Mathematical structure derived from the q-multinomial coefficient in Tsallis statistics

0.25 MB·English

by  Hiroki Suyari

Mathematical structure
derived from the q-multinomial coefficient in Tsallis statistics
4
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0 Hiroki Suyari∗
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Department of Information and Image Sciences,
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J Faculty of Engineering, Chiba University, 263-8522, Japan

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2 (Dated: February 6, 2008)

] Abstract
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e We present the conclusive mathematical structure behind Tsallis statistics. We obtain mainly
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- thefollowing five theoretical results: (i) theone-to-one correspondencebetween theq−multinomial
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t coefficient and Tsallis entropy, (ii) symmetry behindTsallis statistics, (iii) the numerical computa-
s
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t tionsrevealingtheexistenceofthecentrallimittheoreminTsallisstatistics, (iv)Pascal’strianglein
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Tsallis statistics andits properties, (v)the self-similarity of the q−product⊗ leading to successful
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n applications in Tsallis statistics. In particular, the third result (iii) provides us with a mathemat-
o
c ical representation of a convincible answer to the physical problem: “Why so many power-law
[

behaviors exist in nature universally ?”
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4 PACS numbers: 02.50.-r,05.45.-a
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1 Keywords: Tsallis entropy, q−product, q−Stirling’s formula, q-multinomial coefficient, q−binomial coeffi-
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4 cient, the central limit theorem in Tsallis statistics, self-similarity of the q−product, Pascal’s triangle in
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t Tsallis statistics
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∗
Electronic address: [email protected],[email protected]
1
I. INTRODUCTION
Tsallis entropy introduced in 1988 [1][2]:
n
1− pq
i
S (p ,··· ,p ) := i=1 (1)
q 1 n q −P1
has been considered to obtain and provide new possibilities to construct the generalized sta-
tistical physics recovering not only the traditional Boltzmann-Gibbs statistical mechanics
but also the so-called Tsallis statistics to describe power-law behaviors systematically. Until
now, themaximum entropy principle(MEP forshort) hasbeenmainlyappliedtomathemat-
ical foundations in Tsallis statistics along the same lines of Jaynes’ ideas [3][4]. The MEP
for Tsallis statistics yields equilibrium states exhibiting power-law behaviors, which provide
us many satisfactory descriptions of complex systems such as self-gravitating systems [5][6],
pure electron plasma [7][8], full developed turbulence [9][10], high-energy collisions [11][12]
and so on. Through the history of sciences, we have learned an important lesson that there
always exists a beautiful mathematical structure at the birth of a new physics. This lesson
stimulates us to finding it behind Tsallis statistics. On the way to the goal, we obtain some
theoretical results coming from the mathematical structure in Tsallis statistics. The key
concept leading to our results is “the q−product” uniquely determined by Tsallis entropy,
which is first introduced by Borges in [13].
The q−product determined by Tsallis entropy has been successfully applied to the deriva-
tions of “the law of error [14]” and “the q−Stirling’s formula[15]” in Tsallis statistics. The
q−product is defined as follows:
1
[x1−q +y1−q −1]1−q , if x > 0, y > 0, x1−q +y1−q −1 > 0,
x⊗ y := (2)
q

0, otherwise.

The definition of the q−product originates from the requirement of the following satisfac-

tions:
ln (x⊗ y) = ln x+ln y, (3)
q q q q
exp (x)⊗ exp (y) = exp (x+y), (4)
q q q q
where ln x and exp (x) are the q−logarithm function:
q q
x1−q −1
ln x := x > 0,q ∈ R+ (5)
q
1−q
(cid:0) (cid:1)
2
and its inverse function, the q−exponential function:
1
[1+(1−q)x]1−q if 1+(1−q)x > 0,
exp (x) := x ∈ R,q ∈ R+ . (6)
q 
0 otherwise

(cid:0) (cid:1)
These functions, lnx and exp (x), are originally determined by Tsallis entropy and its
q q
maximization [3][16]. As shown in our previous paper [15], exp (x) is rewritten by means
q
of the q−product.
x ⊗n
q
exp (x) = lim 1+ (7)
q n→∞ n
(cid:16) (cid:17)
This representation (7) is a natural generalization of the famous definition of the usual
exponential function: exp(x) = lim 1+ x n. This fundamental property (7) reveals the
n→∞ n
conclusive validity of the q−product (cid:0)in Tsa(cid:1)llis statistics. See the appendix of [15] for the
proof.
Besides the fundamental property (7), until now we have presented two successful appli-
cations of the q−product in Tsallis statistics: “the law of error[14]” and “the q−Stirling’s
formula[15]”.
In the former application [14], the q−product is applied to the likelihood function L (θ)
q
in maximum likelihood principle (MLP for short) instead of the usual product.
L (θ) = L (x ,x ,··· ,x ;θ) := f (x −θ)⊗ f (x −θ)⊗ ···⊗ f (x −θ) (8)
q q 1 2 n 1 q 2 q q n
where x ,x ,··· ,x ∈ R are results of n measurements for some observation. The require-
1 2 n
ment for taking the maximal value of L (θ) at
q
x +x +···+x
θ = θ∗ := 1 2 n (9)
n
uniquely determines a Tsallis distribution:
exp (−β e2)
q q
f (e) = (β > 0) (10)
q
exp (−β e2)de
q q
as a nonextensive generalization of Ra Gaussian distribution.
In the latter application [15], using the q−product, we obtain the q−Stirling’s formula.
For the q−factorial n! for n ∈ N and q > 0 defined by
q
n! := 1⊗ ···⊗ n, (11)
q q q
3
the q−Stirling’s formula (q 6= 1) is
n + 1 n1−q−1 + − n + 1 −δ if q > 0 and q 6= 1,2
ln (n! ) ≃ 2−q 2 1−q 2−q 2−q q (12)
q q
(cid:16)n− 1 −(cid:17)lnn− 1 −(cid:16)δ (cid:17) (cid:16) (cid:17) if q = 2
 2n 2 2
where δ isa q−dependent parameter which does not depend onn. Slightlyroughexpression
q
of the q−Stirling’s formula (q 6= 1) is
n (ln n−1) if q > 0 and q 6= 1,2
ln (n! ) ≃ 2−q q . (13)
q q

n−lnn if q = 2

On the other hand, a more strict expression of the q−Stirling’s formula (q 6= 1) are
n n 1 1 1
ln n− + ln n+ − < ln (n! )
q q q q
2−q 2−q 2 2−q 8
n n 1 1
< ln n− + ln n+ (if q > 0 and q 6= 1,2) (14)
q q
2−q 2−q 2 2−q
and
1 5 1 1
n−lnn− − < ln (n! ) < n−lnn− − (if q = 2). (15)
q q
2n 8 2n 2
These q−Stirling’s formulas recover the famous Stirling’s formulas when q → 1.
Along the lines of these successful applications of the q−product, we introduce the
q−multinomial coefficient using the q−product in section II. The present q−multinomial
coefficient and the Tsallis entropy has a surprising relationship between them, which indi-
cates a symmetry behind Tsallis statistics. Moreover, the numerical computations of the
q−binomial coefficients are shown in section III. When n goes infinity, a set of the nor-
malized q−binomial coefficients converge to its corresponding Tsallis distribution with the
same q. These numerical results indicate the existence of the central limit theorem in Tsallis
statistics as a nonextensive generalization of that of the usual probability theory. In section
IV, we reveal the reason why the q-product can be successfully applied to Tsallis statistics.
The self-similarity of the q−product plays essential roles for these successful applications in
Tsallis statistics. In the appendix, Pascal’s triangle in Tsallis statistics and its properties
are presented using the present q−binomial coefficient.
4
II. q−MULTINOMIAL COEFFICIENTIN TSALLIS STATISTICS AND ITS ONE-
TO-ONE CORRESPONDENCE TO TSALLIS ENTROPY
n
We define the q−binomial coefficient by
 k 
q
 
n
:= (n! )⊘ [(k! )⊗ ((n−k)! )] (n,k(≤ n) ∈ N) (16)
q q q q q
 k 
q
 
where ⊘ is the inverse operation to ⊗ , which is defined by
q q
1
[x1−q −y1−q +1]1−q , if x > 0, y > 0, x1−q −y1−q +1 > 0,
x⊘ y := . (17)
q

0, otherwise

⊘ is also introduced by the following satisfactions as similarly as ⊗ [13].
q  q
ln x⊘ y = ln x−ln y, (18)
q q q q
exp (x)⊘ exp (y) = exp (x−y). (19)
q q q q
Applying the definitions of ⊗ , ⊘ and n! given by
q q q
1
n 1−q
n! = 1⊗ ···⊗ n = k1−q −(n−1) (20)
q q q
" #
k=1
X
to (16), the q−binomial coefficient is explicitly written as
1
n n k n−k 1−q
= ℓ1−q − i1−q − j1−q +1 . (21)
 k  " #
ℓ=1 i=1 j=1
q X X X
 
In general, when q < 1, n ℓ1−q − k i1−q − n−kj1−q + 1 > 0. Thus, we can plot
ℓ=1 i=1 j=1
the probability distributioPn determinePd by the norPmalized q−binomial coefficients in case
0 < q < 1. (See Fig.1). From the definition (16), it is clear that
n n n!
lim = = . (22)
q→1 k   k  k!(n−k)!
q
   
n
On the other hand, when n ℓ1−q− k i1−q − n−kj1−q+1 < 0, takes complex
ℓ=1 i=1 j=1  k 
P P P q
numbers in general, which divides the formulations and discussions oftheq−binomial coef-
n
ficient into two cases: it takes a real number or a complex number. In order to avoid
 k 
q
 
5
such separate formulations and discussions, we consider the q-logarithm of the q−binomial
coefficient:
n
ln = ln (n! )−ln (k! )−ln ((n−k)! ). (23)
q q q q q q q
 k 
q
 
For simplicity, we consider the only case q 6= 1 and q > 0 throughout the paper.
00..003355
ntnt qq == 00..99 qq==00..11
ee
cici qq==00..55
fifi 00..0033
ff qq==00..88
ee
oo
cc qq==00..99
mial mial 00..002255 qq == 00..88
oo
nn
bibi
e q-e q- 00..0022 qq == 00..55
hh
y ty t
ed bed b 00..001155 qq == 00..11
nn
mimi
erer 00..0011
etet
dd
y y
bilitbilit 00..000055
aa
bb
oo
rr
pp 00
00 2200 4400 6600 8800 110000
kk nn == 110000
(( ))
FIG. 1: probability distribution determined by the normalized q−binomial coefficients
The above definition (16) is artificial because it is defined fromthe analogywith the usual
n
binomial coefficient . However, when n goes infinity, the q−binomial coefficient (16)
 k 
has a surprising relationto Tsallis entropy as follows:
n n2−q ·S k, n−k if q > 0, q 6= 2
ln ≃ 2−q 2−q n n , (24)
q
 k   −S (n)+S(cid:0) (k)+(cid:1)S (n−k) if q = 2
q  1 1 1
 
where S is Tsallis entropy(1) and S (n) is Boltzmann entropy:
q 1
S (n) := lnn. (25)
1
6
Applying the rough expression of the q−Stirling’s formula (13), the above relations (24)
are easily proved as follows:
if q > 0, q 6= 2,
n n k n−k
ln ≃ (ln n−1)− (ln k −1)− (ln (n−k)−1)
q q q q
 k  2−q 2−q 2−q
q
  n k n−k
= ln n− ln k − ln (n−k)
q q q
2−q 2−q 2−q
n2−q −k2−q −(n−k)2−q
=
(2−q)(1−q)
n2−q 1− k 2−q − n−k 2−q
= · n n
2−q (2−q)−1
(cid:0) (cid:1) (cid:0) (cid:1)
n2−q k n−k
= ·S , , (26)
2−q 2−q n n
(cid:18) (cid:19)
if q = 2,
n
ln ≃ (n−lnn)−(k −lnk)−((n−k)−ln(n−k))
q
 k 
q
 
= −lnn+lnk +ln(n−k)
= −S (n)+S (k)+S (n−k). (27)
1 1 1
The above correspondence (24) between the q−binomial coefficient (16) and Tsallis en-
tropy convinces us of the fact the q−binomial coefficient (16) is well-defined in Tsallis statis-
tics. Therefore, we can construct Pascal’s triangle in Tsallis statistics. See the appendix for
the detail.
The above relation (24) is easily generalized to the case of the q−multinomial coefficient.
The q−multinomial coefficient in Tsallis statistics is defined in a similar way as that of the
the q−binomial coefficient (16).
n
:= (n! )⊘ [(n ! )⊗ ···⊗ (n ! )] (28)
q q 1 q q q k q
n ··· n 
1 k
q
 
where
k
n = n , n ∈ N (i = 1,··· ,k). (29)
i i
i=1
X
7
Applying the definitions of ⊗ and ⊘ to (28), the q−multinomial coefficient is explicitly
q q
written as
1
n n n1 nk 1−q
= ℓ1−q − i1−q···− i1−q +1 . (30)
n ··· n  " 1 k #
1 k q Xℓ=1 iX1=1 iXk=1
 
Along the same reason as stated above in case of the q−binomial coefficient, we consider
the q-logarithm of the q−multinomial coefficient given by
n
ln = ln (n! )−ln (n ! )···−ln (n ! ). (31)
q q q q 1 q q k q
 n ··· n 
1 k
q
 
From the definition (28), it is clear that
n n n!
lim = = . (32)
q→1 n1 ··· nk   n1 ··· nk  n1!···nk!
q
   
When n goes infinity, the q−multinomial coefficient (28) has the similar relation to Tsallis
entropy as (24):
n2−q ·S n1,··· , nk if q > 0, q 6= 2
n 2−q 2−q n n
lnq ≃  k . (33)
n ··· n  −S (n)+ (cid:0) S (n ) (cid:1) if q = 2
1 k q  1 i=1 1 i
 
P

This is a natural generalization of (24). In the same way as the case of the q−binomial
8
coefficient, the above relation (33) is proved as follows:
if q > 0, q 6= 2,
n n n n
ln ≃ (ln n−1)− 1 (ln n −1)−···− k (ln n −1)
q n ··· n  2−q q 2−q q 1 2−q q k
1 k
q
  n n n
= ln n− 1 ln n −···− k ln n
2−q q 2−q q 1 2−q q k
n2−q −n2−q −···−n2−q
= 1 k
(2−q)(1−q)
n2−q 1− n1 2−q −···− nk 2−q
= · n n
2−q (2−q)−1
(cid:0) (cid:1) (cid:0) (cid:1)
n2−q n n
= ·S 1,··· , k ,
2−q 2−q n n
(cid:16) (cid:17)
if q = 2, (34)
n
ln ≃ (n−lnn)−(n −lnn )−···−(n −lnn )
q 1 1 k k
 n ··· n 
1 k
q
 
= −lnn+lnn +···+lnn
1 k
k
= −S (n)+ S (n ). (35)
1 1 i
i=1
X
When q → 1, (33) recovers the well known result [17]:
n n n
ln ≃ nS 1,··· , k (36)
n ··· n  1 n n
1 k (cid:16) (cid:17)
 
where S is Shannon entropy.
1
The present relation (33) tells us some significant messages about Tsallis statistics. In
particular, we take the following three of them.
1. There always exists a one-to-one correspondence between Tsallis entropy and the
q−multinomial coefficient. In particular, from (33) we obtain the following equiva-
lence which makes sense in statistical physics.
n n
“Maximization of S 1,··· , k is equivalent to
2−q n n
(cid:16) (cid:17)
n
that of the q-multinomial coefficient when q < 2 and n is large. ”
n ··· n 
1 k
q
 
(37)
9
2. When q 6= 2, the relation (33) is rewritten by
n nq′ n n
ln ≃ ·S 1,··· , k (38)
q q′
 n ··· n  q′ n n
1 k q (cid:16) (cid:17)
 
where
q +q′ = 2, q > 0. (39)
In general, when Tsallis entropy S is applied, we consider the case q′ > 0 only,
q′
because Tsallis entropy when q′ ≤ 0 loses its concavity which plays important roles in
nonextensive systems. Thus,
“q′ = 2−q > 0, q > 0 ” ⇔ “0 < q < 2, 0 < q′ = 2−q < 2 ”. (40)
This is in good agreement with the fact that the parameter q′ of Tsallis entropy S
q′
takes a value in (0,2) in most of the successful physical models using Tsallis entropy.
More precisely, in case of q ∈ (0,1] (i.e., q′ ∈ [1,2)), the q-multinomial coefficient
n
takes real numbers because n ℓ1−q− n1 i1−q···− nk j1−q+1 >
n ··· n  ℓ=1 i=1 j=1
1 k
q P P P
0in (30). Onthe other hand, in case of q ∈ (1,2) (i.e., q′ ∈ (0,1)), the q-logarithm of
the q-multinomial coefficients take real numbers, but the q-multinomial coefficient does
not take real numbers in general because n ℓ1−q− n1 i1−q···− nk j1−q+1 < 0
ℓ=1 i=1 j=1
in (30). Thus, we summarize these considPerations asPfollows. P
q ∈ (0,1] (i.e., q′ ∈ [1,2))
n
⇒ the q-multinomial coefficients uniquely corresponding to
n ··· n 
1 k
q
 
Tsallis entropy S = S take real numbers as their values, (41)
q′ 2−q
q ∈ (1,2) (i.e., q′ ∈ (0,1))
n
⇒ the q-multinomial coefficients uniquely corresponding to
n ··· n 
1 k
q
 
Tsallis entropy S = S does not take real numbers as their values in general,
q′ 2−q
n
but ln take real numbers. (42)
q
 n ··· n 
1 k
q
 
10