Tsallis' entropy maximization procedure revisited

The proper way of averaging is an important question with regards to Tsallis' Thermostatistics. Three different procedures have been thus far employed in the pertinent literature. The third one, i.e., the Tsallis-Mendes-Plastino (TMP) normalization procedure, exhibits clear advantages with respect to earlier ones. In this work, we advance a distinct (from the TMP-one) way of handling the Lagrange multipliers involved in the extremization process that leads to Tsallis' statistical operator. It is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the Tsallis-Mendes-Plastino formalism.


I. INTRODUCTION
Tsallis' thermostatistics [1][2][3][4][5][6] is by now recognized as a new paradigm for statistical mechanical considerations. One of its crucial ingredients, Tsallis' normalized probability distribution [1], is obtained by following the well known MaxEnt route [7]. One maximizes Tsallis' generalized entropy [3,8] (k ≡ k(q) tends to the Boltzmann constant k B in the limit q → 1 [2]) subject to the constraints (generalized expectation values) [2] w i=1 where p i is the probability assigned to the microscopic configuration i (i = 1, . . . , w) and one sums over all possible configurations w. O (the observation level [9]), whose generalized expectation values O j q are (assumedly) a priori known.
The Lagrange multipliers recipe entails maximizing [1] yielding where is the so-called configurational characteristic [10] and stands for the partition function.
The above procedure, originally employed in [1], overcomes most of the problems posed by the old, unnormalized way of evaluating Tsallis'generalized mean values [1,11]. Some hardships remain, though. One of them is that numerical difficulties are sometimes encountered, as the p i expression is explicitly self-referential. An even more serious problem is also faced: a maximum is not necessarily guaranteed. Indeed, analyzing the concomitant Hessian so as to ascertain just what kind of extreme we face, one encounters the unpleasant fact that this Hessian is not diagonal.
In the present effort we introduce an alternative Lagrange route, that overcomes the above mentioned problems.

II. THE NEW LAGRANGE MULTIPLIERS' SET
We extremize again (1) subject to the constraints (3), but, and herein lies the central idea, rephrase (4) by recourse to the alternative form with j = 1, . . . , n. We have now so that, following the customary variational procedure and eliminating α 0 we find that the probabilities are, formally, still given by (5). However, in terms of the new set of Lagrange multipliers, the configurational characteristics do not depend explicitly on the probabilities Comparing (10) with (6), it is clear that the Lagrange multipliers λ j of the Tsallis-Mendes-Plastino formalism (TMP) [1] and λ ′ j (present treatment) can be connected via which leads to the nice result f ′ i = f i . The probabilities that appear in (11) are those special ones that maximize the entropy, not generic ones. The ensuing, new partition function is also of the form (7), with f i > 0 the well known Tsallis' cut-off condition [3,8]. Notice that now the expression for the MaxEnt probabilities p i is NOT explicitly self-referential.
In order to ascertain the kind of extreme we are here facing we study the Hessian, that now is of diagonal form. The maximum condition simplifies then to the requirement The above derivatives are trivially performed yielding which formally coincides with the maximum requirement one finds in the case of Tsallis' unnormalized formalism. Since the f i are positive-definite quantities, for a maximum one should demand that q > 0.
Extremes found by following the celebrated Lagrange procedure depend only on the nature of the constraints, not on the form in which they are expressed. Thus, the two sets of multipliers lead to the same numerical values for the micro-state probabilities. Via (11) one is always able to establish a connection between both treatments.
The present algorithm exhibits the same nice properties of the TMP formalism, namely: • The MaxEnt probabilities are invariant under uniform shifts of the Hamiltonian's energy spectrum (see, for instance, the illuminating discussion of Di Sisto et al. [12]).
Indeed, after performing the transformation on equation (5), with f i given by (10), we trivially find that the probabilities p i keep their forms invariant if the λ ′ j do not change. Due to relation (11), the λ j are invariant too.
• The mean value of unity equals unity, i.e., 1 q = 1, which is not the case with the unnormalized expectation values [3,8].
• One easily finds that, for two independent subsystems A, B, energies add up:

III. OLM TREATMENT AND RÈNYI'S MEASURE
The above OLM recipe has the purpose of diagonalizing the Hessian characterizing the kind of extreme one finds after a Lagrange treatment. It is natural to ask whether the concomitant recipe works also for other information measures in addition to the Tsallis' one. We discuss next the Rènyi's measure (R q ) instance [11,[13][14][15] in an OLM context. We We extremize now R q subject to the constraints (8) with Lagrange multipiers µ * j . The ensuing Lagrange functional reads whose first derivatives are The entropy term will lead to a self referencial probability distribution. We have with a functional characteristic given by and the partition function We see that the OLM treatment leads to a self-referential solution that the "Rènyi-TMP" treatament does not exhibit. Indeed, on account of the fact both the constraints and the entropy share the i p q i denominator, the TMP-Rènyi distribution has characteristics of the form [15] Comparing Eqs. (21) and (23), one observes that the Lagrange multipliers are connected via that is the same relation found for Tsallis' measure (see Equation (11)). The following correspondence can be appreciated since the configurational characteristics of Equation (23) is identical to the one of Equation (10).
Consider now, for the second derivatives, the non-diagonal terms. One has. in the OLM instance, If one uses the standard TMP approach of Ref. [1] one finds, instead, for these nondiagonal terms, There is an important distinction to be made between (25) and (26), that refers to the origin of the non-diagonal terms. In the first case, they have as a source just the entropy term of the Lagrangian. In the second, the constraints also contribute.
As a consequence we gather that the OLM approach will work as nicely as in the Tsallis case only for those measures that do not give rise to off-diagonal terms in the associated Hessian.

IV. THERMODYNAMICS
We pass now to the question of writing down the basic mathematical relationships of Thermodynamics, as expressed with respect to the new set of Lagrange multipliers λ ′ j . In order to do this in the most general quantal fashion we shall work in a basisindependent way. This requires consideration of the statistical operator (or density operator) ρ that maximizes Tsallis' entropy, subject to the foreknowledge of M generalized expectation values (corresponding to M operators O j ). These take the form To these we must add, of course, the normalization requirement The TMP formalism, where relations are written in terms of the "old" Lagrange multipliers λ j , yields the usual thermodynamical relationships [1], namely where and [1] so that the essential mathematical structure of Thermodynamics is preserved.
Following the standard procedure [5,16] one getŝ whereZ q stands for the partition function Enters here Tsallis'cut-off condition [5,16]. The form (33) does not a priori guarantee that we will have a positive-definite operator. Some additional considerations are requested.

Consider the operator
enclosed within parentheses in (33). One must ensure its positive-definite character. This entails that the eigenvalues ofÂ must be non-negative quantities. This can be achieved by recourse to an heuristic cut-off procedure. We replace (33) bŷ withZ q given byZ where Θ(x) is the Heaviside step-function. Equations (36)-(37) are to be re-interpreted as follows. Let |i and α i , stand, respectively, for the eigenvectors and eigenvalues of the operator (35), whose spectral decomposition is then In the special basis used aboveρ adopts the appearancê with f (x) defined as and Notice that f (x) possesses, for 0 < q < 1, a continuous derivative for all x. Moreover, In terms of the statistical operator, Tsallis' entropy S q reads whereÎ is the unity operator.
Obviously,ρ commutes withÂ. The product of these two operators can be expressed in the common basis that diagonalizes them which entails, passing from the special basis |i to the general situation, that and, consequently Since the last term of the right-hand-side vanishes, by definition (8), we finally arrive at Now, from the very definition (in terms ofρ) of Tsallis' entropy S q [5,16], we find so that (47) and (48) lead to and where ln qZq has been introduced in (31).
Using (49), equation (11) can be rewritten as Following [1] we define now which leads finally to (see (29) and (30)) where (see equation (52)) Proponemos que β ′ Finally, the specific heat reads We conclude that the mathematical form of the thermodynamic relations is indeed preserved by the present treatment. Both sets of Lagrange multipliers accomplish this feat and they are connected via (11). The primed one, however, allows for a simpler treatment, as will be illustrated below.

V. SIMPLE APPLICATIONS
We consider now some illustrative examples. They are chosen in such a manner that each of them discusses a different type of situation: classical and quantal systems, the latter in the case of both finite and infinite number of levels.

A. The classical harmonic oscillator
Let us consider the classical harmonic oscillator in the canonical ensemble. We can associate with the classical oscillator a continuous energy spectrum ǫ(n) = ǫ n with ǫ > 0 and n ∈ R + compatible with the cut-off condition. The ensuing MaxEnt probabilities adopt the appearance where andZ with u q = U q /ǫ and t ′ = k ′ T /ǫ. We have introduced also n max as the upper integration limit on account of Tsallis' cut-off condition. One appreciates the fact that n max → ∞ if q > 1.
The normalization condition reads or, using (59), Due to the form of f q , equation (63) constitutes a well-defined expression. By explicitly performing the integrals for 1 < q < 2 (for q ≥ 2 the integrals diverge) we obtain After a little algebra, the above equation leads to the simple result Replacing now u q = U q /ǫ and t ′ = k ′ T /ǫ, we obtain U q = k ′ T , so that the specific heat It is worthwhile to remark that, in the case of this particular example, we formally regain the usual expressions typical of the q = 1 case. Due to fact that we possess a degree of freedom in the definition of k ′ , we can set k ′ = k B and thus recover Gibbs' Thermodynamics.
Performing the pertinent integral and using (65), the partition function becomes According to equation (51), t ′ can be written in terms of t andZ q , allowing us to recover and, consequently, These results are identical to those of [1], but are here derived in a remarkably simpler fashion.

B. The two-level system and the quantum harmonic oscillator
Let us consider the discrete case of a single particle with an energy spectrum given by E n = ǫn, where ǫ > 0 and n = 0, 1, ..., N. If N = 1, we are facing the non degenerate two level system, while, if n → ∞, the attendant problem is that of the quantum harmonic oscillator.
The micro-state probabilities are of the form, once again The configurational characteristics take the form where again (see (V A)), t ′ = k ′ T /ǫ and u q = U q /k ′ .
Using (71), the mean energy can be written as which, using the explicit form of f n and rearranging terms, allows one to write down the following equation which implicitly defines u q . Notice that one does not arrive to a closed expression. However, in order to numerically solve for u q , we just face (75). This equation is easily solved by recourse to the so-called "seed" methods (cut-off always taken care of), with quick convergence (seconds). This is to be compared to the TMP instance [1]. In their case, one faces a non-linear coupled system of equations in order to accomplish the same task. This coupled system can be recovered from (75) and (71), writing t ′ in terms of t.

C. Magnetic Systems
Consider now a very simple magnetic model, discussed, for instance, in [17]: the total (collective) spin operator. The simultaneous eigenvectors of S 2 and S z constitute a basis of the concomitant 2 N -dimensional space. We have |S, M , with that leads to an effective particle number as in [18]. Following the same mechanism and using (83), one finds that Remembering that x ′ = x/T r(ρ q ), it is possible to recover the TMP normalized solution and For x ′ → ∞, and for 0 < q < 1, expression (82) leads to an equation identical to that of whose solution reads Ŝ z q = N/2.

VI. CONCLUSIONS
In order to obtain the probability distribution p i that maximizes Tsallis' entropy subject to appropriate constraints, Tsallis-Mendes-Plastino extremize [1] and obtain andZ q is the partition function. Two rather unpleasant facts are thus faced, namely, • p i explicitly depends upon the probability distribution (self-reference).
• The Hessian of F is not diagonal.
In this work we have devised a transformation from the original set of Lagrange multipliers {λ j } to a new set {λ ′ j } such that • Self-reference is avoided.
• The Hessian of F becomes diagonal.
As a consequence, all calculations, whether analytical or numerical, become much simpler than in [1], as illustrated with reference to several simple examples. The primed multipliers λ ′ j = λ j i p q i incorporate the p i 1 in their definition. Since one solves directly for the primed multipliers, such a simple step considerably simplifies the TMP treatment. Finally, we remark on the fact that the two sets of multipliers lead to thermodynamical relationships that involve identical intensive quantities (??).

ACKNOWLEDGMENTS
The financial support of the National Research Council (CONICET) of Argentina is gratefully acknowledged. F. Pennini acknowledges financial support from UNLP, Argentina.
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