Opening Pandora's Box: Maximizing the q -entropy with Escort Averages
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Opening Pandora’s Box: Maximizing the q -entropy with Escort Averages Aruna Bidollina, Thomas Oikonomou, ∗ and G. Baris Bagci Department of Mathematics, School of Science and Technology,Nazarbayev University, Astana 010000, Kazakhstan Department of Physics, School of Science and Technology,Nazarbayev University, Astana 010000, Kazakhstan and Department of Physics, Mersin University, 33110 Mersin, Turkey (Dated: April 2, 2019)It is currently a widely used practice to write the constraints in terms of escort averages when thegeneralized entropies are employed in the maximization scheme. We show that the maximizationof the nonadditive q -entropy with escort averages leads either to an overall lack of connection withthermodynamics or violation of the second and third laws of thermodynamics if one adopts theClausius definition of the physical temperature. If an alternative definition of physical temperatureis chosen by respecting the divisibility of the total system into independent subsystems, thermody-namic relations are restored albeit at the cost of transforming the nonadditive q -entropy into theR´enyi entropy. These results are illustrated by studying the quantum mechanical free particle. PACS numbers: 05.20.-y; 05.20.Dd; 05.20.Gg; 51.30.+iKeywords: nonadditive q -entropy; R´enyi entropy; escort average; entropy maximization; free particle I. INTRODUCTION
The nonadditive q -entropy (historically called as the nonextensive entropy) [1, 2], despite its numerous applicationsin many diverse fields [3–14], still presents some open problems such as its connection to thermodynamics and thedefinition of a physical temperature [15, 16], stability of the averaging schemes [17, 18] and its axiomatic foundations[19, 20]. One such issue still under debate is how the entropy maximization is to be carried out [21–23]. In particular,this issue revolves around the averaging scheme one should adopt concerning the internal energy constraint if oneaims to have a consistent equilibrium distribution and associated thermodynamical structure.Historically, the internal energy in nonextensive theory was defined as usual i.e. U = P i p i ε i in terms of the ordinarylinear averaging scheme [22]. Having understood it to be problematic for various reasons, a second choice has been theexpression U = P i p qi ε i for the internal energy in the functional associated with the entropy maximization (MaxEnt)procedure. Due to its severe drawbacks such as violation of the energy conservation for example, the third and so farfinal choice has been the so-called escort averaged internal energy expression which reads U q = P i p qi ε i P k p qk [22].However, the issue of the escort averaged internal energy presents two distinct problems. First, there emergestwo different expressions of the equilibrium distribution and its concomitant partition function even though oneuses the same constraint for the internal energy i.e. U q = P i p qi ε i P k p qk [22]. Therefore, any discussion of the q -entropymaximization with the escort averages should consider both of these distributions to assess the feasibility of this typeof maximization. Second problem is the vagueness of the physical (inverse) temperature when one employs the escortconstraints [15, 16]. If one assumes the divisibility of the total system into its subsystems despite the nonadditivityof the q -entropy, then one is forced to use β P i p qi ( β being the Lagrange multiplier associated with the internal energyconstraint) as the physical (inverse) temperature for thermodynamic inconsistency. On the other hand, if one considersthe Clausius entropy as the point of departure, one should instead simply use β as the physical (inverse) temperature[15, 16]. Therefore, one should consider both approaches to the physical temperature to ensure a correct assessmentof the issue.Our aim in this work is to attempt a detailed study of the nonadditive q -entropy maximization with escort averagedinternal energy expression. We will consider alternative forms of the MaxEnt probability distribution and also twodistinct physical temperature expressions. The prevalent inconsistencies thereby found are also illustrated throughthe free particle model. Concluding remarks are presented in section III. ∗ Electronic address: [email protected]
II. THE NONADDITIVE q -ENTROPY, ESCORT AVERAGES AND THERMODYNAMICS The functional to be maximized readsΦ = S q k B − α "X i p i − − β (cid:20) P i p qi ε i P k p qk − U q (cid:21) , (1)where S q = k B P i p i ln q (1 /p i ) is the nonadditive q -entropy written in terms of the q -deformed logarithm ln q ( x ) = x − q − − q [24], U q is the internal energy and k B denotes the Boltzmann constant. Taking the partial derivative of theabove functional with respect to p i and then equating it to zero, we obtain qp q − i − q (cid:20) − (1 − q ) β P k p qk ( ε i − U q ) (cid:21) − − q − α = 0 (2)which can be cast into the form below p i = (cid:20) − q ) αq (cid:21) q − (cid:20) − (1 − q ) β P k p qk ( ε i − U q ) (cid:21) − q . (3)Probability normalization yields then p i = 1 Z q (cid:20) − (1 − q ) β P k p qk ( ε i − U q ) (cid:21) − q (4)with the q -partition function Z q is given as Z q = (cid:20) − q ) αq (cid:21) − q = X i (cid:20) − (1 − q ) β P k p qk ( ε i − U q ) (cid:21) − q . (5)If one multiplies Eq. (2) with p i and sums over the all i ’s, one obtains X i p qi = ( Z q ) − q , (6)where Eq. (5) is taken into account. Thus, the nonadditive q -entropy of the equilibrium distribution is calculated as S q = k B X i p i ln q (1 /p i ) = k B P i p qi − − q = k B ( Z q ) − q − − q (7)which, in terms of the deformed q -logarithm, reads S q = k B ln q ( Z q ) . (8)This is an important result worth pondering. First of all, the expression above implies that the essential link betweenthe statistical mechanics and thermodynamics is severely missing, since one cannot construct a relation between theentropy, partition function and the average energy but only between the former two. As a result, one cannot trusta consistent connection to exist between the statistical mechanics and thermodynamics based on the nonadditive q -entropy if one employs the escort averaged internal energy constraints. Note that it has recently been shown thatthe R´enyi entropy is equal to the (natural) logarithm of the partition function when one uses the ordinary internalenergy definition so that it has been concluded that there is no R´enyi thermodynamics (see Ref. [25] and in particularEq. (4.15) therein).Second, the inspection of Eq. (5) in the q → Z = e βU P i e − βε i for the partitionfunction in this particular limit. We recall that one very general feature of the nonextensive theory as a generalizationscheme is that one should obtain the expressions in ordinary statistical mechanics whenever this limit is invoked.A comparison of Z with the ordinary canonical partition function Z = P i e − βε i explicitly shows that they aredifferent from one another, the former including a multiplicative e βU term i.e. Z = e βU Z . Note, however, that thecancellation of this extra term in both nominator and denominator of Eq. (4) in the limit q = 1 enables one to obtainthe ordinary canonical distribution [26]. In other words, employing escort averaged internal energy constraint yieldsa generalized partition function which does not warrant the textbook canonical partition function in the appropriatelimit.In order to illustrate the viewpoint above, we now consider a quantum mechanical free particle [27]. The densityoperator for the q -entropy with escort distributions is given byˆ ρ q = ˆ A q Z q ( β ) , ˆ A q := h − (1 − q ) β q ( ˆ H − U q ) i − q , (9)where we confine ourselves only to q > Z q and the energy factor β q aregiven by Z q ( β ) = Tr n ˆ A q o , β q := β Tr { (ˆ ρ q ) q } = β [ Z q ( β )] − q . (10)Using the integral representation of Z q [30], we can write it as Z q ( β ) = 1Γ (cid:16) q − (cid:17) Z ∞ d t e − t [1+ β q ( q − U − U q )] Z [ tβ q ( q − t q − − , (11)where Z ( β ) = Tr n e − β ( ˆ H − U ) o . (12)In particular, for the free particle of mass m confined in one-dimensional length L , we have Z ( β ) = e βU L (cid:16) m πβ ~ (cid:17) so that its substitution into the equation above yields Z q ( β ) = ( β q ) − e L q [1 − ( q − β q U q ] − q + , e L q := L (cid:16) m π ~ (cid:17) Γ (cid:16) q − − (cid:17) Γ (cid:16) q − (cid:17) √ q − q ∈ (1 , β q U q < / ( q −
1) and e L = L (cid:0) m π ~ (cid:1) . The substitution of the relation β q = β [ Z q ( β )] − q into theequation above gives [ Z q ( β )] q = β − e L q (cid:20) − ( q − β [ Z q ( β )] − q U q (cid:21) − q + . (14)Solving Eq. (14) with respect to U q we have U q ( β ) = [ Z q ( β )] − q β ln q [ Z q ( β )] q − q β − q [ e L q ] − q ! . (15)Calculating the derivative with respect to β and taking into account the relation ∂∂β ln q ( Z q ) = β ∂∂β U q we have1 β ∂ ln q ( Z q ( β )) ∂β = ∂∂β ( [ Z q ( β )] − q β ln q [ Z q ( β )] q − q β − q [ e L q ] − q !) . (16)After some simple algebra, this differential equation reduces to (cid:20) Z q ( β ) − − q ) β ∂Z q ( β ) ∂β (cid:21) q − β e L q ! − q − q [ Z q ( β )] (1 − q )(1+ q )3 − q = 0 . (17)As can be seen here there are two distinct solutions of the differential equation in Eq. (17), namely Z (1) q ( β ) = c [2(1 − q )] − q ) β − q ) , (18a) Z (2) q ( β ) = (cid:16)e L q (cid:17) q (cid:18) − q (cid:19) − q − q β − q , (18b)where c is the integration constant.This is indeed a very strange situation, since the partition function uniquely describes a physical system. In otherwords, it does not make sense to have two partition functions for the same physical system. The first solution doesnot converge in the q → q → q → Z (2) q ( β ) conforms to thisrequirement. Thus, for the sake of simplicity, we will denote Z (2) q as Z q . However, there now emerges another problem,since the partition function Z q does not yield the correct canonical expression in the q → Z ( β ) = √ e Z ( β ) . (19)In fact, this is to be expected, since the ordinary average internal energy for the free particle is U = U = β so thatthe aforementioned relation Z = e βU Z ( β ) yields Eq. (19). As previously explained, this is an artefact of severingthe link between thermodynamics and statistical mechanics as a result of employing escort averaged internal energyexpression i.e. Eq. (8).Nevertheless, one can avoid this problem by rewriting the density operator in Eq. (9) asˆ ρ q = [1 − (1 − q ) β q ˆ H ] − q Z q ( β ) , (20)where Z q ( β ) = Z q ( β )exp q ( β q U q ) and β q = β q − q ) β q U q . (21)Now, it can be easily checked that Z q ( β ) yields the correct canonical expression in the q → Z q to be the ultimate partition function associated with the escort averaging procedure.Despite this succesful rewritting, another difficulty emerges now, since one cannot be sure of how thermodynamicobservables will behave under this rewriting. We certainly know that these new expressions will correctly yield therespective canonical expressions, but we have no clue whatsoever on whether they will exhibit consistent behavior forall q values except q → β q = (cid:16)e L q (cid:17) q − q (cid:18) − q (cid:19) q − q +1 β q , (22a) β q = 23 − q β q , Z q ( β ) = Z q ( β )exp q (1 / , (22b) U q = 12 ( e L q ) − q )1+ q (cid:18) − q (cid:19) − qq +1 β − q = 12 β q . (22c)It can be checked that all these quantities correctly recover the respective canonical ones. One can also calculate theheat capacity using the above expressions so that one has C q = ∂U q ∂T = k B q ( e L q ) − q )1+ q (cid:18) − q (cid:19) − qq +1 ( k B T ) − q q , (23)where k B T = β − . In Fig. 1, for the interval q ∈ (1 , C q as a function of both temperatureand the non-additivity parameter q where we set L = m = ~ = k B = 1. It can be seen that the heat capacity C q decays for increasing temperature therefore violating both 2nd and 3rd laws of thermodynamics. Therefore, therewriting of the distribution given in Eqs. (20) and (21) is unphysical. Note that the same unphysical behavior ofthe heat capacity occurs for the same model even though one adopts the ordinary internal energy expression (seeFig. 5 and related explanations in Ref. [27]). To sum up, the maximization of the nonadditive q -entropy with theescort averages, depending on how we choose to write the resulting equilibrium distribution, either severs the link FIG. 1: The heat capacity C q as a function of the temperature T for the interval q ∈ (1 , between the statistical mechanic and thermodynamics (see Eq. (8) and explanations below it) and does not yield theordinary canonical partition function in the appropriate limit or results in the violation of the second and third lawsof thermodynamics (see Fig. 1).In fact, one may even say that this happens because T is not the physical temperature in nonextensive systems.Note that there are two definitions of (inverse) temperature in the literature when the escort averages are used, onestemming from the Clausius relation [16] and the other from the assumption of divisibility of the total system intoindependent subsystems [15]. The former is the Lagrange multiplier β and corresponds to the (inverse) temperaturedefinition we adopted so far. The latter is defined as k B T phys q := ( β q ) − so that the internal energy and the heatcapacity become U q = k B T phys q ⇒ C q = ∂U q ∂T q = k B . (24)We now have both the consistent limits and sensible behaviors for the thermodynamic observables.Despite this improvement though, we face two novel and serious drawbacks. First drawback is that the thermo-dynamic observables such as internal energy and heat capacity have now exactly the same form as in the ordinarycanonical case. As a result, it is not clear at all why one should use the nonextensive theory instead of the ordinarycanonical scheme. In other words, as far as the thermodynamic observables are concerned, nonextensive theory seemsredundant although the underlying, unobservable entropy S q may be different from the ordinary Boltzmann-Gibbs-Shannon entropy.Second and more serious drawback can be noted by realizing that the change of the inverse temperature β to β q used above is not merely a substitution but a transformation implying also a transformation of the entropy expression.To see this explicitly, we begin with the following relation β = 1 k B ∂S q ∂U q = 1 k B ∂S q /∂β∂U q /∂β . (25)Taking into account Eq. (8) i.e. S q = k B ln q ( Z q ), we have1 k B ∂S q ∂β = ∂ ln q ( Z q ) ∂β = ( Z q ) − q ∂ ln( Z q ) ∂β . (26)Combining Eqs. (25) and (26) we have β q = ∂ ln( Z q ) ∂U q . (27)The comparison of the equation above with Eq. (25) shows that the entropy expression related to the temperature β q is not S q anymore but S phys q = k B ln( Z q ). Finally, using Eq. (6) for the expression S phys q = k B ln( Z q ), we identifythis entropy as the R´enyi entropy [31] S phys q = k B ln( Z q ) = k B − q ln X i p qi ! . (28)In other words, the maximization of the entropy S q with escort constraints and the adoption of β q as the physicaltemperature are equivalent to adopting the R´enyi entropy. Therefore, such a combination can not be solely treatedin the context of the nonextensive theory any more [32]. III. CONCLUSIONS
The nonadditive q -entropy is currently maximized by defining the internal energy as U q = P i p qi ε i P k p qk instead of thewell-known expression U = P i p i ε i . This type of averaging is called escort averaging [22] and it has been criticizedin terms of its stability [17, 18] before.In this work, instead of focusing on the escort averaging per se , we consider the maximization scheme and theresulting probability distribution. Written in one form, the distribution results in S q = k B ln q ( Z q ) i.e. Eq. (8)without any explicit appearance of the internal energy expression. As a result, one cannot construct a bridge betweenstatistical mechanics and thermodynamics as in the ordinary case i.e. S/k B = ln Z + βU . Moreover, the canonicallimit of the partition function in this context is found as e βU P i e − βε i instead of the correct expression P i e − βε i .Written in another form, the distribution obtained from the escort averaged internal energy implies a violation ofsecond and third laws of thermodynamics in thermodynamic observables as shown in Fig. 1. However, all these resultsare valid only if one adopts a physical temperature equal to the inverse of the internal energy Lagrange multiplier β .This is the definition one derives by recourse to the Clausius relation as shown in Ref. [16].On the other hand, there is another definition of physical temperature which stems from the assumption of thedivisibility of the total system into independent subsystems Ref. [15]. When this temperature β q is considered to bethe physical one, one recovers the equipartition results so that the thermodynamic consistency is restored as can beseen from Eq. (24). However, when this is the case, we show that the use of this alternative physical temperaturedefinition is tantamount to the adoption of the R´enyi entropy, deeming the nonadditive q -entropy redundant. Infact, it has been previously found that the requirement of the equipartition theorem necessitates the use of the R´enyientropy [32].To conclude, the maximization of the nonadditive q -entropy with escort averages implies either thermodynamicanomalies or a transmutation of the q -entropy into the R´enyi entropy, making the former redundant. Finally, note thatthe ordinary averaging scheme also results in thermodynamic anomalies [27] or foundational inconsistencies [19, 20, 23].Therefore, we conclude that how constraints should be averaged is still an open problem for the nonadditive q -entropy. Acknowledgments
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