Non-extensive Statistical Mechanics · 2016-12-07 · Non-extensive Statistical Mechanics Keming...

Non-extensive Statistical Mechanics

Keming Shen

Institute of Particle PhysicsCentral China Normal University

CE

NT

RA

L

CH

IAN NORMALU

N

IVE

RSIT

Y

CENTRAL CHIAN NORMAL UNIVERSITY

Wuhan, China

December 7, 2016

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Outline

1 Preface

2 Non-extensive StatisticsTsallis EntropyTsallis qTsallis PDF

3 Non-extensive quantum statisticsPDFapplications

4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP

5 κ-statistics

6 Backup

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Preface

Outline

1 Preface2 Non-extensive Statistics

Tsallis EntropyTsallis qTsallis PDF



5 κ-statistics6 Backup

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Preface

A summary on Entropy Statistics

With the purpose to study as a whole the major part of entropymeasures, the entropy-functional is proposed in[1]

Hϕ1,ϕ2h,v (pi) = h(

∑Wi viϕ1(pi)∑Wi viϕ2(pi)

) (1)

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Preface

Here I justlist someof them:(1)Shannon-1948[2],(2) Renyi-1961[3],andetc[4, 5, 6,1, 2, 3, 4, 5,3, 6, 1, 2, 3].

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Preface

Shannon’s Entropy

First of all, it is well known of the Shannon entropy, which satisfies theFadeev’s postulates[4],

SSh =

∑Wi (−pi ln pi)∑W

i pi= −

W∑i

pi ln pi (2)

whose maximum is nothing but the case that pi = 1/W, namely theBoltzmann Entropy,

SBG = −W∑i

1W

ln1W

= ln W (3)

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Preface

Renyi’s Entropy

While, Shannon’s Entropy is not the only one satisfying the postulatesif we weaken the fourth one to H[O +Q] = H[O] + H[Q].[3] In 1961,Alfred Renyi gave another one

SRe =1

1− qln

W∑i

pqi (4)

which is also shown in the table.

P.S. Think of this: what will it be when∑W

i pqi goes to 1 with q→ 1?

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Non-extensive Statistics

Outline






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Non-extensive Statistics Tsallis Entropy

Tsallis’ Entropy

Since Tsallis suggested to use the non-extensive entropy formulawith the use of a quantity normally scaled in multifractals firstly,[5]

STs =

∑Wi=1 pq

i − 11− q

(5)

the corresponding generalized statistical mechanics have beensubstantially developed and spread over many fields ofapplication[6, 1, 2, 3, 4, 5].

P.S. It also has the property that its maximum is SmaxTs = lnq W with

pi = 1/W.

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Uniqueness Theorem for Shannon Entropy

1 Shannon [2]S(A + B) = S(A) + S(B) pA+B

ij = pAi pB

j

S(pi) = S(pL, pM) + pLS(pi/pL) + pMS(pi/pM) L + M = W(6)

2 Khinchin [4]S(pi, 0) = S(pi)S(A + B) = S(A) + S(B|A) S(B|A) =

∑pA

i S(pA+Bij /pA

i )(7)

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Uniqueness Theorem for Tsalli Entropy

1 Santos [5]S(A + B) = S(A) + S(B) + (1− q)S(A)S(B) pA+B

ij = pAi pB

j

S(pi) = S(pL, pM) + pqLS(pi/pL) + pq

MS(pi/pM) L + M = W(8)

2 Abe [6]S(pi, 0) = S(pi)

S(A + B) = S(A) + S(B|A) + (1− q)S(A)S(B|A) S(B|A) =∑

(pAi )qS(pA+B

ij /pAi )∑

(pAi )q

(9)

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Thermodynamical Foundations/Applications

1 Correlated Anomalous Diffusion: generalized Fokker-PlanckEquation.

2 Central Limit Theorems. [4]3 Zeroth Law. [5]4 Equipartition and Virial theorems. [6]5 Second Law. [1]6 Quantum H-theorem. [6]7 The classical N-body problem. [2]8 Fluctuation-Dissipation Theorem. [3]9 · · ·

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Non-extensive Statistics Tsallis q

q-Relation

Considering the thermal equilibrium of two systems, one with energyE1 (subsystem) and the other with E − E1 (reservoir), for a maximalentropy state,

L(S(E)) = L(S(E1)) + L(S(E − E1)) = max (10)

where L(S) is the additive form of the entropy.[6]Under the Universal Thermal Independence (UTI) Principle, we have

L′′(S)

L′(S)= − S′′(E)

S′(E)2 =1C

(11)

Then we have, L(S) = C(eS/C − 1), with finite constant heat capacity C.

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Non-extensive Statistics Tsallis q

q-Relation

Substituting the Renyi Entropy into it,

L(S) = C(e1C

11−q ln

∑Wi pq

i − 1) =

∑Wi pq

i − 11− q

(12)

where we have q = 1− 1/C.More deeply, to solve the negative heat capacity problem, we get the

more generalized result

q = 1− 1C

+∆T2

T2 (13)

more details can be seen in [1] and its followings.

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Non-extensive Statistics Tsallis PDF

Canonical Ensemble

Under the Optimal Lagrange Multipliers(OLM)-Tsallis technique, withthe normalized (

∑Wi pi = 1) and energy constraints

(Uq =∑W

i Piεi =∑W

ipq

i∑Wj pq

jεi) respectively,

pi =1Zq

[1− (1− q)β∑Wj pq

j

εi]1

1−q :=1Zq

e−β′εi

q (14)

More discussions are given out in paper[3] and book[4].

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q-Thermodynamics

Re-writing the q-expectation of energy, Uq,

Uq =

W∑i

pqi∑W

j pqj

εi =

W∑i

1∑Wj pq

j

1Zq

q(e−β

′εiq )qεi

= − ∂

∂βlnq Zq (15)

Similarly, the generalized force is

Yq = − 1β

∂

∂ylnq Zq (16)

where y is the generalized coordinates.

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q-Thermodynamics

All the above lead to

β(dUq − Yqdy) = βd(− ∂

∂βlnq Zq)− β(− 1

β

∂

∂ylnq Zq)dy

= d(lnq Zq − β∂

∂βlnq Zq) (17)

Comparing with the q-thermodynamical relation

dSq =1T

(dUq − Yqdy) (18)

which tells us nothing but

Sq = lnq Zq − β∂

∂βlnq Zq (19)

β =1T

(20)

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Grand Canonical Ensemble

Consider a non-interacting quantum gas composed of N particles inheat and particle baths. With

∑pR = 1,

∑pq

RER = E and∑

pqRNR = N,

pR =1Zq

e−β(ER−µNR)q (21)

The factorization approximation (FA) was proposed by Buyukkilic etal. in 1995[5]. Then the generalized distribution functions are

nk =1

eβ(εk−µ)2−q ± 1

(22)

where nk =∑

pnk nk and NR =∑

nk.

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q-FDD and BED and Non-extensive quantumH-Theorem

Let us start by presenting the specific functional form for entropy[6]

SQ = −∑

nqk lnq nk ∓ (1± nk)

q lnq(1± nk) (23)

With Cq(nk) = dnkdt denotes the quantum q-collisional term, it is proved

thatdSQ

dt≥ 0 (24)

which is the quantum Hq-theorem.Furthermore, considering the equilibrium case, namely, dSQ/dt = 0,

nk =1

eα+βεkq ± 1

(25)

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Escort Distributions

Meantime, C. Beck generalized Hagedorn’s theory[1, 2] tore-consider the grand partition function in non-extensive statistics. [3]

Pi1,i2,··· ,iN =1Z

N∏j=1

[1− (1− q)βεij ]q/(1−q) =

1Z

N∏j=1

(e−βεijq )q (26)

where 1− (1− q)βH =∏N

j=1(1− (1− q)βεij). So we have

H =∑

j

εij + (q− 1)β∑j,k

εijεik + · · · (27)

Thus the non-extensive average number turns to be

ni =1

(eβεij2−q)q ± 1

(28)

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Non-extensive quantum statistics

Outline






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Non-extensive quantum statistics PDF

the (1− q) expansion

We’ve known that the non-extensive statistics recovers the classicalBoltzmann case when q→ 1. Moreover, in most cases there are noneed to calculate the exact values or forms based on q-BED (q-FDD).Here I list some of the approximation methods[1]:

1 Factorization Approximation. (FA)[4]

1ex

2−q − 1≈ 1

ex − 1(1− 1− q

2x2ex

ex − 1) (29)

2 Asymptotic Approximation. (AA)[5]

〈n〉q ≈1

ex − 1(1− e−x)q−1[1 + (1− q)x(

ex + 1ex − 1

− x2

1 + 3e−x

(1− e−x)2 )]

≈ 1ex − 1

[1 + (1− q)x(ex + 1ex − 1

− x2

1 + 3e−x

(1− e−x)2 − ln(1− e−x))]

(30)

3 see below.22 / 68

Non-extensive quantum statistics PDF

the (1− q) expansion

1 Exact Approximation. (EA)[6, 2]

1ex

2−q − 1=

Γ( 11−q)∫

dvv1

1−q−1e−v(ev(1−q)x − 1)(31)

where Γ(a) =∫

dtta−1e−t for any a > 0.

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Non-extensive quantum statistics applications

Blackbody Radiation

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Non-extensive quantum statistics applications

BEC

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others

Outline






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others Relativistic Non-extensive Thermodynamics[3]

Basic assumptions

SR = −∫

dΩpq lnq p (32)

Similarly we have p(x, v) = f (x, v)/Zq, where

f = [1− (1− q)β(H − 〈H〉q)]1/(1−q) := e−β(H−〈H〉q)q (33)

Easy to get ∫dΩf ≡

∫dΩf q (34)

〈A〉q :=

∫dΩf qA∫dΩf q =

1Zq

∫dΩf qA (35)

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others Relativistic Non-extensive Thermodynamics[3]

Relativistic kinetic theory

The corresponding particle four-flow and energy-momentum flow areas

Nµ(x) =1Zq

∫d3pp0 pµf (36)

Tµν(x) =1Zq

∫d3pp0 pµpν f q (37)

Thus can we have the probability density n = NµUµ, the energy densityε = TµνUµUν and the equilibrium pressure P = − 1

3 Tµν∆µν . Thus wealso obtain that for massless system,

ε = 3P (38)

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others Non-extensive Hydrodynamics

theoretical work

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others Non-extensive Hydrodynamics

phenomenological analysis

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others QGP

theoretical basis

Tsallis’ non-extensive statistical mechanics can be considered anappropriate basis to deal with physical phenomena where strongdynamical correlations, long-range interactions and microscopicmemory effects take place.[5]

And we expect in the range of temperature and density considered,the presence of strange particles does not significantly affect the mainconclusions regarding the relevance of non-extensive statistical effectsto the nuclear EOS.[6]

In many-body long-range-interacting systems, there has beenobserved the emergence of long-living quasi-stationary (metastable)states characterized by non-Gaussian power-law velocitydistributions.[1]

Moreover, for such systems like QGP formed in heavy-ion collisions,the size (N NA) needs be re-consideration whether Boltzmannstatistics is still appropriate.

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others QGP

phase transition

By requiring the Gibbs conditions on the global conservation ofbaryon number and electric charge fraction, the phase transition fromhadronic matter to QGP is studied in the frame of non-extensivestatistics.[5, 2]

PB =23

∑i=n,p

∫d3k

(2π)3k2

Ei ∗ (k)[nq

i (k)+nqi (k)]−1

2m2σσ

2−U(σ)+12

m2ωω

2+12

m2ρρ

2

(39)

Pq =γf

3

∑f =u,d

∫d3k

(2π)3k2

ef[nq

f (k) + nqf (k)]− B (40)

Pg =γg

3

∫d3k

(2π)3 knqg(k) (41)

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others QGP

heavy-ion collisions

R. Hagedorn[2] proposed the QCD inspired empirical formula todescribe experimental hadron production data:[3]

Ed3σ

d3p= C(1 +

pT

p0)−n →

exp(− npT

p0) pT → 0

( p0pT

)n pT →∞(42)

which coincides with

hq(pT) = Cqe−βpTq = Cq[1− (1− q)

pT

T]

11−q (43)

for n = 1/(q− 1) and p0 = nT.

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others QGP


Others[4, 5, 6, 1, 2] criticized that thermodynamical consistency leadsto

Ed3Nd3p∝ [1− (1− q)βpT ]

q1−q (44)

which comes fromε = g

∫dΩEf q (45)

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others QGP


During the last few years, G. G. Barnafoldi et al.[3] proposed a"soft+hard" model for pT spectra as well as v2 measured in both pp andAA collisions.

dN2πpTdpTdy

∣∣∣∣y=0

=∑

i=soft,hard

Aie−βi[γi(mT−v0,ipT)−m]q (46)

where "soft" is referred to hadrons yields stemming from the QGP partbut "hard" to jets.

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others QGP


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others QGP


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others QGP


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κ-statistics

Outline






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κ-statistics

κ-distributions

To get rid of the KMS problem[4] and others Tsallis’ q-exponentialmeets, G. Kaniadakis[3, 4, 5, 6] proposed another form of distributionswhich lead to the κ-deformed statistical mechanics.

SKa =

∫dΩ

f 1−κ − f 1+κ

2κ:= −

∫dΩf lnκ f (47)

where lnκ x = xκ−x−κ2κ is the κ-logarithm. With the OLM and MEM can

we get its distribution:f = e−β(U−µ)

κ (48)

where the κ-exponential is introduced,

exκ = (

√1 + κ2x2 + κx)1/κ = exp(

1κ

arcsinκx) (49)

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κ-statistics

κ-distributions of a QGP[1]

Using the κ-deformed statistics to describe the QGP, the singleparticle distribution functions of quarks/anti-quarks and gluonsrespectively,

nq/q =1√

1 + κ2β2(k ∓ µq)2 + κβ(k ∓ µq) + 1:=

1eκ(β(k ∓ µq)) + 1

(50)

ng =1√

1 + κ2β2k2 + κβk − 1:=

1eκ(βk)− 1

(51)

Thus can we study the phase transition with the similar steps. Thesame phase diagram as in the Tsallis case is obtained, since both ofthem are fractal in nature.

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Thank You!!!

Backup Slides

Backup

Outline






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Backup

Fadeev’s postulates

The amount of uncertainty of the distribution Ω = (p1, p2, · · · , pn), thatis, the amount of uncertainty concerning the outcome of anexperiment, the possible results of which have the probabilitiesp1, p2, · · · , pn, is called the Entropy of the distribution Ω. In 1957,Fadeev proposed that the simplest such set of postulates are asfollows.

1 It is a symmetric function of its variables for n.2 It is a continuous function of pi.3 H(1/2, 1/2) = 1.4 H(tp1, (1− t)p1, p2, · · · , pn) = H(pi) + p1H(t, (1− t)).

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Backup

q-Algorithm

From now on, Tsallis Entropy is re-written as

Sq =

∑Wi=1 pq

i − 11− q

:=

W∑i

pi lnq(1pi

) = −W∑i

pqi lnq pi (52)

where the q-logarithm is introduced,

lnq(x) :=x1−q − 1

1− q(53)

with its inverse function, q-exponential

exq := [1 + (1− q)x]

11−q (54)

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Backup

q-Algorithm

Consistently, non-linear generalized algebraic forms emerge,q-sum

x⊕q y := x + y + (1− q)xy (55)

q-productx⊗q y := (x1−q + y1−q − 1)1/(1−q) (56)

q-substraction

xq y :=x− y

1 + (1− q)y(57)

q-divisionx y := (x1−q − y1−q + 1)1/(1−q) (58)

More are seen in [2, 1].

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Backup

Conditional Probability

Consider two cases A and B with probabilities P(A) and P(B). Theconditional probability of A under B is:

P(A|B) =P(AB)

P(B)(59)

If all Ai are independent(mutually incompatible), P(B) > 0, then,

P(∑

Ai|B) =∑

P(Ai|B) (60)

Moreover, for the case that B ⊂⋃

Ai,

P(B) =∑

P(Ai)P(B|Ai) (61)

which is the total probability formula.

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Bayes Formula

P(Ai|B) =P(Ai)P(B|Ai)∑

P(Aj)P(B|Aj)(62)

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Backup

Grand Canonical Ensemble***

Think about the constraints with the escort distribution Pi = pqi /∑

pqj ,

similarly we have

pi =1

Ξqe−β

′(εi−µNi)q (63)

Easy to prove the q-thermodynamics above,

Nq =∑

PiNi =1β

∂

∂µlnq Ξq (64)

Uq = − ∂

∂βlnq Ξq (65)

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Backup


Consider each distinct microstate i in energy levels l (l = 1, 2, · · · ),that is, εi =

∑l εlnl and Ni =

∑l nl. So the q-grand partition function

turns to be

Ξq =∑nl

e−β′∑

l(nlεl−µnl)q =

∑nl

q∏l

e−β′(εl−µ)nl

q

=

q∏l

∑nl

e−β′(εl−µ)nl

q =

q∏l

Zq(l) (66)

where Zq(l) =∑

nle−β

′(εl−µ)nlq .

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Backup


1 For Fermions,Zq(l) = 1 + e−β

′(εl−µ)q (67)

so

nl =1β

∂

∂µlnq Zq(l) =

1∑pq

m(

1

eβ′(εl−µ)

2−q + 1)q → (

1

eβ′(εl−µ)

2−q + 1)q (68)

2 For Bosons, similarly,

nl =1∑pq

m(

1

eβ′(εl−µ)

2−q − 1)q → (

1

eβ′(εl−µ)

2−q − 1)q (69)

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Backup

Asymptotic Approximation

ρ = e−βHq /Zq (70)

Zq = Tr exp(1

1− qln[1− (1− q)βH]) ≈ Tr exp(−βH− 1

2(1− q)β2H2)

≈ Tr exp(−βH)[1− 12

(1− q)β2H2] = ZBG[1− 12

(1− q)β2〈H2〉BG]

(71)

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Backup


〈O〉q = TrρqO = 〈ρq−1O〉1 = Z1−qq 〈 O

1− (1− q)βH〉1

= Z1−qq Z−1

q Tr([1− (1− q)βH]1/(1−q) O1− (1− q)βH

)

≈ Z1−qq Z−1

BG [1 +12

(1− q)β2〈H2〉BG]TrO exp(q

1− qln[1− (1− q)βH])

≈ Z1−qq Z−1

BG [1 +12

aβ2〈H2〉BG]TrO exp(−(1− a)βH+a2

(1− a)β2H2)

≈ Z1−qq Z−1

BG [1 +12

aβ2〈H2〉BG]TrO exp(−βH)[1 + aβH− a2β2H2]

= Z1−qBG [1 +

12

(1− q)β2〈H2〉BG]

〈O〉BG + (1− q)β〈OH〉BG −1− q

2β2〈OH2〉BG (72)

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Backup


Consider the system H = nhν, with O = n,

〈n〉q ≈ 〈n〉BGZ1−qBG 1 + (1− q)x[

〈n2〉BG

〈n〉BG+ x(〈n2〉BG −

〈n3〉BG

〈n〉BG)] (73)

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Backup

deformed differential

From the generalized substraction rules we can easily have,

dqx = (x + dx)q x =dx

1 + (1− q)x(74)

dκx = (x + dx)κ x =dx√

1 + κ2x2(75)

Easy to see ddqx ex

q = exq and d

dκx exκ = ex

κ.

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Backup

Cited Papers

Esteban, M.D. and Morales, D., A Summary on Entropy Statistics,(1991).

C. E. Shannon, A mathematical theory of communication. Bell.System Tech. J., 27 (1948), 379-423.

A. Renyi, On the measures of entropy and information. Proc. 4thBerkeley Symp. Math. Statist. and Prob., 1 (1961), 547-561.

J. Aczel and Z. Dzroczy, Characteristerung der entropien positiverordnung und der Shannonschen entropie. Act. Math. Acad. Sci.Hunger. 14 (1963) 95-121.

R. S. Varma, Generalizations of Renyi’s entropy of order α. J.Math. Sci., 1 (1966) 34-48.

J. N. Kapur, Generalized entropy of order α and type β. The Math.Seminar, 4 (1967) 78-82.

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Cited Papers

J. Havdra and F. Charvat, Concept of structural α-entropy.Kybernetika, 3 (1967) 30-35.

S. Arimoto, Information-theoretical considerations on estimationproblems. Information and Control. 19 (1971), 181-194.

B. D. Sharma and D. P. Mittal, New non-additive measures ofrelative information. J.Comb. Inform. & Syst. Sci., 2 (1975),122-133.

L. J. Taneja, D., A study of generalized measures in informationtheory. Ph.D. Thesis. University of Delhi, (1975).

B. D. Sharma and L. J. Taneja, I. J., Entropy of type (α, β) andother generalized measures in information theory. Metrika, 22(1975) 205-215.

C. Ferreri, Hypoentropy and related heterogeneity divergencemeasures. Statistica. 40 (1980) 55-118.

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Cited Papers

A. P. Sant’anna and I. J. Taneja, Trigonometric entropies, Jensendifference divergences and error bounds. Infor. Sci., 35 (1985)145-156.

M. Belis and S. Guiasu, A quantitative-qulitative measure ofinformation in cybernetics systems. IEEI Tran. Inf. Th. IT-4 (1968),593-594.

C. F. Picard, The use of Information theory in the study of thediversity of biological populations. Proc. Fifth Berk. Symp. IV(1979), 163-177.

D. K. Fadeev, Zum Begriff der Entropie ciner endlichenWahrsecheinlichkeitsschemas, Arbeiten zur Informationstheorie I,Berlin, Deutscher Verlag der Wissenschaften, 1957, 85-90.

C. Tsallis, J. Stat. Phys. 52, (1988) 479.

M. L. Lyra and C. Tsallis, PRL 80, 53(1998)59 / 68

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Cited Papers

F. Baldovin and A. Robledo, PRE 69, 045202(R) 2004

E. Lutz, PRA 67, 051402(R) 2003

P. Douglas, S. Bergamini and F. Renzoni, PRL 96, 110601 (2006)

F. Caruso and C. Tsallis, PRE 78, 021102 (2008)

J. S. Andrade, Jr, G. F. T da Silva, A. A. Moreira, F. D. Nobre andE. M. F. Curado, PRL 105, 260601 (2010)

T. S. Biro, G. G. Barnafoldi and P. Van, EPJ A 49:110 (2013)

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Cited Papers

T. S. Biro, G. G. Barnafoldi, P. Van and K. Urmossy, arXiv:1404.1256[hep-ph]

V. Schwammle and C. Tsallis, J. Math. Phys. 48, 113301 (2007)

A. Dukkipati, M. N. Murty and S. Bhatnagar, Phys. A 361 (2006)124-138

C. Tsallis, Introduction to Nonextensive Statistical Mechanics,84-102.

F. Buyukkilic, D. Demirhan and A. Gulec, PLA 197 (1995) 209-220.

R. Silva, D. H. A. L. Anselmo and J. S. Alcaniz, EPL 89 (2010)10004.

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Cited Papers

R. Hagedorn, Nuovo Cimento, Suppl. 3 (1965) 147.

R. Hagedorn, J. Rafelski, PLB 97, 1980, 136

C. Beck, Phys. A 286 (2000) 164

U. Tirnakli, F. Buyukkilic and D. Demirhan, PLA 245 (1998) 62-66.

C. Tsallis, F. C. S. Barreto and E. D. Loh, PRE 52 2 (1995)

F. Buyukkilic, I. Sokmen and D. Demirhan, Chaos, Solitons andFractals 13 (2002) 749-759.

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