Physica A 340 (2004) 95–101www.elsevier.com/locate/physa

A possible deformed algebra and calculusinspired in nonextensive thermostatistics

Ernesto P. Borgesa;b;∗aEscola Politecnica, Universidade Federal da Bahia Rua Aristides Novis 2,

Salvador-BA 40210-630, BrazilbCentro Brasileiro de Pesquisas F$sicas R. Dr. Xavier Sigaud, 150, Urca,

Rio de Janeiro-RJ 22290-180, Brazil

Abstract

We present a deformed algebra related to the q-exponential and the q-logarithm functions thatemerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistentlya q-integral) for which the q-exponential is an eigenfunction. The q-derivative and the q-integralhave a dual nature, that is also presented.c© 2004 Elsevier B.V. All rights reserved.

PACS: 02.10.−v; 02.70.Rr; 05.90.+m

Keywords: Nonextensive statistical mechanics; q-Algebra; q-Calculus

1. Introduction

The de2nition of the so-called nonextensive entropy, Sq ≡ k(∑W

i pqi − 1)=(1 − q)

(q∈R is the nonextensive entropic index, k is a positive constant which de2nes theunity in which Sq is measured), and its applications to nonextensive thermostatistics(NEXT) [1,2] have opened new possibilities for the thermodynamical treatment ofcomplex systems [3], with a continuously increasing number of works (for an updatedbibliography, see Ref. [4]). Nonextensive statistical mechanics has been treated basi-cally along three (complementary) lines: formal mathematical developments [5,6] (inwhich this paper is included), observation of consistent concordance with experimental

∗ Corresponding author. Escola Polit9ecnica, Universidade Federal da Bahia Rua Aristides Novis 2,Salvador-BA 40210-630, Brazil.

E-mail address: ernesto@ufba.br (E.P. Borges).

0378-4371/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.03.082

96 E.P. Borges / Physica A 340 (2004) 95–101

(or natural) behavior [5], and theoretical physical developments [7] (see Ref. [8] forsome epistemological remarks on NEXT).

The basis of the formal developments inspired in NEXT is the de2nition ofdeformed expressions for the beautiful and ubiquitous logarithm and exponentialfunctions, namely, the q-logarithm and the q-exponential, 2rst proposed within thiscontext in 1994 [9]:

lnq x ≡ x1−q − 11 − q (x¿ 0); eq(x) ≡ [1 + (1 − q)x]1=(1−q)

+ (x; q∈R) ;

(1)

where [A]+ ≡ max{A; 0}. Note that eq(x)=+∞ if q¿ 1 and 1+(1−q)x6 0. Of coursethere are in2nitely many ways of generalizing a function, and many have already beenproposed, for instance (see Ref. [10]), expq x ≡ ∑∞

n=0 xn=[n]q!, with [n]q! =

∏nj=1 [j]q

and [j]q=(qj−1)=(q−1) and also [0]q!=1, but we focus here speci2cally on those onesthat naturally emerge from NEXT (Eq. (1)). Some works have appeared developingproperties of such q-deformed functions, e.g., Refs. [11–16], and have also stimulatedthe de2nition of other (alternative) deformations, as in Refs. [17–19].

It is remarkable that in less than one decade (since Ref. [9]) these functions (lnq xand eq(x)) have become so popular among the practitioners of NEXT that in manypapers they are referred to as if they were known since long, which is a very posi-tive symptomatic behavior. And they are known since long, indeed: the q-exponential isa particular solution of Bernoulli’s equation, known by Leibniz (according toRefs. [20,21]). (The independent re-discovering of laws, natural phenomena, etc., ismuch more frequent in science than one outsider would be inclined to think.)

This paper presents some formal, and curious, properties of q-exponentials andq-logarithms, hopefully inspiring for further developments. In some sense, we followthe work of Kaniadakis, who has explored his own �-deformation of the exponentialand logarithm functions [17–19].

2. q-Algebra1

We focus on the (now well known) following relations:

lnq(xy) = lnq x + lnq y + (1 − q)lnq x lnq y (2)

(which justi2es the terminology “nonextensive”) and

eq(x)eq(y) = eq(x + y + (1 − q)xy) ; (3)

which holds if eq(x) and eq(y) diLer from zero and from +∞. These properties inspireus to de2ne a generalization of the sum operation between two numbers x and y:

x ⊕q y ≡ x + y + (1 − q)xy ; (4)

1We point out that parts of the contents of this section have appeared independently and almost simul-taneously in the early versions of Ref. [22] and also in the early version of the present work, as explainedin the Acknowledgments.

E.P. Borges / Physica A 340 (2004) 95–101 97

which brings the usual sum as a particular case ⊕1 ≡ +. The q-sum is commutative(x⊕q y= y⊕q x), associative (x⊕q (y⊕q z) = (x⊕q y) ⊕q z), but it is not distributivein relation to the usual multiplication (a(x ⊕q y) �= (ax ⊕q ay)). The neutral elementof the q-sum is zero, x ⊕q 0 = x. We can de2ne the opposite (or inverse additive)of x (calling it qx) as the element that, when q-summed with x, yields the neutralelement: x ⊕q (qx) = 0. So, we have

q x ≡ −x1 + (1 − q)x (x �= 1=(q− 1)) : (5)

This de2nition permits us to de2ne the q-diLerence, as the q-sum with the q-opposite

x q y ≡ x ⊕q (qy) =x − y

1 + (1 − q)y (y �= 1=(q− 1)) : (6)

The q-diLerence obeys x q y = qy ⊕q x and x q (y q z) = (x q y) ⊕q z, buta(x q y) �= (ax q ay).

We now search for a generalization of the multiplication operation in such a waythat we can compactly rewrite Eqs. (2) and (3) as lnq(x ⊗q y) = lnq x + lnq y andeq(x) ⊗q eq(y) = eq(x + y). This leads us to the de2nition of the q-product betweentwo numbers

x ⊗q y ≡ [x1−q + y1−q − 1]1=(1−q)+ (x; y¿ 0) : (7)

The q-product is commutative (x⊗q y= y⊗q x) and associative (x⊗q (y⊗q z) = (x⊗q

y) ⊗q z), provided x ⊗q y and y ⊗q z diLer from zero and from in2nity. It is easy tosee that the number one is the neutral element of the q-product (x ⊗q 1 = x), and thispermits us to de2ne the inverse multiplicative, 1 �q x, by means of x⊗q (1 �q x) ≡ 1.We 2nd

1 �q x ≡ [2 − x1−q]1=(1−q)+ (x¿ 0) : (8)

The relation 1 �q (1 �q x) = 1 holds only if 06 x1−q6 2. It is curious to note that1 �q 0 does not diverge for q¡ 1. The q-ratio is thus de2ned by

x �q y ≡ [x1−q − y1−q + 1]1=(1−q)+ (x; y¿ 0) (9)

and satis2es x�q y=1�q (y�q x), provided x1−q6 1+y1−q, and also x�q (y�q z)=(x�q y) ⊗q z= (x⊗q z) �q y, provided z1−q − 16y1−q6 x1−q + 1. These q-algebraicrelations permit us to express the properties of the q-logarithm and the q-exponentialin a more compact form:

lnq(xy) = lnq x ⊕q lnq y; eq(x)eq(y) = eq(x ⊕q y) ; (10)

lnq(x ⊗q y) = lnq x + lnqy; eq(x) ⊗q eq(y) = eq(x + y) ; (11)

lnq(x=y) = lnq x q lnq y; eq(x)=eq(y) = eq(x q y) ; (12)

lnq(x �q y) = lnq x − lnq y; eq(x) �q eq(y) = eq(x − y) : (13)

98 E.P. Borges / Physica A 340 (2004) 95–101

The equalities hold only under certain restrictions, as indicated in the following table:

x¿ 0; y¿ 0 x¿q 0 or y¿q 0

x1−q + y1−q¿ 1 x¿q 0 and y¿q 0

x¿ 0; y¿ 0 y¿q 0

x1−q + 1¿y1−q x¿q 0 or y¿q 0

(14)

Here, the notation x¿q 0 means 1 + (1 − q)x¿ 0. Among these relations, we callattention to Eq. (11), that shows how to q-factorize the kinetic and potential parts ofthe Hamiltonian of a system (and further q-factorize each one of them individually).It is straightforward to de2ne a q-power,

x⊗nq ≡ x ⊗q x ⊗q x ⊗q · · · ⊗q x︸ ︷︷ ︸

n times

= [nx1−q − (n− 1)]1=(1−q)+ ; (15)

with x⊗n1 ≡ n. It follows also straightforwardly the q-sum of n identical terms

(see Eq. (7) of Ref. [23]):

x ⊕q x ⊕q x ⊕q · · · ⊕q x︸ ︷︷ ︸n terms

=1

1 − q {[1 + (1 − q)x]n − 1} : (16)

We call attention that this equation may be the basis for the de2nition of a diLerent,non-commutative, q-product (that we could note q): n qx �= x qn, with 1 qx=x andn 1x=nx. There is another possibility of generalizing the q-algebra, along nonextensivelines, recovering the distributivity, as done in Ref. [22], following Ref. [19]. We donot explore these possibilities in the present work.

3. q-Calculus

We now focus on a possible q-calculus associated with NEXT. Abe [24] has shownthat the Jackson’s q-derivative, Dqf(x) ≡ [f(qx) −f(x)]=[(q− 1)x] (that is related todilations) is intimately connected to nonextensive entropy, as the usual (Newtonian)derivative (that is related to translations) is connected to Boltzmann–Gibbs entropy.But this Jackson’s operator is not the proper one we are searching here. The usualexponential function is invariant in relation to derivation, dy=dx=y, or, in other words,ex is eigenfunction of the derivative operator. The q-exponential satis2es dy=dx=yq. Sowe can ask: what is the operator for which the q-exponential is an eigenfunction? Theanswer to this question may be found by following the steps of Kaniadakis [17–19].Let us de2ne the q-derivative as

D(q)f(x) ≡ limy→x

f(x) − f(y)x q y

= [1 + (1 − q)x]df(x)dx

: (17)

E.P. Borges / Physica A 340 (2004) 95–101 99

It is promptly veri2ed that the q-exponential is eigenfunction of D(q). We have thecorresponding q-integral (noted as

∫(q)) given by∫

(q)f(x)dqx =

∫f(x)

1 + (1 − q)x dx ; (18)

where

dqx ≡ limy→x

x q y =1

1 + (1 − q)x dx : (19)

Naturally∫

(q) D(q)f(x) dqx = D(q)∫

(q) f(x) dqx = f(x). Eqs. (17)–(19) are restricted to

[1 + (1 − q)x] �= 0. There is a dual derivative operator D(q), associated with D(q),de2ned by

D(q)f(x) ≡ limy→x

f(x) q f(y)x − y =

11 + (1 − q)f(x)

df(x)dx

; (20)

and its corresponding dual q-integral∫ (q)

f(x) dx ≡∫

[1 + (1 − q)f(x)]f(x) dx : (21)

Also here∫ (q) D(q)f(x) dx=D(q)

∫ (q) f(x) dx=f(x). Similarly to the restrictions of D(q)

and∫

(q), Eqs. (20) and (21) are valid if 1+(1−q)f(x) �= 0. We have D(q) lnq x=1=x,

while D(1) lnq x = d=dx lnq x = 1=xq. The usual derivative and the usual integral areobviously auto-dual. The dual q-derivatives are related by

D(q)f(x) =1

[1 + (1 − q)x] [1 + (1 − q)f(x)]D(q)f(x) : (22)

The q-derivatives follow the chain rule in the forms:

D(q)[f(x)g(x)] = D(q)[f(x)]g(x) + f(x)D(q)[g(x)] (23)

and

D(q)[f(x)g(x)] =1

[1 + (1 − q)f(x)g(x)]

× {[1 + (1 − q)f(x)]D(q)[f(x)] g(x)

+ [1 + (1 − q)g(x)]f(x)D(q) [g(x)]} : (24)

4. Concluding remarks

The relations presented are compact and relatively simple, though there are manyexceptions that have to be taken into account originated from the cutoL that appears in

100 E.P. Borges / Physica A 340 (2004) 95–101

the q-deformed exponential function (see, for instance, restrictions (14)), and it is ofcourse important that we are aware of them. Despite these limitations, it is remarkablethat many phenomena seem to be fairly 2tted by q-exponentials, or functions thatbelong to its family. Examples range over physical, biological, economical and socialsystems (all of them may be classi2ed into the category of complex systems). Besidesthe observation of fair description of phenomena, there are increasingly many works ontheoretical developments of nonextensive ideas. Just to cite an important and very recentexample, let us mention the exact analytical results by Baldovin and Robledo [25] aboutthe q-Pesin identity for the dynamics at the edge of chaos of the logistic map. For theinterested reader, we give the address of the web page [4]. Of course, all these elements(mathematical framework, experimental evidence, theoretical developments) are notyet enough to be conclusive, and many eLorts have been made in order to clarify thedomain of validity of Boltzmann–Gibbs statistical mechanics, of nonextensive statisticalmechanics, or even others that might happen in nature.

Acknowledgements

We gratefully acknowledge Giorgio Kaniadakis for calling us attention to the exis-tence of the duality in the q-derivatives. We also thank Q. Alexandre Wang (togetherwith his coauthors) for sending to us their paper Ref. [22], while it was in its e-printform [http://arXiv.org/math-ph/0303061], soon after he took notice of the earlye-print version of the present work [http://arXiv.org/cond-mat/0304545]. Theseworks present many similarities, specially regarding the q-algebra (Section 2). I amdeeply indebted to Jan Naudts, who introduced many valuable improvements on thiswork. Last but not least, the unforgettable hospitality of the organizers of NEXT 2003is warmly acknowledged.

References

[1] C. Tsallis, J. Stat. Phys. 52 (1988) 479.[2] E.M.F. Curado, C. Tsallis, J. Phys. A: Math. Gen. 24 (1991) L69;

E.M.F. Curado, C. Tsallis, Corrigenda, 24 (1991) 3187 and 25 (1992) 1019.[3] A. Cho, Science 297 (2002) 1268.[4] http://tsallis.cat.cbpf.br/biblio.htm[5] S. Abe, A.K. Rajagopal, Science 300 (2003) 249.[6] A. Plastino, Science 300 (2003) 250.[7] V. Latora, A. Rapisarda, A. Robledo, Science 300 (2003) 250.[8] C. Tsallis, E.P. Borges, F. Baldovin, Physica A 305 (2002) 1.[9] C. Tsallis, Quimica Nova 17 (1994) 468.

[10] C. Kassel, Quantum Groups, Springer, New York, 1995.[11] E.P. Borges, J. Phys. A: Math. Gen. 31 (1998) 5281.[12] C. Tsallis, Braz. J. Phys. 29 (1999) 1.[13] E.K. Lenzi, E.P. Borges, R.S. Mendes, J. Phys. A: Math. Gen. 32 (1999) 8551.[14] C. Tsallis, Nonextensive statistical mechanics and thermodynamics: Historical background and present

status, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and Its Applications, SeriesLecture Notes in Physics, Springer, Heidelberg, 2001.

E.P. Borges / Physica A 340 (2004) 95–101 101

[15] T. Yamano, Physica A 305 (2002) 486.[16] J. Naudts, Physica A 316 (2002) 323.[17] G. Kaniadakis, Physica A 296 (2001) 405.[18] G. Kaniadakis, A.M. Scarfone, Physica A 305 (2002) 69.[19] G. Kaniadakis, Phys. Rev. E 66 (2002) 056125.[20] W.E. Boyce, R.C. DiPrima, Elementary DiLerential Equations and Boundary Value Problems, 2nd

Edition, Wiley, New York, 1969.[21] E. Kreyszig, Advanced Engineering Mathematics, 7th Edition, Wiley, New York, 1993.[22] L. Nivanen, A. Le M9ehaut9e, Q.A. Wang, Rep. Math. Phys. 52 (2003) 437.[23] E. Vives, A. Planes, Phys. Rev. Lett. 88 (2002) 020601.[24] S. Abe, Phys. Lett. A 224 (1997) 326.[25] F. Baldovin, A. Robledo, 2003, http://arXiv.org/cond-mat/0304410.