Towards a superstatistical SU2 Yang-Mills eos Tamás S. Bíró (KFKI RMKI Budapest / BME) and Zsolt Schram (DTP University of Debrecen) 1. Superstatistics: Euler-Gamma T 2. Monte Carlo with random spacing 3. Ideal gas limit, effective action 4. First numerical results towards SU2 eos Dense Matter 2010, , Stellenbosch, South-Africa

Entropy formulas, distributions

Laws of thermodynamics 0. Equilibrium  temperature ; entanglement 1.T dY(S) = dX(E) + p dU(V) - µ dZ(N) 2. dS ≥ 0 3. S = 0 at T = 0 4. thermodynamical limit: associative composition rule

Example: Gibbs-Boltzmann

Example: Tsallis

Example: Kaniadakis

Compisition in small steps: asymptotic rule

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

Superstatistics

a.Kinetic simulation (NEBE) b.Monte Carlo simulation c.Superstatistics: effective partition function

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

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

Gamma deviate random spacing asymmetry  A  = DU dt w (t) e t A(U) ∫ ∫ -S(t,U) c DU dt w (t) e ∫ ∫ -S(t,U) c v Expectation values of observables: t = a / a = T / T asymmetry parameter ts Action: S(t,U) = a(U) t + b(U) / t av

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

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

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

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

Test of Gamma deviates

Lattice spacing asymmetry

Asymmetry parameter for c = 5.5

Euler-Gamma random deviates statistics

Equipartition of action

Compare action equipartition

Electric / Magnetic ratio

Random deviate spacing per link update

Action difference at c = 1024

Action difference at several c

Zsolt Schram, Debrecen

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

Action sum at c = 1024

Action sum at several c-s

Wilson Loops at c = 1024

Creutz Ratios at c = 1024

Re Polyakov Loop

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

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

hcbm.kfki.hu Aug Hot and Cold Baryonic Matter