Non-Extensive Quantum Statistics with Particle Hole Symmetry T.S. - - PowerPoint PPT Presentation

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary

Chinese-Hungarian bilateral research project

Non-Extensive Quantum Statistics

with Particle – Hole Symmetry T.S. Biró1

• K. M. Shen2
• B. W. Zhang2

1Heavy Ion Research Group

MTA Research Centre for Physics, Budapest

2Institute of Particle Physics

Central China Normal University, Wuhan

October 7, 2014

Talk given by T. S. Biró, ACHT 2014, Oct.08., Balatonfüred, Hungary TSB KMS BWZ NEXT Q STATISTICS 1 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary

Content

1

Kubo–Martin–Schwinger Relation

2

Generalized Thermodynamics

3

Particle-Hole Symmetry within q- and κ-Statistics

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

Content

1

Kubo–Martin–Schwinger Relation Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

2

Generalized Thermodynamics

3

Particle-Hole Symmetry within q- and κ-Statistics

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

KMS relation and quantum statistics 1

Kubo – Martin – Schwinger: At B0 = = Tr

• e−βHeitHA e−itHB
• =

Tr

• ei(t+iβ)HA e−i(t+iβ)He−βHB
• =

Tr

• e−βHB ei(t+iβ)HA e−i(t+iβ)H

=

• B0 At+iβ
• (1)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

KMS relation and quantum statistics 2

Apply KMS for At = e−iωt a and Bt = A†

t = eiωt a†.

• a a†

=

• a† a
• eβω

(2) meaning 1 + n(ω) = n(ω) eβω. (3) delivers Bose statistics n(ω) = 1 eβω − 1.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

CPT

Missing negative energy particle = positive energy hole. −n(−ω) = (1 + n(ω)) > 0. (4) Connection to canonical thermodynamical weight −n(−ω) n(ω) = eβω − → eq(βω) (5) to be generalized...

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Particle to Hole Ratio Boltzmann - Gibbs ”knows it”

KMS in general

reaction to Cheng-Bin’s question

The KMS relation - based on re-shuffling - holds very generally. At B0 = Tr

• ρ UAU−1 B
• =

= Tr

• ρ UAU−1 ρ−1ρ B
• =

= Tr

• ρB (ρU)A(ρU)−1

=

• B0 At⊕iβ
• (6)

For nonlinear H-dependence the ⊕ operation is energy dependent → curved spectra.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Superstatistics: Fluctuating Temperature Deformed Exponentials

Content

1

Kubo–Martin–Schwinger Relation

2

Generalized Thermodynamics Superstatistics: Fluctuating Temperature Generalized Entropy Formulas Deformed Exponentials

3

Particle-Hole Symmetry within q- and κ-Statistics

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Superstatistics: Fluctuating Temperature Deformed Exponentials

Continous β scale

Euler - integral:

∞

• Θke−αΘ dΘ =

k! αk+1 (7) Gamma distribution: γ(Θ) = αk+1 k! Θke−αΘ : Θ = k + 1 α . (8)

• Char. fct.:
• e−βωΘ

=

• 1 + βΘ ω

k + 1 −(k+1) (9)

this is a q = 1 + 1/(k + 1) Tsallis – Pareto distribution.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Superstatistics: Fluctuating Temperature Deformed Exponentials

KMS with Generalized Weights

Let it be x = βΘ ω and K = k + 1. n 1 + n =

• 1 + x

K −K =: 1 eq(x). (10) From this the Tsallis – Bose would be given by n = 1 eq(x) − 1. (11)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Superstatistics: Fluctuating Temperature Deformed Exponentials

Check negative energy states

n(−ω) = 1 eq(−x) − 1 (12) therefore − (1 + n(ω)) = 1 1/eq(x) − 1 (13) generalized KMS eq(x) · eq(−x) = 1.

It does not hold for Tsallis!

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Content

1

Kubo–Martin–Schwinger Relation

2

Generalized Thermodynamics

3

Particle-Hole Symmetry within q- and κ-Statistics Bosons, Fermions and ”Boltzmannons” Occupancy of Negative Energy States Properties of the Arithmetic Mean

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Kappa Distribution

• G. Kaniadakis

eκ(x) :=

• 1 + κ2x2 + κx

1/κ (14) It knows: eκ(−x) · eκ(x) = 1. (15) It automatically satisfies KMS statistics.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Linear Combination with Holes

Assume nKMS(x) = A (nq(x) + nq∗(x)) + B. For Tsallis distr. eq(−x) = 1/eq∗(x) with q∗ = 2 − q. Finally we get B = A − 1/2 and for having ”zero for zero” we get KMS (Ke-Ming Shen) ansatz nKMS(x) = 1 2

• nq(x) + nq∗(x)
• .

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

KMS with deformed exponentials

Requirements:

1

ρ(H) is the analytic continuation of U(H),

2

U(H) is unitary For U = g(H) from unitarity, U−1 = U†, and from time reversal CPT, it follows 1 g(ω) = g∗(ω) = g(−ω). (16)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

General CPT solution to KMS

We need to have a function (not necessarily the exponential), g(ω) satisfying g(ω) · g(−ω) = 1. The general solution with the best small ω - behavior is g(ω) = eq(ω/2) eq(−ω/2). (17)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Proper Quantum Statistics with Deformed Exponentials

Using the notation x = βω/2 we have nB,F(ω) = eq(−x) eq(x) ∓ eq(−x). (18) Now either with Tsallis or Kaniadakis deformed exp eq(x) we have power-law tail and perfect CPT. But Tsallis eq(−x) has a maximal x-value.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Graphs of Tsallis – exponentials

Tsallis – Pareto deformed exponential distributions (1/eq(x)) with q = 0.95, 1.00, 1.05. Curvature in the log-plot: sign of q − 1

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Graphs of original Bose distributions

Blue: −1 − n(ω), Green: n(−ω), Red: n(ω) Blue and Green coincide.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Graphs of Tsallis – based Bose distributions

q=1.05

Blue: −1 − n(ω), Green: n(−ω), Red: n(ω) Differences between holes and antiparticles?

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Graphs of Tsallis – based Bose distributions

q=1.10

Blue: −1 − n(ω), Green: n(−ω), Red: n(ω) Differences between holes and antiparticles?

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Graphs of Tsallis – based Bose distributions

q=1.20

Blue: −1 − n(ω), Green: n(−ω), Red: n(ω) Differences between holes and antiparticles?

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Log plots of kappa – exponentials (Kaniadakis)

All kappa deformed exponential distributions (1/eκ(x)) with 1 + κ = q = 0.95, 1.00, 1.05. Kappa knows eκ(−x) = 1/eκ(x)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Comparison of deformed Bose distributions

Blue: Kaniadakis (kappa), Green: Tsallis (q = 1 + κ), Red: Boltzmann (q = 1, κ = 0) Parallel asymptotics for K and T

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Massive deformed pressure curves

Blue: K, Green: T, Red: B. No big difference...?

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Occupancy of Negative Energy States Properties of the Arithmetic Mean

Interaction measure

(ǫ − 3p)/T 4 for Blue (K), Green (T) and Red (B). Nonzero only for massive bosons...

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Summary

The (measured and calculated) temperature surely fluctuates, but is not Gaussian. In general Deformed Exponentials describe particle/hole ratios. Tsallis Exponential does not satisfy KMS (kappa-exponential does) But the exact Tsallis distribution with q > 1 is physical (comes from NBD particle number fluctuations) Arithmetic Mean of q and 2 − q does the job.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

BACKUP SLIDES

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Ideal Gas: microcanonical statistical weight

The one-particle energy, ω, out of total energy, E, is distributed in a one-dimensional relativistic jet according to a statistical weight factor which depends on the number of particles in the reservoir, n: P1(ω) = Ω1(ω) Ωn(E − ω) Ωn+1(E) = ρ(ω) · (E − ω)n En (19) HEP Superstatistics: E fix, n has a distribution (based on the

physical model of the reservoir and on the event by event detection of the spectra).

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Ideal Reservoir: (Negative) binomial n-distribution

n particles among k cells: bosons n+k

n

• fermions

k

n

• A subspace (n, k) out of (N, K)

Limit: K → ∞ , N → ∞; average occupancy f = N/K is fixed.

Bn,k(f) := lim

K→∞

n+k

n

N−n+K−k

N−n

• N+K+1

N

• =

n + k n

• f n (1 + f)−n−k−1.

(20) Fn,k(f) := lim

K→∞

k

n

K−k

N−n

• K

N

• =

k n

• f n (1 − f)k−n.

(21)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Norm and Pascal triangle

Binomial expansion: (a + b)k =

∞

• n=0

k n

• anbk−n

(22) Replace k by −k − 1 and a by −a, noting that

−k − 1 n

• =

(−k − 1)(−k − 2) . . . (−k − n) n! = (−1)n (k + 1)(k + 2) . . . (k + n) n! = (−1)n n + k n

• .

we arrive at (b − a)−k−1 =

∞

• n=0

n + k n

• anb−n−k−1

(23)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

BD recursion: Pascal Triangle

Fn,k = f Fn−1,k−1 + (1 − f) Fn,k−1 (24)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

NBD recursion: Tilted Pascal Triangle

Bn,k = f 1 + f Bn−1,k + 1 1 + f Bn,k−1 (25)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Bosonic reservoir

Reservoir in hep: E is fixed, n fluctuates according to NBD.

∞

• n=0
• 1 − ω

E n Bn,k(f) =

• (1 + f) − f
• 1 − ω

E −k−1 =

• 1 + f ω

E −k−1 (26) Note that n = (k + 1)f for NBD. Then with T = E/ n and q − 1 =

1 k+1 we get

• 1 + (q − 1) ω

T −

1 q−1

This is exactly a q > 1 Tsallis-Pareto distribution.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Fermionic reservoir

E is fixed, n is distributed according to BD:

∞

• n=0
• 1 − ω

E n Fn,k(f) =

• (1 − f) + f
• 1 − ω

E k =

• 1 − f ω

E k (27) Note that n = kf for BD. Then with T = E/ n and q − 1 = − 1

k we get

• 1 + (q − 1) ω

T −

1 q−1

This is exactly a q < 1 Tsallis-Pareto distribution.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Boltzmann limit

In the k ≫ n limit (low occupancy in phase space)

n + k n

• f n(1 + f)−n−k−1 −

→ kn n!

• f

1 + f n . . . k n

• f n(1 − f)k−n −

→ kn n!

• f

1 − f n . . . (28)

After normalization this is the Poisson distribution: Πn = n n n! e− n with n = k f 1 ± f (29) The result is exactly the Boltzmann-Gibbs weight factor:

∞

• n=0
• 1 − ω

E n Πn( n ) = e−ω/T. (30)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Experimental NBD distributions PHENIX PRC 78 (2008) 044902

Au + Au collisons at √sNN = 62 (left) and 200 GeV (right). Total charged multiplicities.

k ≈ 10 − 20 → q ≈ 1.05 − 1.10.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Summary of reservoir fluctuation models

1 − ω

E

n

Bernoulli

= Tsallis(q < 1) 1 − ω

E

n

Possion

= Boltzmann(q = 1) 1 − ω

E

n

NBD

= Tsallis(q > 1) (31)

In all the three above cases T = E n , and q = n(n − 1) n 2 (32)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Ideal gas with general n-fluctuations

Canonical approach: expansion for small ω ≪ E.

Tsallis-Pareto distribution as an approximation:

• 1 + (q − 1) ω

T −

1 q−1 = 1 − ω

T + q ω2 2T 2 − . . . (33) Ideal reservoir phase space up to the subleading canonical limit: 1 − ω E n = 1 − n ω E + n(n − 1) ω2 2E2 − . . . (34)

To subleading in ω ≪ E T = E n , q = n(n − 1) n 2 = 1 − 1 n + ∆n2 n 2 . (35)

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

General system with general reservoir fluctuations

Canonical approach: expansion for small ω ≪ E. Ωn(E − ω) Ωn(E)

• =
• eS(E−ω)−S(E)

=

• e−ωS′(E)+ω2S′′(E)/2−...

= 1 − ω

• S′(E)
• + ω2

2

• S′(E)2 + S′′(E)
• − . . .

(36) Compare with expansion of Tsallis

• 1 + (q − 1) ω

T −

1 q−1 = 1 − ω

T + q ω2 2T 2 − . . . (37)

Interpret the parameters 1 T = S′(E) , q = 1 − 1 C + ∆T 2 T 2 (38) S′′(E) = −1/CT 2

expressed via the heat capacity of the reservoir,1/C=dT/dE

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Understanding the parameter q

in terms fluctuations

Opposite sign contributions from

• S′ 2

− S′ 2 and from S′′ . In all cases approximately q = 1 − 1 C + ∆T 2 T 2 . q > 1 and q < 1 are both possible for any relative variance ∆T/T = 1/ √ C it is exactly q = 1 for nT = E/dim = const it is ∆T/ T = ∆n/ n . for ideal gas and n distributed as NBD or BD, the Tsallis form is exact

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Soft and Hard Tsallis fits:

ALICE PLB 720 (2013) 52; PHENIX PRL 101 (2008) 232301

the knick is around pT ≈ 4 − 5 GeV.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Hard and Soft Trends with Npart arxiv: 1405.3963

C = k + 1 powers of the power law and fitted T parameters (ALICE).

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

Soft Powers vs Npart arxiv: 1405.3963

Only the soft ( ”statistical” ) branches for PHENIX and ALICE:

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

n-distribution from superstatistics

We demand

• e−βω γ(β) dβ =
• n

P

n(E)

• 1 − ω

E n Note that e−βω = e(1− ω

E ) βE e−βE

Using the Taylor series of the first exponential one concludes P

n(E) =

(βE)n n! e−βE γ(β) dβ The converter factor is a Poissonian with the parameter n = βE.

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Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q- and κ-Statistics Summary Effects due to Particle Number Fluctuations

superstatistics from n-distribution

Apply the correspondence for ω = E:

• e−βE γ(β) dβ = P0(E).

Inverse Laplace transformation delivers the superstatistical factor γ(β) = L−1 [P0(E)] Expanding for small ω one gets β = n E and

• β2

= n(n − 1) E2 leading to 1 + ∆β2 β 2 = 1 + ∆n2 n 2 − 1 n so for some ”super-distributions” ∆β2 would have to be negative...

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