Grand-Canonical Ensemble Within Tsallis Thermostatistics

Home , Tsallis entropy

CBPF-NF-060/95

Sergio Curilef

Centro Brasileiro de Pesquisas Fsicas - CBPF

Rua Dr. Xavier Sigaud, 150

22290-180 { Rio de Janeiro, RJ { Brasil

Abstract

Within Tsallis generalized thermostatistics, the grand canonical ensemble is derived

for quantum systems. The generalized partition function is also obtained. In addition,

the Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann statistics are de ned and the dis-

tribution function is generalized as well.

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

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

{1{ CBPF-NF-060/95

1 Intro duction

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

way of reference to some interesting generalizations of traditional concepts. A genera-

lization of the Boltzmann-Gibbs statistics has b een recently prop osed by Tsallis [1, 2, 3 ]

for non-extensive systems. This generalization relies on a new form for the entropy, namely

1 Tr( )

S k ;

1 q

where q 2<; k is a p ositive constant. Its standard form intro duced by Boltzmann and

Gibbs (which yields the correct results for the thermo dynamic prop erties of standard

systems) is recovered in the limit q ! 1.

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

one [4]; its connection with thermo dynamics is now established and suitably generalizes

the standard additivity (it is non-extensiveif q 6=1)aswell as the Shannon theorem [5].

In general, the micro-canonical and canonical formulations have b een quite well studied

up to now and this formalism has received imp ortant applications.

Some asp ects of the generalized statistical mechanics in relation to the N-b o dy clas-

sical [6, 7] and quantum [8] problems were discussed, in order to treat more general

situations than the collisionless one. Now, there exists an attempt to generalize the quan-

tum (Fermi-Dirac and Bose-Einstein) statistics [9], but it was not taken into accountthe

diculty asso ciated with the concomitant partition function owing to the factorization

pro cess shown in [6]. Consequently, the quantum ideal gas has not yet b een adequately

discussed within generalized statistics.

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

Section 2, the generalized grand partition function is obtained; Hilhorst transformations

for the partition function and the q-exp ectation values of the energy and particle num-

b er are written. In Section 3, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein

statistics are de ned; the distribution function is generalized in the Hilhorst manner.

2 Op en Systems

An op en system can exchange heat and matter with its surroundings; therefore, the

energy and the particle numb er will uctuate. However, for systems in equilibrium we

can require that b oth the average energy and the average particle numb er b e xed. To nd

the probability distribution, we need to get an extremum of the entropywhich satis es

the ab ove mentioned conditions. We pro ceed by the metho d of Lagrange multipliers with

three constraints.

2.1 Generalized Grand-Partition Function

We can require that the generalized density op erator b e normalized over some set of basis

states. Thus, the normalization condition takes the form

Tr =1 ; (1)

{2{ CBPF-NF-060/95

the generalized average energy is de ned

Tr( H)=U ; (2)

it is also called q-expectation value [3] of the energy and H refers to the Hamiltonian of

the system. The generalized average particle numb er is de ned

Tr( N )=N : (3)

or q-expectation value of N (particle numb er op erator).

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

tremum of the Tsallis entropy sub ject to the ab ove constraints. The obtained density

op erator for the grand-canonical ensemble is the following

=[1 (1 q )(HN )] = ( ; ); (4)

where =1=k T . The generalized grand partition function is obtained from Eq.(4) with

the aid of Eq.(1)

( ; )= Tr [1 (1 q )(HN )] : (5)

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

takes the following form

= kT ; (6)

1 q

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

2.2 Hilhorst Integral Transformations

The so called Hilhorst integral transformations [2] are imp ortant b ecause they connect

a thermo dynamic or statistical generalized quantity to its resp ective standard quantity.

Therefore, an extension for q<1 of the Hilhorst integral in the Prato style [7] to the

grand-canonical ensemble is derived. We obtain

2 q i

( ; )= ( ) dz (z ) e ( (1 q )z; ); (7)

q 1

1 q 2

for q<1. The contour C in the complex plane is depicted in Figure 1. By taking

F (z; )= e ( (1 q )z; )and =1=(1 q ); the integral in Eq.(7) can b e written

I Z Z Z

1 1

dz (z ) F (z; )= + + dz (z ) F (z; ); (8)

C ab bcd de

where ab, bcd and de are lines of C shown in Figure 1. If weuse z = for the integral

i 2i

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

I Z

1 1 i

d e ( (1 q ); ) (9) dz (z ) F (z; )= e

1 C

{3{ CBPF-NF-060/95

i i i i 1 e

( (1 q )e ;) e d(e )(e ) 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); ): (10) ) d

q 1

C 0

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

(p)(1 p)= : (11)

sin p

Using the Eqs.(10) and (11) into Eq.(7), may b e written as

q 1

e ( (q 1); ); (12) ( ; )= d

1 q

( )

q 1

the Hilhorst transformation for q>1.

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

obtained

( )

e ( (1 q )z; )U ( (1 q )z ) (13) U = dz (z )

1 1 q

[ ( )] 2

for q<1; and

q 1

U = d e ( (q 1); )U ( (q 1) ) (14)

q 1 1

[ ( )] ( )

q 1

for q>1 (for the canonical ensemble it is shown in [10 ]).

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

( )

N = dz (z ) e ( (1 q )z; )N ( (1 q )z ) (15)

q 1 1

[ ( )] 2

for q<1; and

q 1

d N = e ( (q 1); )N ( (q 1) ) (16)

q 1 1

) [ ( )] (

q 1

for q>1.

3 Quantum Ideal Gases

Wecho ose as basis states the numb er representation where H and N are diagonal whose

(N )

eigenvalues are E and N resp ectively. The numb er of particles N =0; 1; 2:::, and

(N )

represents the N-particle energy sp ectrum (characterized by the quantum number E L

{4{ CBPF-NF-060/95

or set of quantum numb ers L). Thus, we can obtain the equilibrium distribution for the

grand-canonical ensemble from Eq. (4), this is

h i

(N ) (N )

p = 1 (1 q )(E N ) = ( ; ): (17)

L L

It is convenient to remark that in general

# "

1=q

i h

X X

(N ) (N )

(N )

p (18) 6= p p p

L L

(N )

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

i h

(N )

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

N =0

(N )

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

N N

N =o

(N )

p 6=1 ).

N =0

In this representation Eq.(5) is

h i

X X

(N )

1 (1 q )(E N ) ( ; )= : (19)

N =0 L

3.1 Quantum Statistics

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

mo dynamic b ehavior at very low temp erature. It is known however that, in the standard

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.

When evaluating the trace in Eq.(5) or the set of quantum numbers L in Eq.(19) we

must b e careful to count each p ossible state of the system only once. If the quantum state

of the system is sp eci ed by the one-particle states. The total energy is given by

(N )

E = + + ::: + ;

l l l

L 1 2

is the energy of the single particle i. where

Then, the generalized grand partition function for the Maxwell-Boltzmann statistics

can b e written as

X X X X

M B

[1 (1 q )( + + ::: + N)] ::: ; (20) =

l l l

1 2 q

N !

N =0

l l l

2 1

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.

We consider particles with p erio dic b oundary conditions as a simple application of

Eq.(7) and Eq. (20). If the volume is large enough, the particle energies will b e closely

spaced and we can replace the sum over l byanintegral over a continuous variable k .

Thus,

{5{ CBPF-NF-060/95

D=2

` 2

D 1 h k =2m

e ! dk k e ; (21)

2 (D=2)

l=1

where D is the dimension. Hence, when q = 1, Eq.(20) b ecomes

! "

ND=2 N

1 1

N 2

X X X

e m` 1

N M B

; (22) e = e =

N ! N !

2 h

l N =0 N =0

Replacing Eq.(22) into Eq.(7), we obtain:

1 ND

ND=2

1q 2 2

( )[1 + (1 q )N ]

M B

= ; (23)

2q 2

DN=2

N !(1 q ) ( + ) 2 h

N =0

1q 2

for q<1, this is the high-temp erature approach. Eq.(22) is recovered from Eq.(23) in the

limit q ! 1.

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

exclusion principle, is given by

1 1 1 1

X X X X

F D

= ::: [1 (1 q )( + + ::: N)] : (24)

l l l

1 2

q N

N =0 l l =l +1 l =l +1

1 2 1

N N 1

Each di erent set of o ccupation numb ers corresp onds to one p ossible state.

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

1 1 1 1

X X X X

B E

: (25) = ::: [1 (1 q )( + + ::: + N)]

l l l

1 2

q N

N =0 l l =l l =l

1 2 1

N N 1

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

state.

Low-temp erature approach for fermions with p erio dic b oundary conditions and chem-

ical p otential computations as a function of the temp erature for the particles in a b ox

problem are shown in [8].

3.2 Generalized Distribution Function

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

We remark that a multiple sum app ears byevaluating the trace in Eqs.(1)-(3) and (5).

That multiple sum can b e transformed into a unique sum over some set of basis states of

non-interacting particles, by standard metho ds.

Now, let us remember that

N = n ; (26)

1 1l

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.(26) into Eq.(15)

and Eq.(16), we obtain the generalized distribution functions. We de ne

) (

n = e ( (1 q ) )n ( (1 q ) ) (27) d ( )

ql 1 1l

[ ( )] 2

C q

{6{ CBPF-NF-060/95

for q<1; and

q 1

n = d e ( (q 1) )n ( (q 1) ) (28)

ql 1 1l

) [ ( )] (

q 1

for q>1. Therefore, wehave de ned the generalized distribution functions in connection

with the standard distribution and partition functions through Eq.(27) and Eq.(28). In

addition, wehave

N = n (29)

q ql

which is the generalization of the Eq.(26).

Conclusions

Clearly, the statistical and thermo dynamic quantities recover their standard forms in the

q ! 1 limit.

The connection b etween generalized statistical mechanics in the grand canonical en-

semble and thermo dynamics is well established through the relation given by Eq.(6).

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.

Acknowledgments

The author is very indebted to Constantino Tsallis for valuable discussions and to Celia

Anteneo do for their helpful comments on the draft version of this pap er. The author has also b ene tted from nancial supp ort from the CLAF/CNPq.

{7{ CBPF-NF-060/95

{8{ CBPF-NF-060/95

References

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

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

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

451.

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

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

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

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

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

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

[7] D. Prato, Phys. Lett. A203(1994) 165.

[8] S. Curilef, \Generalized Statistical Mechanics for the N-body Quantum Problem {

Ideal Gases" (1995) preprint.

[9] F. Buy ukkili c and D. Demirhan, Phys. Lett. A 181 (1993) 24; F. Buy ukkili c, D.

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

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