Canonical ensemble in non-extensive statistical mechanics when q>1
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Canonical ensemble in non-extensive statistical mechanics when q > Julius Ruseckas ∗ Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goˇstauto 12, LT-01108 Vilnius, Lithuania
The non-extensive statistical mechanics has been used to describe a variety of complex systems.The maximization of entropy, often used to introduce the non-extensive statistical mechanics, is aformal procedure and does not easily leads to physical insight. In this article we investigate thecanonical ensemble in the non-extensive statistical mechanics by considering a small system inter-acting with a large reservoir via short-range forces and assuming equal probabilities for all availablemicrostates. We concentrate on the situation when the reservoir is characterized by generalizedentropy with non-extensivity parameter q >
1. We also investigate the problem of divergence inthe non-extensive statistical mechanics occurring when q > q . I. INTRODUCTION
The standard, Boltzmann-Gibbs statistical mechanics has been successfully applied to describe a huge variety ofsystems. The cornerstone of the standard statistical mechanics is the functional form of the entropy S BG = − k B X µ p ( µ ) ln p ( µ ) , (1)where p ( µ ) is the probability of finding the system in the state characterized by the parameters µ . However, thereare systems exhibiting long-range interactions, long-range memory, and anomalous diffusion, that possess anomalousproperties in view of traditional Boltzmann-Gibbs statistical mechanics. To understand such systems a generalizationof statistical mechanics has been proposed by Tsallis [1]. The non-extensive statistical mechanics has been used todescribe phenomena in many physical systems: dusty plasmas [2], trapped ions [3], spin-glasses [4], anomalous diffusion[5, 6], high-energy physics [7], Langevin dynamics with fluctuating temperature [8, 9], cold atoms in optical lattices[10], turbulent flows [11]. This generalized framework has found applications also in chemistry, biology, geology, andeconomics [12–15]. Instead of Eq. (1) the non-extensive statistical mechanics is based on the generalized functionalform of the entropy [1] S q = k B − P µ p ( µ ) q q − . (2)Here the parameter q describes the non-extensiveness of the system. The Boltzmann-Gibbs entropy can be obtainedfrom Eq. (2) in the limit q → q -logarithmln q x = x − q − − q (3)and its inverse, the q -exponential [1] exp q ( x ) ≡ [1 + (1 − q ) x ] − q + . (4)Here [ x ] + = x if x >
0, and [ x ] + = 0 otherwise. For example, using the q -logarithm one can write Eq. (2) in a formsimilar to the Boltzmann-Gibbs entropy (1) [1]: S q = k B X µ p ( µ ) ln q p ( µ ) . (5) ∗ The exponential Boltzmann factor in the non-extensive statistical mechanics is replaced by a q -exponential. In thelimit q → q -logarithm becomes an ordinary logarithm and the q -exponential function becomes the ordinaryexponential e x .In the non-extensive statistical mechanics the canonical ensemble is often described in a formal way, starting fromthe maximization of the generalized entropy (2) [1]. The physical content enters as a form of constraints in themaximization procedure. In Ref. [19] the canonical ensemble in the non-extensive statistical mechanics has beenconsidered starting from a physical situation of a small system interacting with a large reservoir via short-rangeforces. Assuming that the q -heat capacity of the reservoir instead of the ordinary heat capacity is large, the equationsof the non-extensive statistical mechanics have been obtained. However, in Ref. [19] only the case of q < q > q > q <
1. For example, let usconsider the microcanonical ensemble where the probability of a microstate µ is p ( µ ) = 1 /W , with W being thenumber of microstates. If the generalized entropy S q is extensive and proportional to the number of particles N in thesystem, the number of microstates W behaves as (1 − ( q − AN ) − / ( q − . Thus the number of microstates becomesinfinite when the number of particles N approaches a finite maximum number N crit and the macroscopic limit N → ∞ cannot be taken. In this situation one can try to take a different limit, N → N crit , instead of the limit N → ∞ .Additional problem is that the q -exponential distributions with q > q > II. CANONICAL ENSEMBLE IN NON-EXTENSIVE STATISTICAL MECHANICS WHEN q > As in Ref. [19] we will consider a composite system consisting of a small system S interacting with a large reservoirR. We assume that the interaction between the system S and the reservoir R is via short-range forces, however thereservoir R is not described by the Boltzmann-Gibbs statistics. We require that the q -heat capacity C (R) q of thereservoir, defined by Eq. (14), instead of standard heat capacity should be large. In this article we consider only thesituation when q > E R grows as a power-law of E R , the q -exponential distribution of the energy of the system has been obtained. InRef. [22] the parameter q tends to 1 when the number of particles of the reservoir increases. Here we do not assumeany particular dependence of the parameter q on the number of particles in the reservoir.The probability of the microstate of the system S can be obtained similarly as for the case of q <
1, consideredin Ref. [19]. The total number of microstates W tot ( E tot ) of the combined system can be expressed as a sum over allavailable energies of the system S, W tot ( E tot ) = X E W ( E ) W R ( E tot − E ) , (6)where W ( E ) is the number of microstates in the system S having the energy E and W R ( E R ) is the number of mi-crostates in the reservoir. Assuming that in the non-extensive statistical mechanics the postulate of equal probabilitiesof microstates in the equilibrium remains valid, the probability of the system S being in the microstate µ and thereservoir being in the microstate µ R is equal to p ( µ ⊗ µ R ) = 1 W ( E tot ) . (7)The probability of the microstate µ of the system S is obtained by summing over microstates of the reservoir, p ( µ ) = X µ R p ( µ ⊗ µ R ) . (8)When the energy of the microstate µ is E µ , the number of possible microstates of the reservoir is W R ( E tot − E µ ) andthe expression for the probability of the microstate becomes p ( µ ) = W R ( E tot − E µ ) W ( E tot ) . (9)As Eq. (9) shows, from the postulate of equal probabilities of microstates follows that the statistics of the system Sis determined by the reservoir. Therefore, even an ordinary system interacting with the reservoir having large q -heatcapacity can be described by the q -entropy.In terms of the generalized entropy of the system S q ( E ) = k B ln q W ( E ) and the generalized entropy of the reservoir S (R) q ( E R ) = k B ln q W R ( E R ) Eq. (6) reads W tot ( E tot ) = X E e k B S q ( E ) q e k B S (R) q ( E tot − E ) q . (10)Differently from the case of q <
1, this sum can be approximated by the largest term when q >
1. Approximation ofa sum of large q -exponentials is investigated in Appendix A.Since each microstate of the composite system has the same probability, the largest term in the sum (10) correspondsto the most probable state of the composite system. As in Ref. [19], the condition of the maximum probability leadsto the inverse temperature 1 T = ∂∂U S q ( U )1 − q − k B S q ( U ) = ∂∂E tot S (R) q ( E tot − U )1 − q − k B S (R) q ( E tot − U ) . (11)where U is the most-probable energy of the system. From Eq. (11) it follows that the heat capacity of the reservoircan be expressed as C R = 1 TT ( R ) q C (R) q − q − k B , (12)where 1 T (R) q = ∂∂E R S (R) q ( E R ) (13)is the auxiliary q -temperature of the reservoir and C (R) q = − T (R) q ) ∂ ∂E S (R) q ( E R ) (14)is the q -heat capacity of the reservoir, defined similarly to the heat capacity in standard statistical mechanics. Equationsimilar to Eq. (12) has been obtained in Ref. [23]. In the formulation of the non-extensive statistical mechanics basedon maximization of entropy, the auxiliary temperature T q appears as the inverse of the Lagrange multiplier associatedwith the energy constraint. This temperature can have a physical meaning in systems with long-range interactions.For example, temperature T q is related to the density of vortices in type II superconductors [24].If we introduce the entropy of the combined system as S (tot) q ( E tot ) = k B ln q W ( E tot ) then approximating the sum(10) by the largest term we get that the entropy of the combined system is a pseudo-additive combination of theentropies of the system S and the reservoir R: S (tot) q ( E tot ) ≈ S q ( U ) + S (R) q ( E tot − U ) − q − k B S q ( U ) S (R) q ( E tot − U ) . (15)According to Eq. (11), in the ensemble considered in this section the physical temperatures of the system and thereservoir are equal, whereas the corresponding q -temperatures are not. If one requires equality of q -temperature,the additivity of energies does not apply [25]. However, when the interactions between the system S and R are longrange and, consequently, the energy is not additive, then the pseudo-additivity of entropies together with equality of q -temperatures can be valid [26].Similarly as in Ref. [19] for the q < q -entropy of the reservoir isvery small, ∂ ∂E S (R) q ( E tot ) ≈
0, and, consequently, the q -heat capacity of the reservoir, defined by Eq. (14), is verylarge. Taking the limit C (R) q → ∞ in Eq. (12) we obtain the heat capacity of the reservoir C R = − k B q − . The heatcapacity is negative when q >
1. Increase of the energy of the reservoir with very large q -heat capacity by ∆ E leadsto the new temperature of the reservoir T ′ = T − q − k B ∆ E . (16)The temperature of the reservoir decreases by increasing the energy.Possibility of negative heat capacity in the case of q > q -ideal gas [43, 44] and for two-level systems [45].Assuming very small second derivative of q -entropy of the reservoir, the number of microstates of the reservoir canbe approximated as W R ( E tot − E ) = e k B S (R) q ( E tot − E ) q ≈ e k B S (R) q ( E tot − U ) − k B ( E − U ) ∂∂E tot S (R) q ( E tot − U ) q . (17)Using Eqs. (9) and (17) we obtain that the probability of the microstate of the system S is proportional to the factor˜ P ( E ) = exp q (cid:18) − k B T ( U ) ( E − U ) (cid:19) , (18)where the temperature T ( U ) is given by Eq. (11). In contrast to the thermostat with the very large heat capacity,the temperature T ( U ) depends not only on the reservoir but also on the properties of the system. Therefore, it isconvenient to introduce the temperature of the isolated reservoir1 T (0) = ∂∂E tot S (R) q ( E tot )1 − q − k B S (R) q ( E tot ) . (19)Using Eqs. (11), (19) together with the assumption ∂ ∂E S (R) q ( E R ) ≈ T ( U ) ≈ T (0) + q − k B U . (20)This equation shows that the interaction with the system raises the temperature of the reservoir. However, due tothe large q -heat capacity the q -temperature of the reservoir, defined by Eq. (13), remains constant. Inserting Eq. (20)into Eq. (18) we get that the probability of the microstate of the system S is proportional to the factor P ( E ) = exp q (cid:18) − k B T (0) E (cid:19) . (21)An expression similar to Eq. (21) has been obtained in Ref. [21]. Using the factor (21) we can write the normalizedprobability of the microstate as p ( µ ) = 1 Z q e − k B T (0) E µ q , (22)where Z q = X µ e − k B T (0) E µ q (23)is the generalized partition function.On the first sight the factor (21) is not invariant to the change of zero of energies. However, as in Ref. [19], we canargue that the shift of the energy zero of the system by ∆ E is equivalent to the decrease of the energy of the reservoirby ∆ E leading to the increase of the temperature. From the requirement that the probability of the microstate shouldremain the same follows that the new factor should be proportional to the old, P ′ ( E ) = exp q (cid:18) − k B T ′ (0) E (cid:19) ∼ P ( E + ∆ E ) = exp q (cid:18) − k B T (0) ( E + ∆ E ) (cid:19) . (24)Consequently, the new temperature of the reservoir should be equal to T ′ (0) = T (0) + q − k B ∆ E . (25)This equation is consistent with Eq. (16).
III. DIVERGENCES IN CANONICAL ENSEMBLE APPROACH
In the canonical ensemble approach the description of the reservoir is simplified to just one number, the temperature.The validity of such a simplification depends the system interacting with the reservoir. Namely, it is assumed that thesystem should be much smaller that the reservoir; the precise requirement depends on statistics. Let us consider thestandard, Boltzmann-Gibbs statistical mechanics at first. In the derivation of the Boltzmann factor an assumption ismade that the number W ( E ) of microstates of the system having energy E µ = E should not grow fast with increasingenergy and the distribution of the energy p ( E ) should be normalizable, Z W ( E ) exp (cid:18) − k B T E (cid:19) dE < ∞ . (26)For a hypothetical system where the number of microstates W ( E ) grows with increasing energy as fast as E − e k B T E or faster, this assumption is not satisfied and the reservoir cannot be considered as a thermostat. Such a system isnot smaller than the reservoir. In order to get normalizable probabilities in this situation one should consider thereservoir as a finite system having finite energy. Thus the canonical ensemble leading to the exponential Boltzmannfactor is not applicable when the number of microstates grow exponentially.Now let us examine the situation described in the previous Section, when the q -heat capacity of the reservoir islarge when q >
1. Similarly as in the Boltzmann-Gibbs statistical mechanics the description using canonical ensemblecan be applied only when the system is small and the number W ( E ) of microstates having energy E µ = E grows withincreasing energy slow enough. Using the factor (21) we get that probability is normalizable when W ( E ) grows withincreasing energy slower than E q − − . That is, at large energies W ( E ) should grow slower than exp q lim ( aE ) with q lim = 2 − − q . (27)When q = 1 we get q lim = 1, which coincides with the limit on the growth in the Boltzmann-Gibbs statisticalmechanics. According to Eq. (27), q lim < < q <
2. If the number of microstates of the system W ( E ) growswith increasing energy faster than this limit then to get finite probabilities the reservoir should be described as a finitesystem having finite energy and the generalized canonical ensemble is not applicable. When q >
1, this situation canoccur for conventional physical systems, e.g. for classical Hamiltonian systems in the thermodynamic limit [20]. Thisproblem has been first noticed by Abe [43] by trying to describe ideal gas where the effects of the interaction arereplaced by the introduction of q = 1.Similar limitation occurs also in the case of q <
1. Since the distribution of energies when q < E max ,this allows for the number of microstates of the system to grow with increasing energy even faster than in the case ofBoltzmann-Gibbs canonical ensemble. However, if the number of microstates is singular when E approaches E max ,the probability can become unnormalizable. To get finite probabilities the number of microstates W ( E ) when energyapproaches E max should grow slower than exp q lim ( aE ), where a = 1 / [( q lim − E max ]. Here the value of q lim is givenby the same equation (27). Thus we can conclude that for all possible values of q the number of microstates W ( E )should grow with increasing energy slower that the q -exponential with the limiting value of q (27).The simplest way to take into account the finiteness of the reservoir is to introduce a cut-off energy E max into theprobability of the microstate: p ′ ( µ ) = 1 Z ′ q e − k B T (0) E µ q Θ( E max − E µ ) . (28)Here Θ is the Heaviside step function. The cut-off energy E max has the meaning of the finite energy of the isolatedreservoir. Similar possibility has been suggested in Ref. [20]. The cut-off using the step function is only the simplestpossibility, the specific form of the cut-off depends on the details of the reservoir. Such a description is outside of theformalism of canonical ensemble where reservoir is characterized only by temperature.In Ref. [46] it was suggested to remove the divergences occurring in the case of q > q -partitionfunction Z q as a q -Laplace transform of the energy density. However, this proposal is problematic, as it is pointedout in Ref. [47], because the introduction of the q -Laplace transform only removes divergences in the averages of thefunctions of energy.Note, that we obtained the non-applicability of the canonical ensemble in the non-extensive statistical mechanicsfor the systems where the growth of the number of microstates with the energy is faster than q -exponential with q = q lim using the assumption of short-range interactions between the system and and the reservoir. In the caseof long-range interactions this result is not necessarily valid. When interactions are long-range, the systems can benon-ergodic and not all available microstates can be reached. In this situation the description using microcanonicalensemble should be modified, for example, assigning equal probabilities only to reachable microstates. The effectivenumber of reachable microstates can grow slower than in the ergodic case and the probability proportional to the q -expoential with q > IV. GENERALIZED THERMODYNAMICAL QUANTITIES
As for the case of q <
1, considered in Ref. [19], there are several different possibilities to generalize the free energy.All equations of Ref. [19] where no approximations have been made remain valid also for q >
1. In this Section wehighlight only the differences between q > q < Z q , given by Eq. (23). The distribution of the energy of thesystem E is equal to the probability p ( µ ) multiplied by the number W ( E ) = e k B S q ( E ) q of microstates having energy E µ = E . Thus the generalized partition function Z q can be written as a sum over energies Z q = X E e k B S q ( E ) q e − k B T (0) Eq = X E e k B T ( E ) T (0) S q ( E ) − k B T (0) Eq , (29)where T ( E ) = T (0) + q − k B E . (30)From the properties of q -logarithm (3) with q > S q ( E ) is smaller than the q -dependentmaximum value, S q ( E ) < k B q − . (31)When q > S q ( U ) corresponding to the most-probable energy of the system U is close to the limitingvalue k B / ( q − q -exponentials with q > q -logarithm of the sum in Eq. (29) can beapproximated as ln q Z q ≈ k B S q ( U ) − − q − k B S q ( U ) k B T (0) U . (32)The unnormalized q -average energy of the system ¯ U q = X µ E µ p ( µ ) q (33)can be calculated using the equation [19] ¯ U q = k B T (0) ∂∂T (0) ln q Z q . (34)From the approximation (32) we get ¯ U q ≈ (cid:18) − q − k B S q ( U ) (cid:19) U . (35)As have been shown in Ref. [19], the entropy ¯ S q = k B − P µ p ( µ ) q q − . (36)can be obtained using the equation ¯ F q = ¯ U q − T (0) ¯ S q , (37)where ¯ F q = − k B T (0) ln q Z q . (38)is the generalized free energy corresponding to the temperature T (0). Using the approximation (32) and Eqs. (35),(37) we obtain ¯ S q ≈ S q ( U ) . Thus, similarly as in Boltzmann-Gibbs statistics and differently than in the case with q <
1, the average entropy ¯ S q for q > S q ( U ). The approximation ¯ S q ≈ S q ( U ) is consistent withEq. (36). Indeed, we have ¯ S q = k B − P µ p ( µ ) q q − k B − P E (cid:16) − q − k B S q ( E ) (cid:17) p ( E ) q q − . (39)When the maximum of the entropy S q ( U ) is close to the limiting value, approximating the sum by the largest termcorresponding to E = U we get ¯ S q ≈ S q ( U ).The normalized q -average of the energy U q = P µ E µ p ( µ ) q P µ p ( µ ) q (40)is related to the unnormalized q -average as [19] U q = ¯ U q − q − k B ¯ S q . (41)Using the approximation (35) we obtain U q ≈ U . (42)Differentiating the expression for the average energy of the system¯ U = X µ E µ p ( µ ) (43)with respect to the temperature T (0) and using Eqs. (22), (34), and (41) we can express the difference between theaverage energy and normalized q -average energy as¯ U − U q = ( q − T ( U q ) ∂∂T (0) ¯ U . (44)As this equation shows, when q = 1 the difference between different averages is proportional to the physical tempera-ture T ( U q ). When q > S q ( U ) is close to the limiting value, the probability of theenergy E = U is much larger than the probabilities of other energy values. In this case ¯ U ≈ U q and from Eq. (44)follows that ∂ ¯ U /∂T (0) ≈ q -temperature T q of the system S, defined as1 T q = ∂ ¯ S q ∂U q (45)is related via the equation T ( U q ) = T q (cid:18) − q − k B ¯ S q (cid:19) (46)to the temperature T ( U q ) = T (0) + q − k B U q of the reservoir corresponding to the energy of the system equal to U q [19].Since ¯ S q >
0, the q -temperature is always larger than the physical temperature T ( U q ). In contrast, the q -temperatureis smaller than the physical temperature when q < C , obtained as the derivative of U q with respect to the physical temperature T ( U q ), C = ∂U q ∂T ( U q ) (47)is related to the q -heat capacity of the system C q = ∂U q ∂T q = T q ∂ ¯ S q ∂T q (48)via the equation [19] C = 1 T ( U q ) T q C q − q − k B . (49)Similar equation has been obtained in Ref. [23]. Since T q > T ( U q ) when q >
1, from Eq. (49) follows that the physicalheat capacity C is always larger than the q -heat capacity C q . In contrast, for q < C isalways smaller than the q -heat capacity C q . V. CONCLUSIONS
In summary, we have considered a small system interacting via short-range forces with a large reservoir that haslarge q -heat capacity with q >
1. Such a system can be described by the non-extensive statistical mechanics, withthe probability of the microstate of the system given by the q -exponential (21) instead of the usual Boltzmann factor.The reservoir can be described using the generalized entropy and exhibit large q -heat capacity only when long-rangeinteractions and long-range correlations are present. Since we assumed short-range interactions of the system underconsideration with the reservoir, the approach presented in this paper is not applicable to a subsystem of such areservoir.The assumption of large q -heat capacity leads to a negative physical heat capacity, thus the description usingcanonical ensemble with q > T depends both on the properties of the reservoir and the properties of the system. On the other hand,the auxiliary q -temperature T (R) q (13) remains constant due to large q -heat capacity of the reservoir.The requirement that the system interacting with the reservoir should be small limits the growth of the numberof microstates of the system W ( E ) with increasing energy. We obtained that the description using the canonicalensemble is applicable only when W ( E ) grows slower than q -exponential with the value of q given by Eq. (27). Thislimit is valid for all values of q , for q > q = 1 and q < Appendix A: Sum of large q -exponentials Let us consider the sum of large q -exponentials Z q = W X i =1 e Nφ ( i ) q (A1)with q >
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