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The thermodynamical limit of abstract composition rules

The thermodynamical limit of abstract composition rules. Non-extensive thermodynamics Composition rules and formal log-s Repeated rules are associative Examples. T. S. Bíró, KFKI RMKI Budapest, H. Talk by T. S. Biro at Varo š Rab, Dalmatia, Croatia, Sept. 1. 2008.

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The thermodynamical limit of abstract composition rules

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  1. The thermodynamical limit of abstract composition rules • Non-extensive thermodynamics • Composition rules and formal log-s • Repeated rules are associative • Examples T. S. Bíró, KFKI RMKI Budapest, H Talk by T. S. Biro at Varoš Rab, Dalmatia, Croatia, Sept. 1. 2008.

  2. From physics to composition rules Entropy is not a sum: correlations in the common probability Energy is not a sum: (long range) interaction inside the system Thermodynamical limit: extensive but not additive?

  3. Short / long range interaction vs. extensivity short range long range

  4. Short / long range correlation vs. extensivity short range long range

  5. Typical g( r ) functions liquid gas stringy crystal

  6. From physics to composition rules chaotic dynamics anomalous diffusion fractal phase space filling Lévy distributions multiplicative noise coupled stochasticequations superstatistics power-law tailed distributions extended logarithm and exponential Abstract composition rule h(x,y)

  7. From composition rules to physics Abstract composition rule h(x,y) h(x,0) = x, general rules associative (commutative) rules Formal logarithm L(x) equilibrium distribution: exp סּ L generalized entropy: L̄¹סּ ln

  8. Thermodynamical limit: repeated rules

  9. Thermodynamical limit: repeated rules N-fold composition

  10. Thermodynamical limit:repeated rules recursion

  11. Thermodynamical limit: repeated rules use the ‘ zero property ’ h(x,0) = x

  12. Thermodynamical limit: repeated rules The N   limit: scaling differential equation

  13. Thermodynamical limit: repeated rules solution: asymptotic formal logarithm Note: t / t_f = n / N finite ratio of infinite system sizes (parts’ numbers)

  14. Thermodynamical limit: repeated rules The asymptotic rule is given by Proof of associativity:

  15. Thermodynamical limit: associative rules are attractors If we began with h(x,y) associative, then it has a formal logarithm, F(x). Proportional formal logarithm  same composition rule!

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

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

  18. Entropy formulae from canonical equilibrium Equilibrium: q – exponential, entropy: q - logarithm All composition rules generate a non-extensive entropy formula in the th. limit

  19. Entropy formulae from canonical equilibrium Dual views: either additive or physical quantities Associative composition rules can be viewed as a canonical equilibrium

  20. Rules and entropies Gibbs, Boltzmann

  21. Rules and entropies Pareto, Tsallis

  22. Rules and entropies Lévy

  23. Rules and entropies Einstein

  24. Classification based on h’(x,0) • constant  addition  Gibbs distribution • linear  Tsallis rule  Pareto • pure quadratic  Einstein rule  rapidity • quadratic  combined Einstein-Tsallis • polynomial  multinomial rule  rational function of power laws

  25. Interaction and kinematics • Assume that the interaction energy can be expressed via the asymptotic, free individual energies. This gives an energy composition law as:

  26. Interaction and kinematics Let U depend on the relative momentum squared:

  27. Interaction and kinematics Average over the directions gives for the kinetic energy composition rule (with F’ = U)

  28. Interaction and kinematics In the extreme relativistic limit (K >> m) it gives

  29. Rule and asymptotic rule Pareto - Tsallis

  30. Interaction and kinematics In the non-relativistic limit (K << m) the angle averaged composition rule has the form:  non-trivial formal logarithm  non-additive entropy formula

  31. Summary • Extensive composition rules in the thermodynamical limit are associative and symmetric, they define a formal logarithm, L • The stationary distribution is exp o L, the entropy is related to L inverse o ln. • Extreme relativistic kinematics leads to the Pareto-Tsallis distribution.

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