Generalization of Shannon’s theorem for Tsallis entropyRoberto J. V. dos Santos Citation: Journal of Mathematical Physics 38, 4104 (1997); doi: 10.1063/1.532107 View online: http://dx.doi.org/10.1063/1.532107 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Upper bounds on Shannon and Rényi entropies for central potentials J. Math. Phys. 52, 022105 (2011); 10.1063/1.3549585 Projections maximizing Tsallis entropy AIP Conf. Proc. 965, 90 (2007); 10.1063/1.2828766 A rooted tree whose lower bound of average description length is given by Tsallis entropy AIP Conf. Proc. 965, 80 (2007); 10.1063/1.2828763 Information theoretical properties of Tsallis entropies J. Math. Phys. 47, 023302 (2006); 10.1063/1.2165744 Fundamental properties of Tsallis relative entropy J. Math. Phys. 45, 4868 (2004); 10.1063/1.1805729

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Generalization of Shannon’s theorem for Tsallis entropyRoberto J. V. dos SantosDepartamento de Fı´sica, Universidade Federal de Alagoas,57072-970, Maceio´, Alagoas, Brazil

~Received 7 January 1997; accepted for publication 7 March 1997!

By using the assumptions that the entropy must~i! be a continuous function of theprobabilities$pi%(piP(0,1); i ), only; ~ii ! be a monotonic increasing function ofthe number of statesW, in the case of equiprobability;~iii ! satisfy the pseudoad-ditivity relation Sq(A1B)/k5Sq(A)/k1Sq(B)/k1(12q)Sq(A)Sq(B)/k

2 ~A andB being two independent systems,qPR andk a positive constant!, and~iv! satisfythe relation Sq($pi%)5Sq(pL ,pM)1pL

qSq($pi /pL%)1pMqSq($pi /pM%), where

pL1pM51(pL5( i51WL pi and pM5( i5WL11

W pi!, we prove, along Shannon’s lines,

that the unique function that satisfies all these properties is the generalized TsallisentropySq5k(12( i51

W piq)/(q21). © 1997 American Institute of Physics.

@S0022-2488~97!03607-4#

I. INTRODUCTION

Nonextensive physical systems are being intensively studied nowadays. A recently introducedformalism, namely Tsallis statistics,1 has been proposed in order to cover many of such anomaloussystems. Indeed, it has been successfully applied to Le´vy-type2–4 and correlated-type5–8 anoma-lous diffusions, turbulence in electron plasmas,9 the solar neutrino problem,10 quantumgroups,11–13 nonlinear dynamical systems,14 cosmology,15 linear response theory,16 as well as tooptimization technics.17–23

This thermo-statistical formalism is based upon the so-called Tsallis entropy formula

Sq5k12( i51

W piq

q21~qPR!, ~1!

wherek is a positive constant~which we shall from now on take equal to 1!, q is a real number,W is the total number of microscopic configurations, and$pi% is the set of associated probabilities(( i51

W pi51). It is easily seen that in the limitq→1, one recovers the well-known Boltzmann–Gibbs–Shannon formula21

S52(i51

W

pi ln pi , ~2!

which successfully accounts for extensive problems.It is known thatSq satisfies the following conditions:

~i! Sq is, for 0,pi,1, a continuous function of$pi%, only.~ii ! For a given set ofW equiprobable states, i.e.,pi51/W, Sq is a monotonic increasing

function of W, namelySq5(W12q21)/(12q).~iii ! For two independent systemsA andB, the generalized entropy of the composed system

A1B satisfies the pseudoadditivity relation~see, for instance, Ref. 11!:

Sq~A1B!5Sq~A!1Sq~B!1~12q!Sq~A!Sq~B!. ~3!

~iv! With

0022-2488/97/38(8)/4104/4/$10.004104 J. Math. Phys. 38 (8), August 1997 © 1997 American Institute of Physics

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W5WL1WM , ~4!

pL5(i51

WL

pi ~WL terms!, ~5!

pM5 (i5WL11

W

pi ~WM terms!, ~6!

~hence, pL1pM51! ~7!

we have:1,20

Sq~$pi%!5Sq~pL ,pM !1pLqSqS H pipLJ D1pM

qSqS H pipMJ D . ~8!

In this letter we show that theuniquefunction that simultaneously satisfies all these properties isthe Tsallis generalized entropy formula~1!. By so doing, we are generalizing, for the case ofnonextensive systems, the famous Shannon’s theorem.24

II. PROOF

Let us decompose a choice fromsm equally likely possibilities into a series ofm choices withs equally likely possibilities each. It is straightforward to show that using condition~iii ! one gets:

Sq~sm!5

@11~12q!Sq~s!#m21

12q. ~9!

This expression is the generalization forqPR of the extensivity condition of the Boltzmann–Gibbs–Shannon entropy, and, in the limitq→1, yields the well-known result24

S1~sm!5mS1~s!. ~10!

For a large enoughm ands it is always possible to find a pair of integer numbers (t,n) such that

sm<tn<sm11. ~11!

Now, from condition~ii !, we have, for all values ofq:

Sq~sm!<Sq~ t

n!<Sq~sm11! ~12!

so, using Eq.~9! for q,1,

@11~12q!Sq~s!#m<@11~12q!Sq~ t !#n<@11~12q!Sq~s!#m11. ~13!

Taking the logarithm of this inequality we get

m

n<ln@11~12q!Sq~ t !#

ln@11~12q!Sq~s!#<m

n11

n~14!

or equivalently,

Umn 2ln@11~12q!Sq~ t !#

ln@11~12q!Sq~s!#U<e[

1

n. ~15!

4105Roberto J. V. dos Santos: Shannon’s theorem for Tsallis entropy

J. Math. Phys., Vol. 38, No. 8, August 1997

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Now, from Eq.~11!, we have

m

n<ln t

ln s<m

n11

n~16!

and, as before,

Umn2ln t

ln sU<e[1

n. ~17!

Combining Eqs.~15! and ~17! there comes

U ln t

ln s2ln@11~12q!Sq~ t !#

ln@11~12q!Sq~s!#U<2e. ~18!

If we now allow e→0 we get

ln@11~12q!Sq~s!#

ln s5ln@11~12q!Sq~ t !#

ln t5p~q!, ~19!

wherep(q) is a quantity which at most can depend onq.So we get the functional form ofSq(t) given by:

Sq~ t !5tp~q!21

12q. ~20!

Let us now consider a choice fromW partitions, each one with probability

pi5ni

( i51W n1

, ~21!

whereni is the number of possibilities in thei th partition, each one with equal probability. Usingcondition ~iv! expressed in Eq.~8! we get

SqS H 1

( i51W n1

J D 5Sq~p1 ,p2 ,...,pW!1(i51

W

piqSqS H 1ni J D ~22!

or, using the functional form of Eq.~20!,

~( i51W ni !

p21

12q5Sq~p1 ,p2 ,...,pW!1(

i51

W

piqS niq21

12q D ~23!

so

SqS p1 ,p2 ,...,pWD51

12q H S (i51

W

ni D p211(i51

W

piq2(

i51

W

piqni

pJ ~24!

but

nip5pi

pS (i51

W

ni D p. ~25!

4106 Roberto J. V. dos Santos: Shannon’s theorem for Tsallis entropy

J. Math. Phys., Vol. 38, No. 8, August 1997

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Thus, substituting into Eq.~24!, we see that, in order to satisfy condition~i! so thatSq($pi%) mustdepend only on$pi%, one must have

p512q. ~26!

Therefore

Sq~$pi%!512( i51

W piq

q21~27!

and, fort equiprobable choices,

Sq~ t !5t12q21

q21. ~28!

In this way we have generalized, for the Tsallis entropy, Shannon’s remarkable theorem.

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Dr. Constantino Tsallis, for fruitful discussions.This work was partially supported by CNPq and FINEP, Brazilian agencies.

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