principal goal

is to pay attention to _some unusual features of the

system

in the Tsallis statistics and to

get

a more

physical insight

to the fractal index _q. The results that _we obtained _are based _on the

properties

of the Hurwitz zeta function

[6, 7].

^The

zeta function method allow _us to obtain

analytic

results both in the

factorization hypothesis

_[5]

as well in the

general

_case. We also believe that the realistic

interpretation

^of the fractal index q is

given by

the Nernst theorem.

2. Tsallis Distributions

In the

following

_we

adopt

the

general setting

of the second

quantization theory. Therefore,

let us consider _a

non-interacting quantum

system

composed

of N

particles.

We consider

(* ^e-mail soprisantluaic.ro

©

Les

#ditions

_de

_Physique

₁₉₉₇

854 JOURNAL DE

PHYSIQUE

I N°7

the system ^{in the heat bath. Then the}

steady

states of the system are solutions of the

Schrbdinger equation

H1fiR

"

ERi~R, Ii)

where ifiR ^{and H} are the _wave function and

respectively

the Hamiltonian of the

system.

Let

(ni,

_n2,..

, nk,.

)

be the

occupation

^{number of the}

quantum

state R. The _nk

quantities

_are the number of

particles

in the state k.

The Tsallis entropy ^is defined

by _[1,

_5]

s)

=

~~

i

~j Pjl. ^j2)

~

R

For the sake of

simplicity

_we consider the Boltzmann constant

kB

= 1. In order to determine the distribution

probability

let us assume the

validity

of the Boltzmann H theorem.

Using

the

Lagrange's multipliers

method with the

following

constrains

~jPR

=

1,

R

~jERP(

= £,

(3)

R

~jP(NR

=

N,

R

we first write the

expression

Q= 11-~jP( -a~jPR-fI~ERP(-+t~jNRP(, ₍₄₎

~~~

R R R R

that has to be

maximized,

where _a,

fl

and

~y are undetermined

Lagrange multipliers. Setting

the derivative of the

Q

with respect to

PR equals

to _zero then the

probability

of state R is

PR

₌

) _ii

+

fliq I)ER fliq I)NRJ~)'

,

15)

where

Zq

=

~ ii

₊

pjq I)ER piq I)NRI~)I

₁₆₎

R

is the

partition

^{function of the}

grand

canonical ensemble.

Assuming

the

validity

of the funda- mental

equation

of

thermodynamics

then _we can

identify fl

=

1IT,

_+t

=

-fl~t.

The energy of

the state R and the total

particle

number _can be

expressed by

the

occupation

number of the

one-particle

states

ER

= nisi + n2e2 + + nkek +

,

(7)

NR"ni+fl2+.

_{+nk+. ~~~}

Using (7, 8, 5, 6)

_one ^{obtains for} the

generalized partition

function

Zq

=

L _ii

₊

_{fliq i) ml (El}

_{J~) + +}

nN

(EN /~))) ~

, 19)

and for the

generalized probability

Pni,

n~ ⁼

) _ii

+

fliq I) ml iEi

_{J~) +} + _nN

(EN /~))) ' (lo)

When the correlation between

particles

_can be

ignored

the

partition

function

(9)

_can be

fac-

torized

[4,

5]

N m

Zq

=

fl ~ _ii

₊

_{fl(q 1)nk(ek} ^lt))~ _Ill)

k=I n~#0

In these conditions the

partition

function _can be viewed _{as a}

product

of factors each _corre-

sponding

to the k

single-particle

state. In what follows _we refer

only

to the Maxwell-Boltzmann

statistics. Thus the

partition

^function

(9)

becomes

Zq

=

~ Ii

₊

fliq I) _((Ski

_{/~) + +}

_{iEk~ /~)))} ~

,

i12)

where the _sum is _over all

possible

states of the individual

particles (ki, k~,

..,

kN) According

to the

factorization

view

point

the

partition

function of the

grand

canonical ensemble _can be written

Z~

=

z~, ₍₁₃₎

where

z =

f ii

₊

fliq i)

_(Sk

/~)) ' 1i4)

is the

generalized single-particle partition

function. The distribution

probability

from

(10)

becomes

Pk

_"

Ii

₊

fll~ l)l~k

_/L))

^' (is)

3. The Behavior of the Maxwell-Boltzmann Gases in the Factorized Tsallis

Ensemble

Let _us

consider,

_{as a} first step ⁱⁿ our

general approach, only

the

systems having

linear _energy level distribution

Sk = ak + b,

(16)

where _a and b _are constants. As _a

generic systems

we refer here to the harmonic oscillator.

3.I. FACTORIzATION ViEw POINT. From _a consequent

factorization

view

point (14)

be-

cOmeS

m

~ "

~ Ii

₊

fll~ l)Ek) '

₁₁

fll~ l)/L) '

>

Ii?)

k=0

and thus the

probability

distribution leads to

Pk

₌

/~

^~

^~i~

~~~~~

~~

i18) c ii

₊

piq I)Ek

@

856 JOURNAL DE

PHYSIQUE

I N°7

Introducing (16)

into

(17)

_one

gets

« N

z =

~ _ii

₊

_{fliq i) iak}

₊

_b)) ^'

=

ifliq i)a) ^' ~ _ik

₊

_a)

~

,

i19)

where

°

(~~~ _)~~~

~

fl(q 1)a

^~~~~

Let _{us note s}

=

fi.

The

single-particle partition

function

(19)

_can be written

z =

( _IS, a)

'

~~~

^~

(21)

«

where

((s, a)

=

£ (p

+

a)~~

is the Hurwitz zeta function

(or generalized

Riemann zeta func- p=o

tion) _[6,

_7]

(see Appendix). Using (16, 21), equation (18)

_can be put ^{in the} more useful form

jk

₊

a)-s

~k

₌

(js,aj ^i~~~

The

occupation

number of the kth

single-particle

^level is

fik ₌

NPk, ₍₂₃₎

where

N

_is the _mean number of

particles.

As _we

expected

for fixed _a the

occupation

number is _a

monotonically decreasing

function of k

l~

~

~

~$~

~

~~

~~~),~j/~

k

a~

^~~~~

Let k* be the value of k such that

lank /bT)~

= 0. This value _can be found from the

following equation

j~~~~~~

~

j~*j~ ~)

~

~~~'~~

~ ~

~~~

(~fi)

'

(IS

+

1, °~ f _jp

+

a)-s-i

p=o

Therefore,

for _a

given

k _we have

l~ l~

^~~

_(k* la) _)~+

al'

^~~~~

which indicates that when the

temperature

^T decreases the

occupation

^{number of the} _energy level with k < k* increases and vice-versa. The existence of the

spectrum

^of _energy ^{values for} the condensation is reminiscent of the

properties

of the Fermi-Dirac

systems

^where acts the Pauli exclusion

principle.

To be _more

specific

let _us refer to

Figure

1. We observe that there

w..-

~~___,__j

~

..~"'

.~

""'

o

"'

5 lo

~ ~

o ^°

Fig.

I. The

dependence

of the rank of energy level k _on the values of _s and _a.

exists _a minimum rank of the

energetic

levels that determine the Fermi energy. From

Figure

1

we also observe that when _s

approaches

the classic limit _{8 ~ oJ} the values of k* _are almost

constant. To _summarize the results it _can be said that in the

generalized

Maxwell-Boltzmann

statistics the system ^has features similar to Fermi gases. The _mean value of the _{energy can} also be found

~

fl

"

~ _~kPl

"

qj)

~~

~(l~,

°) )(l~

⁺

I,")j ¹²⁸⁾

~~~ ^,

Using

the definition

(2)

the

entropy

becomes

ST

- S

Ii

~

i)

- S

Ii ~ll,li~l ₁₂₉₎

We _agree that the

validity

of the Boltzmann H-theorem is _a reasonable statement, for Tsallis statistics

especially (see

_[4]

).

But this

assumption

^does not resolve the

problem

of the

meaning

of fractal index q,

So,

_we also believe that it is reasonable to assume the

validity

of the Nernst theorem and that _can be the

key

of the

problem. Therefore,

let _us consider that

lim

S)

= 0.

(30)

T~0

From

(29)

^results the

following equation

that

properly

determine the fractal index of the Tsallis statistics

(~~~ Is, ao)

"

(~ Is

₊

_1, ao), (31)

where _{oo =} lim _a. From

Figure

2 we see that

equation (31)

has solutions with _a

high

level of

T~0

confidence.

Using (31)

the _mean

ground

state energy of the

system

^becomes

858 JOURNAL DE

PHYSIQUE

I N°7

f _{~.._}

i

zeta f

'"'

-" 'h,

0 01 "?~' ~'"i'"--.,_

,"" "'"'~,i

fl." ~..,

;""I ;""'j """""~".u~,

,k"' ' ""~""~~..l.

I ."'i ] i""""..,

I""' _.3.;.. "'I

"I j '~"'~'""J....

i '"'"""...._,f '

i 1""-.-..__

""'""1

l

2

,-""'

0.002

_,.

0 15

2

2.2 10

~'~

2.6 5 ^~

s

Fig.

2. The surface

plot

of the functions

(~+~(s, oo) (label 2)

and

(~(s

+ I,

oo) (label 1).

where

k(

= lim k*,

Having

in mind the

meaning

^of the k* it _can be said that

ski

^{is the}

T~o

energy of the Fermi level at 0 K.

Therefore,

_{a new} difference manifests between the classic and

generalized

Maxwell-Boltzmann statistics.

3.2. THE NONFACTORIzABLE CASE. We consider that in the

expression

of the

partition

function

(14)

^{there is} no reason to

accept

the factorization with

respect

to ~

Therefore,

relation

(17)

is _no

longer

valid. In these conditions _we will _use the definitions

(14, 15).

The factorized

partition

^function of the

grand

^{canonical ensemble} can be written in the form

«

~ ~

N

z =

~ _ii

₊

_{fliq i) iak}

_{+ b}

_/~))

~

=

lfliq i)a)

@

~ik

₊

_O*) ', ₁₃₃₎

where

~

~~

_{~~ ~~}

~~

l)q

I)a

~ a

~

~

fllq I)a

^~~~~

That relation _can be written in _a soniewhat _more

transparent

form

z =

~~ ((s, o*). (35)

s

~

Introducing (16)

^and

(35)

into

(18) gives

^the

probability

distribution

p

ik

+

°*)

~~

~

(iS, a*) j~~~

Using

the definition of _mean number of

particles

in

grand

canonical Maxwell-Boltzmann _en- semble

fl

=

~

_nk,

₍₃₇₎

k

and the fact that

fl

_{is constant}

one obtains:

l~i~~ _~~~ ^~~~~,~j/~~

k

la*~

~~~~

Let k/~~ ^{be the value of k} such that

lank /bT)~

= 0. This value _can be found from the

following equation

«

~j~

~~

( ^PIP

+ °" ~

~/v)

"

(j~ ~i

o*)

^{°~ "}

~ oo

~3~)

'

£

_{~p +} a"

)~~~~

p=o

Therefore,

for _a

given

k _we have

Iii1~ _n~1

~~ li)'11

₊

_~~ ¹⁴⁰⁾

~~~

It must be

pointed

out that the value of

k/~~

depends

_on chemical

potential

_~. It _can be said that the behavior of the

system

^is not

changed

but _now the Fermi level

depends

on the chemical

potential

of the gas.

4. The Behavior of the Maxwell-Boltzmann Gases in the Tsallis Ensemble. The

General Solution

The

general

form of the

partition

function is

~Q

~

~~ ~

$~~

_~~~~ki ~ ~k2 ^{~ ~} ~kN_~~^~~~

ki,k2.

=

~ _{(afl(q 1)(ki}

₊

_k2

_{+ +}

_kN)

_{+ +}

_{fl(q 1)Nb)} ^~

hi,k~

=

lafllq i)) ' ~j lki

₊

k?

+ +

kN

+ r~)

I, ₁₄₁₎

ki,k2 u-here

~

~~ ~~~~

'~

apj~ ₁₎

^~~~~

In these conditions the

probability

to realize the

one-particle

k energy level is

f _(ki

+ +

kN

_{+ ~y)}

~ f _C(

~~_3((S,

P + k

+'f)

~~

^~~'

~~~(kl

+ +

kN

₊

'f)

~~~

~

~~i+p-2((~'P ~'f~

~~~~

ki. kN=0 P"°

In the

following

_n>e short

point

out _some of the most

important

features of the

occupation

number.

Thus,

_{as we}

expected

when energy level order increases the

occupation

number decreases

according

to the formula

£

«

C(~~_~((s

+

1,

_p + k +

i)

~ IT

^{~~ ~~}

i _{cj~~_~(js,}

p + r~)

~ °' ~~~~

»=o

862 JOURNAL DE

PHYSIQUE

I N°7

where _al,

, aN, a are constants.

By performing

_a Mellin transform

(55)

becomes

~

«

~l~,~i; ,aN,a)

=

w ~

/t~~e~iaini+.+~NnN+~)dt

nl;.,nN"0 _~

=

) _Itsie-at ⁱⁱ _e-~Nti

^-~

+

~

e~Lt I

ii e~Jti ~ldt

~ ~ ~

o ^{= J=}

m

=

ajj~ ~ ~ _{( (s,}

_{b + bki + +}

bk~,

(56)

p=0 1<ki<...<kp<N-I

where

bj

= aj

laN,

₌

alaN, j

"

1, 2,.

,

N 1. If _al

= a2 = " aN then

( is;

_al,

, aN,

a)

=

a[~ ~ _{C( ~~_~(}

s, p + ^~

(57)

o

~i

We also indicate _some of the most

important properties

^of the Hurwitz zeta function that _we used in the _text.

I

1°~jj~~j

=-

(lp+a)~~Inlp+a) ^<0,

~ p=

2.

l~~jj

^~~ " ~~

f~ ^lfl

⁺

^°)~~~~

^"

^{~~( l~, °)}

^<

^o,

s P"

m ~

3'

(P~ ^lP

+

~)~~

"

L cl j-O)~ ( is

^k

a)

~'" 1=0

' '

cc 1

4.

L P~( _IS,

_{p + Ci)}

"

L cl j-Ci)~ j(

_IS I ₊

j i, O)

+

ii

_b)

(

IS ^t +

j, ~))

p=0 j=0

References

[II

^Tsallis

C.,

J. Stat.

Phys.

52

(1988)

479.

[2] ^Mariz

A.M., Phys.

Lett. A 165

(1992)

409.

[3] Rarnshaw

J.D., Phys.

Lett. A 175

(1993)

169.

[4] Ramshaw

J.D., Phys.

Lett. A198

(1995)

122.

[5]

Biiyiikkili~ F.,

Demirhan

D., Giile~ A., Phys.

Lett. A 197

(1995)

209.

[6] Elizalde

E.,

J.

Phys.

A: Math. Gen. 22

(1989)

931.

[7] Waldschmidt

M.,

Moussa

P.,

Luck J.-M. and

Itzykson C.,

From Number

Theory

to

Physics

(Springer-Verlag,

^Berlin

Heidelberg, 1992).