SU2 Yang-Mills eos with fluctuating Temperature

1. SU2 Yang-Mills eos with fluctuating Temperature • Superstatistics: Euler-Gamma T • Monte Carlo with rnd. spacing • Ideal gas limit, effective action • Numerical results for SU2 Tamás S. Bíró (KFKI RMKI Budapest / ELTE) and Zsolt Schram (DTP ATOMKI Debrecen) Non-Perturbative Methods in Quantum Field Theory, 10-12. 03. 2010 Hévíz, Hungary

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

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

9. q = 1 + 1 / c Canonical distribution: POWER – LAW TAILED -(c+1) f  exp( - U / T ) = ( 1 + E / cT ) This equals to Gamma distributed Gibbs factors: -(c+1) c 1  -t -xt/c ( 1 + x / c ) = dt t e e (c+1) Interpretations: fluctuating temperature, energy imbalance, multiplicative + additive noise, . . .

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

11. Fluctuating spacing 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

12. 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

13. 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)

14. 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

15. Numerical results 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

17. Test of Gamma deviates

18. Lattice 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. Zsolt Schram, Debrecen

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

30. Action sum at c = 1024

31. Action sum at several c-s

32. Composition rule  entropy Power-law not exponential Superstatistics Tsallis-Bose id.gas eos SU2 YM Monte Carlo eos