Towards a superstatistical SU2 Yang-Mills eos

1. Towards a superstatistical SU2 Yang-Mills eos • Superstatistics: Euler-Gamma T • Monte Carlo with random spacing • Ideal gas limit, effective action • First numerical results towards SU2 eos Tamás S. Bíró (KFKI RMKI Budapest / BME) and Zsolt Schram (DTP University of Debrecen) Dense Matter 2010, 5-10. 04. 2010, Stellenbosch, South-Africa

2. Entropy formulas, distributions

3. Laws of thermodynamics • 0. Equilibrium  temperature ; entanglement • T dY(S) = dX(E) + p dU(V) - µ dZ(N) • dS ≥ 0 • S = 0 at T = 0 • thermodynamical limit: • associative composition rule

4. Example: Gibbs-Boltzmann

5. Example: Tsallis

6. Compisition in small steps: asymptotic rule

7. 3. Possible causes for non-additivity • Long range interaction  energy not add. • Long range correlation  entropy not add. • Example: kinetic energy composition rule for massless partons with E - dependent interaction Our view to the forest is blocked by single trees

8. Superstatistics

9. Superstatistics • Kinetic simulation (NEBE) • Monte Carlo simulation • Superstatistics: effective partition function

10. q = 1 + 1 / c POWER _LAW TAILED canonical distribution This equals to Euler-Gamma distributed Gibbs factors: Interpretations: fluctuating temperature, (Wilk-Wlodarczyk) energy imbalance, (Rafelski) multiplicative + additive noise, (Tsallis, Biró-Jakovác) finite step CLT (Beck – Cohen)

11. Gamma distribution max: 1 – 1/c, mean: 1, spread: 1 / √ c

12. Gamma deviate random spacing asymmetry 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 = T / T asymmetry parameter av t s

13. 1. Effective action method Effective action calculation: -S (U,v) DU e A(U) ∫ eff A = -S (U,0) ∫ DU e eff v=0: Polyakov line, v=1: ss Plaquettes, v=-1: ts Plaquettes

14. Lattice theory: effective action ∞ c c+v-1 -(a+c)t - b/t c ∫ S = dt t e - ln G(c) eff 0 Plaquette sums: space-space: a = ∑ (1 – Re tr P ss) space-time: b = ∑ (1 – Re tr P ts) Evaluation methods: • exact analytical • saddle point • numerical (Gauss-Laguerre)

15. Lattice theory: effective action (c+v)/2 c c b ( ( ) ) S = 2K (2  b(a+c) ) - ln a+c c+v eff G(c) Asymptotics: • large a,b finite c: 2  ab • large a,b,c and a-b<< (a+b): a + b

16. 2. Numerical approach Euler Gamma distribution Near to standard: c = 1024.0 Smaller values of c (13.5, 5.5) Asymmetry parameter in MC Action difference and sum -> eos Other quantities

18. Lattice spacing asymmetry

19. Asymmetry parameter for c = 5.5

21. Euler-Gamma random deviates statistics

22. Equipartition of action

23. Compare action equipartition

24. Electric / Magnetic ratio

25. Random deviate spacing per link update

26. Action difference at c = 1024

27. Action difference at several c

28. Ideal Tsallis-Bose gas For c = 5.5 we have 1 / a = 4.5 and e ≈ 4 e_0

29. Action sum at c = 1024

30. Action sum at several c-s

31. Summary Composition rule  entropy formula Power-law (not exponential) Superstatistics (Euler-Gamma) Tsallis-Bose id.gas eos (SB const.) Towards SU2 YM Monte Carlo eos: RND asymmetry equipartition interaction measure

32. hcbm.kfki.hu Aug. 15-20. 2010 Hot and Cold Baryonic Matter