Canonical Distribution and Entropy

Physica A 391 (2012) 31403150

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An equilibrium thermostatistics of a nonextensive finite system:Canonical distribution and entropyJ. Jiang a,b,c,, R. Wang b,e, Y. Lysogorskii d, D. Zvezdov d, D. Tayurskii d, Q.A. Wang ba Complexity Science Center, Institute of Particle, Physics, Hua-Zhong (Central China)Normal University, Wuhan 430079, PR Chinab Institut Suprieur des Matriaux et Mcaniques Avancs, 44, Avenue F.A. Bartholdi, 72000 Le Mans, Francec LPEC, Facult des Sciences et Techniques, Universit du, Maine, Ave. O. Messiaen, 72035 Le Mans, Franced Departement of physics, Kazan State University, Kazan 420008, Russiae College of Information Science and Engineering, Huaqiao University, Quanzhou, 362021, PR China

a r t i c l e i n f o

Article history:Received 27 July 2010Received in revised form 30 December 2011Available online 20 January 2012

Keywords:Nonextensive finite systemCanonical distributionEntropy

a b s t r a c t

A simplemodel is presented to illustrate the equilibrium thermostatistics of a nonentensivefinite system. Interaction between the finite system and the reservoir is taken into accountas a nonextensive term H1H2 in the expression of total energy (H1 and H2 are the energyof the finite system and the reservoir respectively, is nonadditivity parameter). In thepresent paper, a case with harmonic reservoir potential is considered. Energy probabilitydistribution, average energy, heat capacity and entropy function for energy distribution arederived in different finite systems including those with constant density of state in energy,the ideal gas and the phonon gas.

2012 Elsevier B.V. All rights reserved.

A system can be called finite size or a small system when its size is comparable with the interaction scale between itselements. A finite system is a family of complex systems to which the paradigms of statistical physics for large system arechallenged. One challenge is that the hypothesis of extensive energy and entropy may become invalid for systems in whichthe interacting scale is comparative to its size or the surface effect is not negligiblewith respect to the volume effect. Anotherchallengemay be related to the fluctuation of many thermodynamic quantities such as temperature, energy, pressure etc.. Apossible consequence of that is the failure of the statistics theory based on the exponential probability distribution. Hence,more and more interest is transferred to the nonextensive system (see for example Ref. [1]). We also notice many prosand cons around nonextensive statistical theory derived from the Tsallis entropy with the help of the maximum entropyprinciple (maxent) [2].

Finding the probability distribution of energy is one of themost important things towork out for the statistical descriptionof a thermodynamic system. The difficulty for a finite system is that we know nothing about the entropy and the energynonextensivity in general. In other words, the knowledge is generally missing about the entropy function of probability andabout the mutual dependence in energy of interacting parts. Hence, for example, if we look at a closed system in contactwith a reservoir (canonical ensemble), we cannot even write the relationship between total energy H of the union system+reservoir and H1 and H2, respectively the energy of the system and the reservoir. Worse, the thermodynamic limit of largenumber of constituents does not exist. All these hypotheseswe had for a large extensive system aremissing now. It becomesimpossible to derive a probability distribution with the mathematical tricks in textbooks (e.g., Stirling formula, H1 H2or H = H1 + H2 approximation, maxent, etc.). Nevertheless, there are many efforts to derive the probability distribution ofenergy from the first principle (assuming equiprobable microstates of isolated system) [37]. The work of Ref. [7] is the first

Corresponding author at: Complexity Science Center, Institute of Particle Physics, Hua-Zhong (Central China) Normal University, Wuhan 430079, PRChina.

E-mail address: [email protected] (J. Jiang).

0378-4371/$ see front matter 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.01.012

J. Jiang et al. / Physica A 391 (2012) 31403150 3141

tentative step to doing this with nonadditive energy of the form H = H1 + H2 + f (H1,H2), a more general approach thanother tentative approaches with additive energy for a finite system. f (H1,H2) is the nonadditive part of the energy. In thatwork, the authors provided a rigorous proof of nonadditive energy for the system with potential energy (e.g., gravitatingsystem) having f (H1,H2) = H1H2, that is

H = H1 + H2 + H1H2, (1)where is a constant characterizing the nonadditivity of the system [7]. We call it the nonadditivity parameter. Thisresult tells us that the theoretical investigation based on Eq. (1) does have some physical meanings and may be helpful forunderstanding the deviation of such complex system from classical behaviors. For example, one result of the present worksuggests that, BoltzmannGibbs (BG) statistics is not always recovered in the thermodynamical limit, when nonextensiveenergy is considered, which is different from Ref. [5] where conventional BG statistics remains valid regardless of the typeof potential among the interacting particles.

As is well known, there is a nonlinear relationship in the q-nonextensive statistics characterizing the nonadditivity ofentropy and energy. The nonadditivity is given by, for a total system composed of two subsystems 1 and 2:

X(1 2) = X(1)+ X(2)+ X(1)X(2) (2)where X represents entropy, energy or other variables, = 1 q (or q 1) (with positive q). Since characterizes thenonadditivity of the total system, it is expected to be unique for a given system in a given situation. This relationship hasbeen employed in the definition of temperature, pressure and other intensive variables in order to connect the statisticalmechanics for a nonextensive systemwith thermodynamics. This nonadditivity was first mentioned in the Ref. [2] with thehypothesis of joint probability p(1 + 2) = p(1)p(2). The nonadditivity of entropy S given by S(1 2) = S(1) + S(2) +S(1)S(2) was subsequently used in the discussion of the zeroth law of thermodynamics with or without an additionalnonadditivity of energy U given by U(1 2) = U(1)+ U(2)+ U(1)U(2) (see for example a historical review in Ref. [8]).In the Refs. [912], the reader can find some examples of the unique values of derived from the experimental data inthe fields such as astrophysics, high energy collisions, and biology. Here, we simply prove this nonadditivitys validity in aself-gravitation system. The Hamiltonian of the system is given as follows:

Htotal = H1 + H2 + H1H2 25R51

k2 12k5 5k2 + 3 , (3)

where R, R1 radius, H1 = G2 162R515 , H2 = G2 162

3

R55

R31R2

2 +3R5110

, and k = R/R1. = 25R51

k212k55k2+3 = 0. The details

are shown in Ref. [7]. More examples with different potential energy can be found in Ref. [13].Although themathematical coherence of Eq. (1) was questioned in Ref. [14], amore detailed numerical investigation [13]

of three nonextensive systems with potential energy revealed that Eq. (1) could be a good approximation when the totalsystem is not too small and the sizes of the subsystems are comparable.

The present work is a further development in the direction of Ref. [7] in which the authors investigated the simple case ofconstant (and zero) potential energy of the reservoir. Here we consider a more complicated case with harmonic potential inthe reservoir. Furthermore, quantummechanical small system (phonon gas) and classical one (ideal gas and the systemwithconstant density of state in energy) are analyzed in the present paper. Inwhat follows, the canonical probability distribution,the average energy and heat capacity of these systems are calculated to show the influence of nonextensivity energy onthermal properties. The thermodynamic entropy functions are also derived for finite systems with constant and harmonicreservoir potential, respectively. The last section contains the concluding remarks.

1. The model and probability distribution

We consider an adiabatically and mechanically isolated system with energy E containing finite N particles and makeassumptions of equiprobability and ergodicity for this system [3,7]. In this case probability distribution is given by

p(X) = 1(E)

[E H(X)] (4)

where H(X) is the Hamiltonian of having energy E,(E) the total volume of the phase space points satisfying H(X) = E,and X the phase space coordinates of the N particles (dX = idridpi with i = 1, 2, . . . ,N and pi the momentum and ri thecoordinates of the ith particle).

We divide the total system into the systems of interest and reservoir with Hamiltonian H1(X1) and H2(X2) respectively,and each subsystem has N1 and N2 particles. In that case the Hamiltonian of the total system can be expressed as Eq. (1) inourmodel. The following discussion ismade along the line of Refs. [3,7] with the approximationH1(X1) E. The probabilitydistribution of system of interest is given by

3142 J. Jiang et al. / Physica A 391 (2012) 31403150

p(X1) = 1(E)

(X2)

(E H1(X1) H2(X2) H1(X1)H2(X2)) dX2

= 1(E)

(X2)

(E H1(X1) (1+ H1(X1))(K2(P2)+ V2(R2))) dX2

= 1(E)

(R2)

k [E H1(X1) (1+ H1(X1))V2(R2)] dR2, (5)

whereK2(P2) is the kinetic energy andV2(R2) is the potential energy of particles in the reservoir. P2 andR2 are themomentumand the coordinates, respectively.k [] is given by

k[y] =(P2)

(y (1+ H1(X1))K2(P2)) dP2

=(P2)

(y u(,H1, P2)) dP2, (6)

with u(,H1, P2) = (1 + H1(X1))K2(P2). k[y] is a measure of the hypersurface in momentum space which is describedby u(,H1, P2) = y. In other words,k[y] is equal to the derivative of hyper volume of momentum space related to P2 bythe quantity u(,H1, P2), i.e.k[y] = k(y)/y [3] where

k(y) =uy

dP2. (7)

If we introduce the new variable,

Dk =1+ H12mn

(P)n,

where k = 3(n 1) + and mn is the mass of nth particle, then the equation of hypersurface u = y can be written in theform

3N2k=1 D

2k = y. According to some geometrical considerations, the integration in Eq. (7) will be k[y] = ay3N2/2. We can

obtain,

k[y] = b y3N2/21

(1+ H1)3N2/2 . (8)Then Eq. (5) becomes

p(X1) = b(E)

(R2)

(E H1(X1) (1+ H1(X1)) V (R2))3N2/21(1+ H1)3N2/2 dR2 (9)

After some reorganizations, Eq. (9) can be written as

p(X1) = bE3N2/21

(E)(1+ H1)3N2/21 H1(X1)

E

3N2/21 (R2)

1 1+ H1(X1)

E H1(X1) V (R2)3N2/21

dR2. (10)

We will consider the harmonic interaction potential of the reservoir

V (R2) = 3N2i=1

x2i , (11)

and the configuration space of 3N2 dimensions in which harmonic potential energy can be written as

V (R2) = 3N2i=1

x2i = r2, (12)

where r is the length of supervector in that space. Since the potential in Eq. (12) depends only on the supervectors lengthr , the integration over angles in Eq. (10) gives only constant factor.

With respect to the interval of the integration in Eq. (10), because of the quadratic form of the potential, the domain of theintegration should be a 3N2 dimensional hypersphere with radius between the largest possible value of reservoir potentialenergy V2max = H2 with fixed value of full energy E and the system energy H1

V2max = E H11+ H1 = r20 . (13)


In the case of 3N2 dimensional system, the elementary volume is given by dR2 r3N21dr . The integration in Eq. (10) nowbecomes

Z = r00

1 1+ H1

E H1 r23N2/21

r3N21dr, (14)

and after renotations A = 1+H1EH1 , B = 3N2/2 1, C = 3N2 1, we should calculate

Z = r00(1 Ar2)BrCdr. (15)

After changing the integration variable t = Ar2, we get the following expression:

Z = A C+12 10

tC12 (1 t)Bdt

=1+ H1E H1

3N22B(3N2/2, 3N2/2), (16)

where B(a, b) is the Euler beta function. We substitute Eq. (16) into Eq. (10), and obtain the probability distribution

p() (E )3N21 (1+ )3N2

1

E

3N21(1+ )3N2 , (17)

where is the energy of the system. In general case with arbitrary value of and E = 32N2 = 32N2kT (k is the Boltzmannconstant, T is the temperature of system, we set that N1 N2), the probability distribution has the form

p(,N2, , ) = A(N2, , )1

3/2N2

3N21(1+ )3N2

= A(N2, , )1 (1 q) 2

(2 q) 1

1q(1+ ) q21q , (18)

where q = 3N223N21 and A(N2, , ) is the normalization constantwith condition E0 p()()d = 1, where () is the density

of state per energy unity of the system. From Eq. (18), it is obvious that when = 0, the probability distribution is in theq-exponential form, and reduces to exponential function under the thermodynamic limit (N2 , q 1).

In general, different () represents different systems. The following calculations are made with constant () = . Inthis case, A(N2, , ) is given by

A(N2, , ) = 2(3/2N2)3N2

3N2B3/2N2 (3N2, 0) , (19)

where Bz(a, b) is the incomplete Beta function.

Bz(a, b) = z0

ta1(1 t)b1dt. (20)

From Eqs. (18) and (20), we can get the following results: (1) the range of the nonadditivity parameter is 23N2 0. = 0 means the ideal case of the finite system. < 0 means the exclusive interaction among the subsystems; (2) there isone cutoff for energy with the value of1/.

The probability distributions with different particle number of reservoir N2 are shown in Fig. 1. It is found thatthe probability of the system energy increases with the number of particles N2, and the probability distribution is amonotonically decreasing function with energy. Hence the thermodynamic limit N2 doesnt lead to a Boltzmannexponential distribution. This is different from the result of Ref. [5] in which the nonadditive energy of the system was notconsidered.

In order to investigate the influence of the energys nonadditivity to the system, the behavior of probability distributionof system energy with different is shown in Fig. 2. We can see that the probability decreases with the increasing of energyand the distribution with larger is close to Boltzmann form. It suggests that the Boltzmann distribution is in the limit ofadditive energy and large particle number N2 [3].


Fig. 1. Probability distribution of the system energy with constant density of state with different particle number N2 of reservoir at = 0.005, = 1.Black line: N2 = 40; dashed line: N2 = 80; dot dash line: Boltzmann distribution of energy. It is found that thermodynamic limit (N2 ) does notrecover the Boltzmann exponential distribution of energy.

Fig. 2. Probability distribution of system energy with constant density of state under different with N2 = 100. Dashed line: = 0.006; black line: = 0.005; and dot dash line: Boltzmann distribution of energy. It suggests that decreasing the nonadditivity energy of the system could reduce to aBoltzmann distribution of energy.

2. Average energy and heat capacity

2.1. Finite system with constant density of state

From Eq. (18) we can obtain the average energy function of the finite system as

= E0p()d (21)

which can be calculated as follows

= 1.5N2 (2+ 3N2( 3N2))(1+ 3N2)(2+ 3N2)

2F11, 3N2, 2+ 3N2, 32N2

2F11, 3N2, 1+ 3N2, 32N2

, (22)where

2F1(a, b; c; z) = (c) (b) (c b)

10

tb1(1 t)cb1(1 tz)adt. (23)In the limit of 0, we obtain:

= 3N26N2 + 2 . (24)

When N2 , we get the well known result = /2 = kT/2. Fig. 3 shows the caloric curve (the average energy againstthe temperature, here, we call the temperature due to the relationship between and T ) with different at N2 = 2000in loglog plot. From left to right in the picture: = 1/6,1/15,1/30, respectively. It is found that, for each curve,there is a cutoff value for temperature ( = 23N2 = 0.002, 0.005, 0.01), corresponding to the maximum value of averageenergy Emax = 6, 15, 30, respectively. Besides, the bigger ||, the lower the temperature the system has when the average


Fig. 3. Variation of the average energy of the systemwith constant density of state against the temperature with different in loglog plot. From left toright: = 1/6,1/15,1/30, respectively. we can find that there is a cutoff for temperature ( = 23N2 = 0.002, 0.005, 0.01 for each curve), and theaverage energy takes themaximum value Emax = 1/ = 6, 15, 30, respectively. The dashed line means the upper bound of the average system energy.

Fig. 4. Variation of the average energy with constant density of state against the temperature for different particle number N2 at = 1/30 in loglogplot. In the picture, from left to right, N2 is equal to 2000, 200, 20, respectively. It shows that the more particles in the reservoir there are, the faster theaverage energy increases with the temperature.

energy takes the cutoff value. Furthermore, there is a dashed line at the top of the curves, whichmeans that there is an upperbound for the maximum value of the average energy.

In order to analyze the behavior of the average energy with different particle number of reservoir, Fig. 4 is given with = 1/30. From left to right in the picture, N2 is equal to 2000, 200, 20, respectively. It demonstrates that the moreparticles in the reservoir there are, the faster the average energy increases. In addition, all of the functions of average energyare monotonically increasing ones with the temperature and the average energy increasing almost linearly in the range ofthe temperature below the cutoff value. When the temperature is close to its cutoff value, the average energy of the systemincreases dramatically up to maximum value.

Next, the heat capacity C of the finite system with constant density of state is defined by

C = dEd

,

which can be calculated from Eq. (22) and is given by

C = (213N2(4 27N2(N2)3N2B 32N2(3N2, 0)+ (2+ 3N2) B 32N2(3N2,1)(2 27

N2(N2)3N2 + 3 8N2N2 B 32N2(3N2, 0))))/((2+ 3N2)B 32N2(3N2, 0)

2). (25)

In the limit of 0, the heat capacity will be equal to:C = 3N2

6N2 + 2 .IfN2 tends to infinity, the heat capacity C will reduce to the ordinary result for Boltzmann distribution of energy: C = 0.5.

In Fig. 5, we give the distributions of the heat capacity with different at N2 = 2000. From left to right, is equal to1/15,1/30,1/45, respectively. It is shown that the value of the heat capacity will be infinite regardless of the value of when


Fig. 5. Variation of the heat capacity of a finite system with constant density of state against the temperature for different in loglog plot. From left toright, is equal to1/15,1/30,1/45, respectively. In all cases N2 = 2000. It is found that when tends to the cutoff value 23N2 , the heat capacitywill be close to infinity regardless of the value of .

is close to the cutoff value 23N2 . The appearance of the infinite heat capacity may suggest that the potential behavior ofphase transition happened in the system.

2.2. Generalized finite system

In this part, two special finite systems are taken into consideration. One is the ideal gas and another is the phonon gas.

For the former, the expression of density of state () is 2V (2m)32

h3

12 ; for the latter, the expression of () is 4V

h3c32, where

m is the mass of gas molecule, V is the volume of finite system, h is Planck constant, and c is the velocity of light. So, ageneralized density of state () is considered (finite systemswith constant density of state, ideal gas and phonon gascorrespond to = 0, 0.5, 2, respectively). Hence, the expressions of the average energy and the heat capacity of generalizedfinite systems are calculated based on Eq. (21) and are given as follows:

E =3N2(1+ ) 2F1

3N2, 2+ , 2+ 3N2 + , 3N22

(1+ 3N2 + )

2 2F13N2, 1+ , 1+ 3N2 + , 3N22

(2+ 3N2 + )

, (26)

C =3N2(1+ )

9N22 (1+ ) 2F13N2, 2+ , 2+ 3N2 + , 3N22

(2+ 3N2 + )2

2F11+ 3N2, 2+ , 2+ 3N2 + ,3N22

+ 2

F13N2, 1+ , 1+ 3N2 + , 3N22

(1+ 3N2 + )

4 2F1

3N2, 2+ , 2+ 3N2 + , 3N22

(2+ 3N2 + )

9N22 (2+ )2F11+ 3N2, 3+ , 3+ 3N2 + , 3N22

(3+ 3N2 + )

8 2F1

3N2, 1+ , 1+ 3N2 + , 3N22

2 (1+ 3N2 + )2

. (27)In order to explore the difference among various finite systems, wemake a comparison in Fig. 6where the average energy

of three different finite systems versus the temperature withdifferent atN2 = 2104 is shown. Some interesting findingsare deduced: (1) for the same , finite system with phonon gas has a larger average energy than the other two systems. Itsaverage energy increases faster than the others with temperature; (2) when the temperature is close to the cutoff value,each system gets close to their maximum value 1

which are all below the dashed line.

With regard to the heat capacity of different finite systems, it is shown in Fig. 7. We find that, for the same , phonongas has larger heat capacity which increases faster with temperature than the other two systems. Moreover, when thetemperature is close to the cutoff value, the heat capacities of three systems all tend to infinity.


1E-31E-3

0.01

0.1

1

10

100

0.01

Fig. 6. Average energy of three different finite systems versus temperature with different at N2 = 2 104 in loglog plot. = 0, 0.5, 2 correspondsto the systems with constant density of state, ideal gas, and phonon gas, respectively. The dashed line is the upper bound of maximum value of the averageenergy. In this picture, it is found that, for the same , the system with phonon gas has a larger average energy than the other two systems and its averageenergy increases faster with temperature than the others. When the temperature is close to the cutoff value, three systems are close to the samemaximumvalue.

1E-3 0.01

d

d

1

10

100

1000

Fig. 7. The variation of heat capacity of three different finite systems versus the temperature with different atN2 = 2104 in loglog plot. = 0, 0.5, 2corresponds to the systemswith constant density of state, ideal gas, and phonon gas, respectively. In this picture, it is found that, for the same , the systemwith phonon gas has a larger heat capacity which increases faster with temperature than the other two systems. Moreover, when the temperature tendsto the cutoff value, the heat capacities of three systems are close to infinity.

3. Entropy function

The thermodynamic entropy of Clausius for a reversible process can be written as

dI = (d d), (28)where is the inverse temperature of an equilibrium system in canonical ensemble with energy i and the arithmetic mean =i pii. From Ref. [15], we know that this expression of entropy variation as a difference between the variation of themean and the mean of variation (d d) can be considered as a general definition of uncertainty measure and be used toderive an entropy function if the probability function is known. After integration, the expression of entropy is given as:

I =i

pi0

f 1(u)du+ C , (29)

where f 1(u) is the inverse of the probability distribution pi = f (i). The constant C is equal to 10 f

1(u)du in order thatI vanishes for the non probabilistic case pi = 0, . . . , 1, . . . , 0, . . . for all possible states i = 1, 2, . . ..

3.1. Entropy for the system with constant reservoir potential

From Eq. (10) probability distribution for nonextensive finite system with thermostat potential V (R2) = C (C is theconstant here) can be written as


pi = p(i) = 1Z1

(1+ i)3N22

1 (1+ C)i

E C 3N2

2 1, (30)

where Z is the normalization constant:

Z =i

1 (1+C)iEC

3N22 1

(1+ i)3N22

. (31)

For the entropy calculation we will use Eq. (28)

dI = (d d) = i

idpi, (32)

where i(pi) is obtained from Eq. (30):

i(pi) =(E C)

1 p

23N2i Z

23N2

1+ C+ (E C)p

23N2i Z

23N2

. (33)

So, the expression for entropy differential is

dI = i

(E C)1 p

23N2i Z

23N2

1+ C+ (E C)p

23N2i Z

23N2

dpi.

After integration we derive

I = i

pi

1+ C (1+ E) 2F11, 3N22 , 1+ 3N22 , (CE)p

23N2i Z

23N2

1+C

(1+ C) + const. (34)

Constant const can be found from the condition I = 0 without uncertainty. Distribution of entropy I for two-state systemversus the probability for different values of is shown in Fig. 8. From top to bottom, is equal to0.001,0.0008,0.0004,respectively. In all cases, = 1/6,N2 = 2000, C = 1. It is found that the function of entropy has the concavity form withthe maximum value for equiprobability distribution. But the value of the maximum entropy is changeable with different .It decreases with the increasing of , which hints at the decreasing of uncertainty in the finite system.

In the case of = 0, the entropy in Eq. (34) becomes similar to the Tsallis entropy. Under the assumptions that q = 3N2+23N2and E C 32N2 , it can be rewritten as

I = 1q 1

1 Z

q1

qi

pq

i

. (35)

3.2. Entropy for the system with harmonic reservoir potential

In Section One, we have obtained the probability distribution of the energy of finite system with harmonic potential.Now, we make some new notations in Eq. (18):

pi = 1Z1 i

E

3N21(1+ i)3N2 , (36)

where Z is partition sum. We suppose that 3N22 1 3N22 , the inverse function of above expression is obtained as follows:

i = 1 (piZ)1

3N2

(piZ)1

3N2 + 1E. (37)


Fig. 8. The variation of the entropy I for two-state system with constant thermostat potential and different . From top to bottom, is equal to0.001,0.0008,0.0004, respectively with = 1/6, N2 = 2000, C = 1.

Fig. 9. Variation of the entropy I for two-state system with harmonic thermostat potential for different . From top to bottom, is equal to 0.001,0.0008,0.0004, respectively with = 1/6, N2 = 2000.

So the entropy I is given by

I = i

idpi

= i

1 (piZ)

13N2

(piZ)1

3N2 + 1Edpi

=i

pi

(1+ E)2F1

1, 3N2, 1+ 3N2,3N22 p

13N2i Z

13N2

1+ C , (38)

where the constant C can be found in the sameway as in the previous section. Variation of the entropy of a two-state finitesystem with different is given in Fig. 9. From top to bottom, is equal to 0.001, 0.0008, 0.0004, respectively. In allcases, N2 = 2000, = 1/6. We also find that the function of the entropy is in concavity form with maximum value forequiprobability distribution, which is similar to Fig. 8. But they have different entropy values.

In the case of = 0, we set q = 1+3N23N2 , the entropy in Eq. (38) will become as follows:

I = 12(q 1)

1 Z

q1

qi

pq

i

, (39)

which indicates that the Tsallis entropy can be used to describe the entropy of different finite systems without thenonadditivity energy.

4. Concluding remarks

In the present paper, the energy distribution of the finite system with harmonic reservoir potential has been calculated.It can be deduced to Boltzmann form in the condition of neglecting nonadditivity energy ( = 0) and taking the


thermodynamic limit (N2 ). The small probability of high-energy state is related to the large nonadditivity energyH1H2. Moreover, the range of nonadditivity parameter is between 23N2 and 0.

In addition, the average energy and the heat capacity of different finite systems have been derived, containing systemswith constant density of state, ideal gas, andphonongas. For any given system, its average energy ismonotonically increasingwith the temperature and is always smaller than the limit value 3N2/2. The heat capacity of the system will be infinity atthe temperatures cutoff value. Moreover, it is found that the system with phonon gas has the largest average energy andheat capacity among three systems, but the system with a constant density of state behaves oppositely. Within the limit ofthermodynamics, the conventional BG statistical results are recovered.

The thermodynamic entropy functions have been calculated for two-state finite systems with constant and harmonicreservoir potential, respectively. Both of them could recover the Tsallis entropywhen the nonadditivity energy of the systemis not considered.

For the validity of the fundamentals of this model, we want to emphasize that the nonadditive relationship given inEq. (1) could characterize the nonadditivity of at least the potential energy of the finite system. Since the kinetic energyis not contained in our calculation, further consideration is feasible. In addition, this model sheds light on the research ofnonextensive statistical mechanics at least in the theoretical view.

Acknowledgments

This work was supported by ChineseMinistry of Education (2008677010). One of the authors thanks the Region des Paysde la Loire of France for the support under the grant number 2009-9397.

References

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[10] Constantino Tsallis, Domingo Prato, Angel R. Plastino, Astrophys. Space Sci. 290 (2004) 259.[11] C. Anteneodo, C. Tsallis, J. Molecular Liquids 71 (1997) 255.[12] C. Tsallis, G. Bemski, R.S. Mendes, Phys. Lett. A 257 (1999) 93.[13] R. Wang, S. Nganso Djanteng, A; El Kaabouchi, Q.A. Wang, Investigation of an energy nonadditivity for nonextensive systems, Chinese Sci. Bull. 56

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An equilibrium thermostatistics of a nonextensive finite system: Canonical distribution and entropyThe model and probability distributionAverage energy and heat capacityFinite system with constant density of stateGeneralized finite system

Entropy functionEntropy for the system with constant reservoir potentialEntropy for the system with harmonic reservoir potential

Concluding remarksAcknowledgmentsReferences

Canonical Distribution and Entropy

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