1. The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s Repeated rules are associative Examples 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 gas crystal stringy liquid

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

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: L ̄ ¹ סּ ln

8. Thermodynamical limit: repeated rules

9. 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 h(x,y)L(x) L ̄ ¹(t) exp סּ L L ̄ ¹ סּ ln composition ruleformal logarithm formal exponential equilibrium distribution entropy formula Gibbs, Boltzmann

21. Rules and entropies h(x,y)L(x) L ̄ ¹(t) exp סּ L L ̄ ¹ סּ ln composition ruleformal logarithm formal exponential equilibrium distribution entropy formula Pareto, Tsallis

22. Rules and entropies h(x,y)L(x) L ̄ ¹(t) exp סּ L L ̄ ¹ סּ ln composition ruleformal logarithm formal exponential equilibrium distribution entropy formula Lévy

23. Rules and entropies h(x,y)L(x) L ̄ ¹(t) exp סּ L L ̄ ¹ סּ ln composition ruleformal logarithm formal exponential equilibrium distribution entropy formula 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 h(x,y)L(x) L ̄ ¹(t) exp סּ L L ̄ ¹ סּ ln Original composition rule Asymptotic formal logarithm formal exponential equilibrium distribution entropy formula 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.