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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
2 / 68
Preface
Outline
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
3 / 68
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
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
8 / 68
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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Non-extensive Statistics Tsallis Entropy
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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Non-extensive Statistics Tsallis Entropy
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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Non-extensive Statistics Tsallis Entropy
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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Non-extensive Statistics Tsallis PDF
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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Non-extensive Statistics Tsallis PDF
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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Non-extensive Statistics Tsallis PDF
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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Non-extensive Statistics Tsallis PDF
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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Non-extensive Statistics Tsallis PDF
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
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
21 / 68
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
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
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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
heavy-ion collisions
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
heavy-ion collisions
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
heavy-ion collisions
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others QGP
heavy-ion collisions
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others QGP
heavy-ion collisions
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κ-statistics
Outline
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
39 / 68
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
1 Preface2 Non-extensive Statistics
Tsallis EntropyTsallis qTsallis PDF
3 Non-extensive quantum statisticsPDFapplications
4 othersRelativistic Non-extensive Thermodynamics[3]Non-extensive HydrodynamicsQGP
5 κ-statistics6 Backup
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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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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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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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Grand Canonical Ensemble***
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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Grand Canonical Ensemble***
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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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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Asymptotic Approximation
〈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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Asymptotic Approximation
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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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
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