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Thermodynamics of abstract composition rules

This work explores non-extensive thermodynamics, focusing on the statistical macro-equilibrium of long-range correlated and entangled systems. It involves generalizations of entropy formulas and the Boltzmann equation, examining applications within various fields such as astrophysics, granular matter, and finance. The talk presented at the Zimányi School in Budapest outlines the framework for understanding complex systems through additive and non-additive composition rules, aiming to bridge the gap between classical and modern statistical mechanics.

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Thermodynamics of abstract composition rules

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  1. Thermodynamics of abstract composition rules T.S.Biró, MTA KFKI RMKI Budapest Product, addition, logarithm Abstract composition rules, entropy formulas and generalizations of the Boltzmann equation Application: Lattice SU2 with fluctuating temperature Thanks to: G.Purcsel, K.Ürmössy, Zs.Schram, P.Ván Talk given at Zimányi School, Nov. 30. – Dec. 4. 2009, Budapest, Hungary

  2. Non-extensive Thermodynamics The goal is to describe: statistical macro-equilibrium irreversible properties of long-range correlated (entangled) systems

  3. Non-extensive Thermodynamics This is a dream ! The goal is to describe: statistical macro-equilibrium irreversible properties of long-range correlated (entangled) systems

  4. Non-extensive Thermodynamics This is a theory... Generalizations done (more or less): entropy formulas kinetic eq.-s: Boltzmann, Fokker-Planck, Langevin composition rules Most important: fat tail distributions canonically

  5. Applications (fits) • galaxies, galaxy clusters • anomalous diffusion (Lévy flight) • turbulence, granular matter, viscous fingering • solar neutrinos, cosmic rays • plasma, glass, spin-glass • superfluid He, BE-condenstaion • hadron spectra • liquid crystals, microemulsions • finance models • tomography • lingustics, hydrology, cognitive sciences

  6. Logarithm: Product  Sum additive extensive

  7. Abstract Composition Rules EPL 84: 56003, 2008

  8. Repeated Composition, large-N

  9. Scaling law for large-N

  10. Formal Logarithm

  11. Asymptotic rules are associative and attractors among all rules…

  12. Asymptotic rules are associative

  13. Associative rules are asymptotic

  14. Scaled Formal Logarithm

  15. Deformed logarithm Deformed exponential

  16. Entropy formulas, distributions Boltzmann – Gibbs Rényi Tsallis Kaniadakis … EPJ A 40: 325, 2009

  17. Entropy formulas from composition rules Joint probability = marginal prob. * conditional prob. The last line is for a subset

  18. Entropy formulas from composition rules Equiprobability: p = 1 / N Nontrivial composition rule at statistical independence

  19. Entropy formulas from composition rules 1. Thermodynamical limit: deformed log

  20. Entropy maximum at fixed energy

  21. Generalized kinetic theory

  22. Boltzmann algorithm: pairwise combination + separation With additive composition rule at independence: Such rules generate exponential distribution

  23. Boltzmann algorithm: pairwise combination + separation With associative composition rule at independence: Such rules generate ‘exponential of the formal logarithm’ distribution

  24. Generalized Stoßzahlansatz

  25. General H theorem

  26. General H theorem: entropy density formula

  27. Detailed balance: G = G 12 34

  28. Examples for composition rules

  29. Example: Gibbs-Boltzmann

  30. Example: Rényi, Tsallis

  31. Example: Einstein

  32. Important example: product class

  33. Important example: product class QCD is like this!

  34. Relativistic energy composition

  35. Relativistic energy composition ( high-energy limit: mass ≈ 0 )

  36. Asymptotic rule for m=0

  37. Physics background: α q > 1 q < 1 Q²

  38. Simulation using non-additive rule PRL 95: 162302, 2005 with Gábor Purcsel • Non-extensive Boltzmann Equation • (NEBE) : • Rényi-Tsallis energy addition rule • random momenta accordingly • pairwise collisions repeated • momentum distribution collected

  39. Evolution in NEBE phase space

  40. Stationary energy distributions in NEBE program x + y x + y + 2 x y

  41. Thermal equilibration in NEBE program

  42. Károly Ürmössy Scaling variable E or X(E)?

  43. Károly Ürmössy Scaling variable E or X(E)?

  44. Microscopic theory in non-extensive approach: questions, projects, ... • Ideal gas with deformed exponentials • Boltzmann and Bose distribution • Fermi distribution: ptl – hole effect • Thermal field theory with stohastic temperature • Lattice SU(2) with Gamma * Metropolis method

  45. As if temperature fluctuated… • EulerGamma  Boltzmann = Tsallis • EulerGamma  Poisson = Negative Binomial

  46. Euler - Gamma distribution max: 1 – 1/c, mean: 1, spread: 1 / √ c

  47. Tsallis lattice EOS Tamás S. Bíró (KFKI RMKI Budapest) and Zsolt Schram (DTP ATOMKI Debrecen) • Lattice action with superstatistics • Ideal gas with power-law tails • Numerical results on EOS

  48. Lattice theory Expectation values of observables: -S(t,U) DU dt w (t) e t A(U) ∫ ∫ v c A = -S(t,U) DU dt w (t) e ∫ ∫ c Action: S(t,U) = a(U) t + b(U) / t t= a / a asymmetry parameter t s

  49. Su2 Yang-Mills eos on the lattice with Euler-Gamma distributed inverse temperature: Effective action method with Zsolt Schram (work in progress) preliminary

  50. Method: EulerGamma * Metropolis • asymmetry thrown from Euler-Gamma • at each Monte Carlo step / only after a while • at each link update / only for the whole lattice • meaning local / global fluctuation in space • c = 1024 for checking usual su2 • c = 5.5 for genuine quark matter

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