Multiplicative duality, q-triplet and (μ,ν,q) -relation derived from the one-to-one correspondence between the (μ,ν) -multinomial coefficient and Tsallis entropy Sq

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Physica A 387 (2008) 71–83 www.elsevier.com/locate/physa

Multiplicative duality, q-triplet and ðm; n; qÞ-relation derived from the one-to-one correspondence between the ðm; nÞ-multinomial coefficient and Tsallis entropy S q Hiroki Suyaria,, Tatsuaki Wadab a

Department of Information and Image Sciences, Chiba University, Chiba 263-8522, Japan Department of Electrical and Electronic Engineering, Ibaraki University, Hitachi, Ibaraki 316-8511, Japan

b

Received 14 February 2007; received in revised form 23 July 2007 Available online 12 September 2007

Abstract We derive the multiplicative duality ‘‘q21=q’’ and other typical mathematical structures as the special cases of the ðm; n; qÞ-relation behind Tsallis statistics by means of the ðm; nÞ-multinomial coefficient. Recently the additive duality ‘‘q22  q’’ in Tsallis statistics is derived in the form of the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy. A slight generalization of this correspondence for the multiplicative duality requires the ðm; nÞ-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the ðm; nÞ-multinomial coefficient and Tsallis entropy S q , which determines a concrete relation among three parameters m; n and q, i.e., nð1  mÞ þ 1 ¼ q which is called ‘‘ðm; n; qÞ-relation’’ in this paper. As special cases of the ðm; n; qÞ-relation, the additive duality and the multiplicative duality are recovered when n ¼ 1 and n ¼ q, respectively. As other special cases, when n ¼ 2  q, a set of three parameters ðm; n; qÞ is identified with the q-triplet ðqsen ; qrel ; qstat Þ recently conjectured by Tsallis. Moreover, when n ¼ 1=q, the relation 1=ð1  qsen Þ ¼ 1=amin  1=amax in the multifractal singularity spectrum f ðaÞ is recovered by means of the ðm; n; qÞ-relation. r 2007 Published by Elsevier B.V. Keywords: Additive duality; Multiplicative duality; q-triplet; Multifractal triplet; ðm; n; qÞ-relation; q-product; Tsallis entrophy; ðm; nÞ-multinomial coefficient

1. Introduction In the last two decades the so-called Tsallis statistics or q-statistics has been introduced [1] and studied as a generalization of Boltzmann–Gibbs statistics with many applications to complex systems [2,3], whose information measure is given by Sq ðp1 ; . . . ; pk Þ ¼

P 1  ki¼1 pqi , q1

Corresponding author.

E-mail addresses: [email protected], [email protected] (H. Suyari), [email protected] (T. Wada). 0378-4371/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physa.2007.07.074

(1)

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where pi is a probability of ith state and q is a real parameter. This generalized entropy S q is nowadays called Tsallis entropy which recovers Boltzmann–Gibbs–Shannon entropy S1 when q ! 1. The above entropic form (1) was first given in Refs. [4,5] from a mathematical motivation, but in 1988 [1] Tsallis first applied the above form (1) to a generalization of Boltzmann–Gibbs statistics for nonequilibrium systems through the maximum entropy principle (MaxEnt for short) along the lines of Jaynes approach [6]. Since then, many applications of (1) to the studies of complex systems with power-law behaviors have been presented using the MaxEnt as a main approach [7]. In fact, the q-exponential function appeared in the MaxEnt plays a crucial role in the formalism and applications [2,3]. For all many applications of the MaxEnt for Tsallis entropy (1), there have been missing a combinatorial consideration in Tsallis statistics until recently [8], whose ideas originate from Boltzmann’s pioneering work [9] (see Ref. [10] for the comprehensive review). By means of the q-product uniquely determined by the qexponential function [11,12] as the q-exponential law, the one-to-one correspondence between the qP multinomial coefficient and Tsallis entropy is obtained as follows [8]: for n ¼ ki¼1 ni and ni 2 N if qa2, " # n n n2q nk  1  S 2q ;...; ’ lnq , (2) n1    nk 2q n n q h i n is the q-multinomial coefficient and lnq is the q-logarithm (see Definitions 1 and 6). The above where n1 n k q

correspondence (2) obviously recovers the well-known correspondence: " # n n nk  1 ;...; ’ n  S1 ln n1    nk n n

(3)

when q ! 1. Moreover, the additive duality ‘‘q22  q’’ in Tsallis statistics is presented in the form of (2). In the MaxEnt formalism for Tsallis entropy, two kinds of dualities ‘‘q22  q’’ and ‘‘q21=q’’ have been observed and discussed [7,13–16], but in the combinatorial formalism the multiplicative duality ‘‘q21=q’’ is still missing. In this paper, we derive the multiplicative duality ‘‘q21=q’’ along the lines of the above correspondence (2), which introduces the ðm; nÞ-factorial as a generalization of the q-factorial. We apply the ðm; nÞ-factorial to the formulation of the ðm; nÞ-multinomial coefficient and ðm; nÞ-Stirling’s formula, which results in the following P correspondence: for n ¼ ki¼1 ni and ni 2 N if q; na0, " # n n 1 nq nk  1 lnm ;...; ’  Sq , (4) n1    nk n q n n ðm;nÞ h i n is the ðm; nÞ-multinomial coefficient and three parameters m; n; q satisfy the relation: where n1 n k ðm;nÞ

nð1  mÞ þ 1 ¼ q

(5)

which is called ‘‘ðm; n; qÞ-relation’’ throughout the paper. Using the additive duality ‘‘q22  q’’ in (2), (2) is rewritten by " # n n nq nk  1 ln2q ;...; ’  Sq . n1    nk q n n

(6)

2q

Hence the above generalized correspondence (4) is found to recover (6) when m ¼ 2  q and n ¼ 1. As will be shown later, " # " # n n ¼ . (7) n1    nk n1    nk ðm;1Þ

m

The ðm; n; qÞ-relation (5) among three parameters m; n; q yields the additive duality ‘‘q22  q’’ when n ¼ 1 and the multiplicative duality ‘‘q21=q’’ when n ¼ q, respectively. As other special cases of the ðm; n; qÞrelation, when n ¼ 2  q, it is shown that the q-triplet ðqsen ; qrel ; qstat Þ recently conjectured by Tsallis [17] is

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identified with the ðm; n; qÞ-relation (5) in the following sense: m¼

1 ; qsen

n¼

1 ; qrel

q ¼ qstat .

(8)

Moreover, when n ¼ 1=q, the relation [18]: 1 1 1 ¼  1  qsen amin amax

(9)

in the multifractal singularity spectrum f ðaÞ is recovered by means of the ðm; n; qÞ-relation in the following sense: m ¼ qsen ;

n¼

1 ; amax

q ¼ amax

(10)

with amax  amin ¼ 1. The above new results are derived in detail in the following sections. This paper consists of the five sections including this introduction. In the next section, we briefly review the fundamental formulas such as the q-product, the q-factorial, the q-multinomial coefficient and q-Stirling’s formula which are applied to the derivation of (2). In Section 3, the correspondence (2) is modified to derive the multiplicative duality ‘‘q21=q’’ in our combinatorial formalism. In this derivation, a slight generalization of the q-factorial is required, which is called ‘‘ðm; nÞ-factorial’’. Similar to Section 2, we formulate the ðm; nÞmultinomial coefficient and ðm; nÞ-Stirling’s formula based on the ðm; nÞ-factorial and apply them to find the generalized correspondence (4). In Section 4, we derive the additive duality and the multiplicative duality as special cases of (4). Moreover, when n ¼ 2  q and n ¼ 1=q, each interpretation of the ðm; n; qÞ-relation shown in (8) and (10) is, respectively, presented. The final section is devoted to our conclusion.

2. Additive duality derived from the q-multinomial coefficient The MaxEnt for Boltzmann–Gibbs–Shannon entropy S1 yields the exponential function expðxÞ which is well known to be characterized by the linear differential function dy=dx ¼ y. In parallel with this, the MaxEnt for Tsallis entropy S q yields a generalization of the exponential function expq ðxÞ [7,19,20] which is characterized by the nonlinear differential function dy=dx ¼ yq [21,22]. In Tsallis statistics, the fundamental functions are the q-logarithm lnq x and the q-exponential expq ðxÞ, respectively, defined as follows: Definition 1 (q-logarithm, q-exponential). The q-logarithm lnq x : Rþ ! R and the q-exponential expq ðxÞ : R ! R are defined by lnq x:¼

x1q  1 , 1q (

expq ðxÞ:¼

(11)

½1 þ ð1  qÞx1=ð1qÞ

if 1 þ ð1  qÞx40;

0

otherwise:

(12)

Then a new product q to satisfy the following identities as the q-exponential law is introduced: lnq ðxq yÞ ¼ lnq x þ lnq y,

(13)

expq ðxÞq expq ðyÞ ¼ expq ðx þ yÞ.

(14)

For this purpose, the new multiplication operation q is introduced in Refs. [11,12]. The concrete forms of the q-logarithm and q-exponential are given in (11) and (12), so that the above requirement (13) or (14) as the q-exponential law leads to the definition of q between two positive numbers.

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Definition 2 (q-product). For x; y 2 Rþ , the q-product q is defined by ( ½x1q þ y1q  11=ð1qÞ if x40; y40; x1q þ y1q  140; xq y:¼ 0 otherwise:

(15)

The q-product recovers the usual product such that limq!1 ðxq yÞ ¼ xy. The fundamental properties of the q-product q are almost the same as the usual product, but aðxq yÞaðaxÞq y ða; x; y 2 RÞ.

(16)

The other properties of the q-product are available in Refs. [11,12]. By means of the q-product (15), the q-factorial is naturally defined in the following form [8]. Definition 3 (q-factorial). For a natural number n 2 N and q 2 Rþ , the q-factorial n!q is defined by n!q :¼1q    q n.

(17)

Thus, we concretely compute q-Stirling’s formula. Theorem 4 (q-Stirling’s formula). Let n!q be the q-factorial defined by (17). The rough q-Stirling’s formula lnq ðn!q Þ is computed as follows: 8 < n lnq n  n þ Oðlnq nÞ if qa2; 2q lnq ðn!q Þ ¼ (18) : n  ln n þ Oð1Þ if q ¼ 2: The above rough q-Stirling’s formula is obtained by the approximation: Z n n X lnq k ’ lnq x dx. lnq ðn!q Þ ¼

(19)

1

k¼1

The rigorous derivation of q-Stirling’s formula is given in Ref. [8]. Similarly as for the q-product, q-ratio is introduced from the requirements: lnq ðxq yÞ ¼ lnq x  lnq y,

(20)

expq ðxÞq expq ðyÞ ¼ expq ðx  yÞ.

(21)

Then we define the q-ratio as follows. Definition 5 (q-ratio). For x; y 2 Rþ , the inverse operation to the q-product is defined by ( ½x1q  y1q þ 11=ð1qÞ if x40; y40; x1q  y1q þ 140; xq y:¼ 0 otherwise

(22)

which is called q-ratio in Ref. [12]. The q-product, q-factorial and q-ratio are applied to the definition of the q-multinomial coefficient [8]. P Definition 6 (q-multinomial coefficient). For n ¼ ki¼1 ni and ni 2 Nði ¼ 1; . . . ; kÞ, the q-multinomial coefficient is defined by " # n :¼ðn!q Þq ½ðn1 !q Þq    q ðnk !q Þ. (23) n1    n k q

From the definition (23), it is clear that " # " # n n n! . ¼ ¼ lim n1    nk nk q!1 n1    n1 !    nk ! q

(24)

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Throughout the present paper, we consider the q-logarithm of the q-multinomial coefficient to be given by " lnq

n1

n 

# nk

¼ lnq ðn!q Þ  lnq ðn1 !q Þ     lnq ðnk !q Þ.

(25)

q

Based on these fundamental formulas, we obtain the one-to-one correspondence (2) between the qmultinomial coefficient and Tsallis entropy as follows [8]. Theorem 7. When n 2 N is sufficiently large, the q-logarithm of the q-multinomial coefficient coincides with Tsallis entropy (1) in the following correspondence: " lnq

#

n n1



nk

q

8 2q n n nk  1 > > > < 2  q  S 2q n ; . . . ; n ’ k P > > > : S1 ðnÞ þ S1 ðni Þ

if q40; qa2; (26) if q ¼ 2;

i¼1

where S q is Tsallis entropy (1) and S 1 ðnÞ:¼ ln n. Straightforward computation of the left side of (26) by means of q-Stirling’s formula (18) yields the above result (26). (See Ref. [8] for the proof.) Clearly the additive duality ‘‘q22  q’’ is appeared in the above one-to-one correspondence (26). In the following sections these fundamental formulas are generalized for the derivation of the multiplicative duality ‘‘q21=q’’ in the similar correspondence as (26). 3. One-to-one correspondence between the ðm; nÞ-multinomial coefficient and Tsallis entropy In this section the correspondence (26) is generalized for the multiplicative duality ‘‘q21=q’’ . For this purpose, replace q in (26) by 1=q at first. Then we obtain " # n n n21=q nk  1 ln1=q  S21=q ;...; ’ , (27) n1    nk 2  1=q n n 1=q

where we consider the case q40 and qa12 only. The left side of (26) is computed as " # n ln1=q ¼ ln1=q ðn!1=q Þ  ln1=q ðn1 !1=q Þ     ln1=q ðnk !1=q Þ. n1    nk

(28)

1=q

Using this formula (28), we will represent the left side of (27) by means of the forms such as lnq or ln2q to find the multiplicative duality. The important relation for this purpose is the following identity: 1 ln1=q ð q Þ ¼ q lnq x. x Each term ln1=q ðn!1=q Þ on the right side of (28) is equal to ! 1 ln1=q ðn!1=q Þ ¼ ln1=q . ðn!1=q Þ1

(29)

(30)

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ðn!1=q Þ1 is expanded in accordance with the definition of the q-product (15): ðn!1=q Þ1 ¼ ð11=q    1=q nÞ1 ¼ ½111=q þ 211=q þ    þ n11=q  ðn  1Þ1=ð11=qÞ 2 3q=ð1qÞ  1=q !1q  1=q !1q  1=q !1q 1 1 1 þ þ  þ  ðn  1Þ5 ¼4 1 2 n "   1=q  1=q #q 1 1=q 1 1 ¼ q  q    q . 1 2 n Then ln1=q ðn!1=q Þ is given by 0

ð31Þ ð32Þ

ð33Þ

1

B ln1=q ðn!1=q Þ ¼ ln1=q @h 1=q 1

11=q

1

C 11=q iq A ð_ð30Þ and ð33ÞÞ

q 2  q    q n "   1=q  1=q # 1 1=q 1 1 ¼  q lnq q  q    q ð_ð29ÞÞ 1 2 n

ð34Þ

1

¼ q

n X

lnq j 1=q .

ð35Þ ð36Þ

j¼1

Thus, substitution of (36) to (28) yields " # ! nk n1 n X X X n 1=q 1=q 1=q ln1=q ¼q  lnq j þ lnq j 1 þ  þ lnq j k . n1    nk j¼1 j ¼1 j ¼1 1=q

(37)

k

1

Applying the general formula: lnq j 1=q ¼ ln2q j 1=q

(38)

to the above result (37), we have " # nk n1 n n X X X 1 1=q 1=q ln1=q ¼ ln2q j 1=q  ln2q j 1      ln2q j k q n1    nk 1=q j¼1 j 1 ¼1 j k ¼1

ð39Þ

¼ ln2q ½ð11=q 2q 21=q 2q    2q n1=q Þ 1=q

2q ð11=q 2q 21=q 2q    2q n1 Þ  1=q

2q ð11=q 2q 21=q 2q    2q nk Þ.

ð40Þ

On the other hand, from (25) the q-logarithm of the q-multinomial coefficient is given by " # nk n1 n n X X X lnq ¼ lnq j  lnq j 1      lnq j k n1    nk q j¼1 j 1 ¼1 j k ¼1 ¼ lnq ½ð1q 2q    q nÞ q ð1q 2q    q n1 Þ  q ð1q 2q    q nk Þ.

ð41Þ

Comparing the argument of ln2q on the right side of (40) with that of lnq on (41), a generalization of the q-factorial (17) is found to be required for our purpose.

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Definition 8 (ðm; nÞ-factorial). For a natural number n 2 N and m; n 2 R, the ðm; nÞ-factorial n!ðm;nÞ is defined by n!ðm;nÞ :¼1n m 2n m    m nn ,

(42)

where na0, Clearly when m ¼ q; n ¼ 1 the q-factorial (17) is recovered: n!q ¼ n!ðq;1Þ .

(43)

Moreover, when m ¼ 1, the ðm; nÞ-factorial n!ðm;nÞ is equal to ðn!Þn because the m-product recovers the usual product: n!ð1;nÞ ¼ 1n 2n    nn ¼ ðn!Þn .

(44)

Thus, throughout the paper we consider the case ma1 only. Using the ðm; nÞ-factorial, we have 11=q 2q 21=q 2q    2q n1=q ¼ n!ð2q;1=qÞ ,

(45)

so that (40) is rewritten by means of the ðm; nÞ-factorial. " # n 1 ln1=q ¼ ln2q ðn!ð2q;1=qÞ 2q n1 !ð2q;1=qÞ    2q nk !ð2q;1=qÞ Þ. n1    nk q

(46)

1=q

Then we define the form of the argument of ln2q on the right side of (46) by the ðm; nÞ-multinomial coefficient as a generalization of the q-multinomial coefficient (23). P Definition 9 (ðm; nÞ-multinomial coefficient). For n ¼ ki¼1 ni and ni 2 Nði ¼ 1; . . . ; kÞ, the ðm; nÞ-multinomial coefficient is defined by " # n :¼ðn!ðm;nÞ Þm ½ðn1 !ðm;nÞ Þm    m ðnk !ðm;nÞ Þ, (47) n1    nk ðm;nÞ

where n!ðm;nÞ is the ðm; nÞ-factorial defined in (42). Clearly when m ¼ q and n ¼ 1 the q-multinomial coefficient (23) is recovered. " # " # n n ¼ . n1    nk n1    nk q

(48)

ðq;1Þ

Using the ðm; nÞ-multinomial coefficient, (46) becomes " # " # n n 1 ln1=q ¼ ln2q n1    nk n1    nk q ð1=q;1Þ

.

(49)

ð2q;1=qÞ

Moreover, ðm; nÞ-Stirling’s formula is computed as the following form: Theorem 10 (ðm; nÞ-Stirling’s formula). Let n!ðm;nÞ be the ðm; nÞ-factorial defined by (42). ðm; nÞ-Stirling’s formula lnm ðn!ðm;nÞ Þ is computed as follows: 8 n < n lnm n  nn þ Oðlnm nÞ if nð1  mÞ þ 1a0; lnm ðn!ðm;nÞ Þ ¼ nð1  mÞ þ 1 (50) : nðn  ln nÞ þ Oð1Þ if nð1  mÞ þ 1 ¼ 0: This formula is computed by the approximation: Z n n X n lnm k ’ lnm xn dx. lnq ðn!ðm;nÞ Þ ¼ k¼1

1

(51)

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Based on these results, we obtain the one-to-one correspondence between the ðm; nÞ-multinomial coefficient and Tsallis entropy as follows. Theorem 11. When n is sufficiently large, the m-logarithm of the ðm; nÞ-multinomial coefficient coincides with Tsallis entropy (1) as follows: 8 q n n nk  1 > > " #  S ; . . . ; if qa0; q > n1    nk n > ðnÞ þ S ðn Þ if q ¼ 0; S > ðm;nÞ 1 1 i : i¼1

where na0, nð1  mÞ þ 1 ¼ q,

(53)

Sq is Tsallis entropy (1) and S 1 ðnÞ:¼ ln n. The proof is given in Appendix A. The generalized correspondence (52) between the ðm; nÞ-multinomial coefficient and Tsallis entropy includes some typical mathematical structures such as the two kinds of dualities and the q-triplet as the special cases, shown in the next section. 4. Multiplicative duality and other typical mathematical structures derived from the combinatorial formalism We consider the case qa0 only. Then, the generalized correspondence (52) is given by " # n n 1 nq nk  1 lnm ;...; ’  Sq . n1    nk n q n n

(54)

ðm;nÞ

In this paper, we call the above relation (53) ‘‘ðm; n; qÞ-relation’’ which provides us interesting features in Tsallis statistics. In particular, we consider the following four cases. ðAÞ n ¼ 1. In this case, from the ðm; n; qÞ-relation (53) m is given by m ¼ 2  q.

(55)

Then the generalized correspondence (54) becomes " # n n nq nk  1 ;...; ’  Sq ln2q n1    nk q n n

(56)

2q

which is equivalent to (6) or (26) representing the additive duality ‘‘q22  q’’. (B) n ¼ q. In this case, from the ðm; n; qÞ-relation (53) m is determined as 1 m¼ . q

(57)

Then the generalized correspondence (54) becomes " # n n nk  1 ln1=q ;...; ’ nq  S q n1    nk n n

(58)

ð1=q;qÞ

which represents the multiplicative duality ‘‘q21=q’’ . Aside from the above representation (58), the multiplicative duality in Tsallis statistics is represented in the form expq ðS q ðpi ÞÞ ¼ exp1=q ðS1=q ðPi ÞÞ

(59)

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for any given probability distribution fpi g and its associated escort distribution fPi g. This simple expression (59) has been unknown until now and provides us with a hint to unify some generalized entropies. See Appendix B for the detail. (C) n ¼ 2  q. In this case, from the ðm; n; qÞ-relation (53) m is obtained as m¼

3  2q . 2q

(60)

Then the generalized correspondence (54) becomes " # n n 1 nq nk  1 lnð32qÞ=ð2qÞ ;...; , ’  Sq n1    nk 2q q n n

(61)

ðð32qÞ=ð2qÞ;2qÞ

where the ðm; n; qÞ-relation for this case is equivalent to the q-triplet ðqsen ; qrel ; qstat Þ recently conjectured by Tsallis [17,24] in the following sense. In [17], Tsallis first conjectured the three entropic q-indices ðqsen ; qrel ; qstat Þ, respectively, for q-exponential sensitivity to the initial conditions, q-exponential relaxation of macroscopic quantities to thermal equilibrium and q-exponential distribution describing a stationary state. More concretely, based on his recent results in Ref. [23] he conjectured the concrete q-triplet ðqsen ; qrel ; qstat Þ satisfying the following relation [24]: qrel þ

1 ¼ 2; qsen

qstat þ

1 ¼ 2. qrel

(62)

From this relation, we immediately obtain 1 qsen

¼

3  2qstat 2  qstat

(63)

which is the same form as m obtained in (60). Therefore, when n ¼ 2  q, the present ðm; n; qÞ-relation is identified with the q-triplet ðqsen ; qrel ; qstat Þ in the following sense: m¼

1 ; qsen

n¼

1 ; qrel

q ¼ qstat .

(64)

As shown in this paper, the above identification (64) is derived from the mathematical discussion only. Besides our analytical derivation, the q-triplet ðqsen ; qrel ; qstat Þ has been already confirmed in the experimental observations in Ref. [25]. Therefore, Tsallis’ conjecture on the q-triplet ðqsen ; qrel ; qstat Þ in Ref. [24] is correct in both theoretical and experimental aspects. Note that in our theoretical derivation of (64) we never use the definition of the three entropic q-indices qsen ; qrel ; qstat , which may be remained as a future work from the theoretical points of view in this case n ¼ 2  q. However, the present identification (64) is just an interpretation of the ðm; n; qÞ-relation. In fact, in our formulation the additive duality, the multiplicative duality and the q-triplet are derived as special cases of the ðm; n; qÞ-relation (53). For example, we present other possible interpretation of the present ðm; n; qÞ-relation (53) for the case n ¼ 1=q, shown in the next subsection. (D) n ¼ 1=q. As an other special case of the ðm; n; qÞ-relation, we consider the case n ¼ 1=q. For this case, we obtain 1 1 1 ¼  . 1m q1 q

(65)

This identity reminds us of the following relationship [18]: 1 1 1 ¼  , 1  qsen amin amax

(66)

where qsen is the same entropic q-index as the case (C) for the q-exponential sensitivity to the initial conditions, amin and amax are the values of a at which the multifractal singularity spectrum f ðaÞ vanishes (with amin oamax ).

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These amin and amax are given by amin ¼

ln b ; z ln aF

amax ¼

ln b , ln aF

(67)

where b stands for a natural scale for the partitions, aF is the Feigenbaum universal scaling factor and z represents the nonlinearity of the map at the vicinity of its extremal point [18]. A choice of the nonlinearity z to satisfy  z b ¼b (68) aF implies amax  amin ¼ 1. In other words, (68) means a rescaling of amax  amin to be 1. Therefore, if the nonlinearity z is determined by the above requirement (68), we have the following identification: m ¼ qsen ;

n¼

1 ; amax

q ¼ amax

(69)

which we call ‘‘the multifractal triplet’’ as one of the interpretations of the ðm; n; qÞ-relation. Note that the above identification (69) implies (68). All results in cases A–D mean that the ðm; n; qÞ-relation is found to be a more general nature in Tsallis statistics to recover these specific mathematical structures.

5. Conclusion We present the one-to-one correspondence between the ðm; nÞ-multinomial coefficient and Tsallis entropy S q to represent both the additive duality ‘‘q22  q’’ and the multiplicative duality ‘‘q21=q’’ in one unified formula (54). In this derivation, ðm; nÞ-factorial, ðm; nÞ-multinomial coefficient and ðm; nÞ-Stirling’s formula are concretely formulated as a generalization of q-factorial, q-multinomial coefficient and q-Stirling’s formula, respectively. In the present one-to-one correspondence (54), when n ¼ 2  q, the ðm; n; qÞ-relation among three parameters m; n; q is shown to be identified with the q-triplet ðqsen ; qrel ; qstat Þ in the sense of (64). In addition, as other interpretation of the ðm; n; qÞ-relation, the multifractal structure 1=ð1  qsen Þ ¼ 1=amin  1=amax is recovered.

Acknowledgements The first author is grateful to Jan Naudts for a short discussion in Trieste conference 2006, which inspires the author to find the ideas in this paper. The authors acknowledge the partial support given by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), 18300003, 2006.

Appendix A. Proof of Theorem 11 When m ¼ 1, m-product recovers the usual product regardless of n. Thus, we consider the case ma1 only. If nð1  mÞ þ 1a0, " # n lnm ¼ lnm n!ðm;nÞ  lnm n1 !ðm;nÞ      lnm nk !ðm;nÞ ðA:1Þ n1    nk ðm;nÞ ’

nk lnm nnk  nnk n lnm nn  nn n1 lnm nn1  nn1     nð1  mÞ þ 1 nð1  mÞ þ 1 nð1  mÞ þ 1

ð_ð50ÞÞ

ðA:2Þ

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1 ðn lnm nn  n1 lnm nn1      nk lnm nnk Þ ¼ nð1  mÞ þ 1 ! k  nð1mÞþ1 X nnð1mÞþ1 ni 1 ¼ ð1  mÞðnð1  mÞ þ 1Þ n i¼1  nð1mÞþ1 n n1 nk  S nð1mÞþ1 ;...; ¼n nð1  mÞ þ 1 n n

_n ¼

k X

If nð1  mÞ þ 1 ¼ 0, " # n lnm ¼ lnm n!ðm;nÞ  lnm n1 !ðm;nÞ      lnm nk !ðm;nÞ n1    nk ðm;nÞ ’ nðn  ln nÞ  nðn1  ln n1 Þ      nðnk  ln nk Þ ! k X _n ¼ ni ¼ nð ln n þ ln n1    þ ln nk Þ ¼ n S1 ðnÞ þ

k X

!

! ni

ðA:3Þ

i¼1

ðA:4Þ ðA:5Þ

ðA:6Þ ð_ð50ÞÞ

ðA:7Þ ðA:8Þ

i¼1

S1 ðni Þ .

ðA:9Þ

i¼1

Appendix B. The multiplicative duality derived from the definition of the escort distribution The multiplicative duality in Tsallis statistics has been presented in several Refs. [7,13–16]. In this appendix, the multiplicative duality is derived in the simplest form from the definition of the escort distribution, which fact has been unknown in the studies of Tsallis statistics until now. The present result (B.3) does not only represent the multiplicative duality but also provides us with a hint to unify several generalized entropies. In the second half of this appendix, it is shown that Sharma–Mittal entropy [27] and Supra–extensive entropy [28] are, respectively, expressed by means of Tsallis entropy and Re´nyi entropy only in the sense of (B.3), which results reveal that Tsallis entropy and Re´nyi entropy are more fundamental than Sharma–Mittal entropy and Supra-extensive entropy. The escort distribution was first introduced in Ref. [26] to scan the structure of a given distribution by using similarities as thermodynamic equilibrium distribution. Definition 12 (escort distribution). For any given probability distribution fpi g, the escort distribution fPi g is defined as pq Pi :¼ Pn i

q j¼1 pj

ðq40Þ.

(B.1)

Note that a given distribution fpi g in the above definition is not necessarily normalized, but in our formulations we require fpi g to be a normalized distribution, that is, a probability distribution. Until now, the multiplicative duality ‘‘q21=q’’ in Tsallis statistics is based on the following property derived from the above definition of the escort distribution: 1=q

P pi ¼ Pn i

1=q j¼1 Pj

.

(B.2)

However, the escort distribution fPi g is originally associated with Tsallis entropy in the following sense: Theorem 13. For any given probability distribution fpi g and its associated escort distribution fPi g, the next identity is satisfied: expq ðS q ðpi ÞÞ ¼ exp1=q ðS1=q ðPi ÞÞ, where S q ðpi Þ is Tsallis entropy (1).

(B.3)

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Proof. Using the definition of the escort distribution (B.1), we have !1=q !1=q n n n n X X X X pqi pi 1=q q Pn q Pi ¼ ¼ pj . Pn q 1=q ¼ j¼1 pj i¼1 i¼1 i¼1 ð j¼1 pj Þ j¼1 Both sides to the power 1=ð1  1=qÞ ¼ q=ð1  qÞ is !1=ð11=qÞ !1=ð1qÞ n n X X 1=q q Pj ¼ pi j¼1

(B.4)

(B.5)

i¼1

which is equivalent to Pk 1=q !1=ð11=qÞ   1 1  j¼1 Pj 1þ 1 ¼ q 1=q  1

!1=ð1qÞ P 1  ki¼1 pqi 1 þ ð1  qÞ . q1

(B.6)

Clearly, this is identical to the simple form exp1=q ðS 1=q ðPj ÞÞ ¼ expq ðSq ðpi ÞÞ:

(B.7)

&

Note that the above result (B.3) obviously represents the multiplicative duality ‘‘q21=q’’ of Tsallis entropy, which is derived from the definition of the escort distribution only. The above relation (B.3) provides us a key to unify several entropies such as Boltzmann–Gibbs–Shannon entropy, Re´nyi entropy [29], Tsallis entropy [1], Gaussian entropy [30], Sharma–Mittal entropy [27] and Supra-extensive entropy [28]. Among these entropies, Sharma–Mittal entropy and Supra-extensive entropy are the two-parameterized entropies including the other entropies as special cases. For these two entropies, the similar identities as (B.3) are satisfied in the forms expr ðSSharma2Mittal Þ ¼ expq ðS Tsallis Þ, q;r q

ðB:8Þ

expq ðS Supraextensive Þ ¼ expr ðSRenyi Þ, q;r q

ðB:9Þ

where ð ðpi Þ:¼ SSharma2Mittal q;r SSupraextensive ðpi Þ:¼ q;r

Pn

q ð1rÞ=ð1qÞ i¼1 pi Þ

1

, 1r P 1r ð1 þ 1q log ni¼1 pqi Þð1qÞ=ð1rÞ  1 1q

ðB:10Þ .

ðB:11Þ

The above identities (B.8) and (B.9) are, respectively, simple mathematical modifications of (2.9) and (2.10) in Ref. [28], but they reveal that Tsallis entropy and Re´nyi entropy are more fundamental than Sharma–Mittal entropy and Supra-extensive entropy. For example, as seen in the above identity (B.8), there is a clear one-toone correspondence between Sharma–Mittal entropy and Tsallis entropy. References [1] [2] [3] [4] [5] [6] [7] [8]

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