The thermodynamical limit of abstract composition rules

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.