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Review of Tsallis distribution applied to RHIC data? NO! Theory about why is it applicable: yes! T.S.Bíró, G.Purcsel and K.Ürmössy MTA KFKI RMKI Budapest Talk given at Zimányi Winter School, 2008. nov. 25-29. Budapest, Hungary
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Transverse momentum spectra
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What all it can… dN/dy rapidity distribution: reduced phase space (q < 1) Multiplicity distribution: negative binomial Temperature – average energy fluctuation Superstatistics Coalescence scaling for (q-1) Theory: thermodynamics with power-law tailed energy distributions
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Power-law tailed distributions and abstract composition rules Extensivity and non-extensivity Composition rules in the large-N limit Entropy formulas and distributions Relativistic kinetic energy composition T.S.Bíró, MTA KFKI RMKI Budapest Talk given at Zimanyi Winter School, 2008. nov. 25-29. Budapest, Hungary
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Extensivity and non-extensivity T.S.Biro, arxiv:0809.4675 Europhysics Letters 2008 T.S.Biro, G.Purcsel, Phys.Lett.A 372, 1174, 2008 T.S.Biro, K.Urmossy, G.G.Barnafoldi, J.Phys.G 35:044012, 2008
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Extensive is not always additive
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Nonextensive is less tahn nonadditive
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Nonextensive as composite sums Pl. x_i=i, L(x)=exp(ax)
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What is a problem? no problem: additive hence extensive problem: non-additive belief: it becomes extensive in the large-N limit What to do if non-additive and non-extensive?
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Example
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Normalizations
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Entropy: contribution of a pair
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Entropy: N ptl-s N (N-1) / 2 pairs
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Energy: N ptl-s N (N-1) / 2 pairs
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Characteristic g(r) and … g(r) - g(r) ln g(r) g(r) v(r) weakly interacting pairs: Vinfo finite confined pairs: Vinfo may diverge
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Large distance behavior
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Composition, large-N limit
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n = t N
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Composition by formal logarithm
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Asymptotic rules are associative
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Associative rules are asymptotic
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Associative rules are attractors among more general rules
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Entropy formulas, distributions
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Formal logarithm
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Deformed logarithm Deformed exponential
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Non-extensive entropy and energy
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Entropy maximum at fixed energy
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Canonical distribution and detailed balance solution in generalized Boltzmann equation:
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Example: Gibbs-Boltzmann
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Example: Tsallis
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Example: Kaniadakis
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Example: Einstein
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Example: Non associative
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Generalized Boltzmann equation
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H theorem
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Relativistic energy composition
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Angle averaged Q dependent composition rule for the relativistic kinetic energies
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Formal logarithm and asymptotic rule for the relativistic kinetic energies
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Asymptotic rule for m=0
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Summary Non-extensive thermodynamics requires non-additive composition rules Large N limit: rules are associative and symmetric, formal logarithm L Entropy formulas, equilibrium distributions, H-theorem based on L Relativistic kinematics Tsallis-Pareto
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Appendix: how to throw new momenta?
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Appendix: how to throw new momenta for BG?
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Appendix: how to throw new momenta for Tsallis?
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