SU2 Yang-Mills eos with fluctuating Temperature

Tamás S. Bíró (KFKI RMKI Budapest / ELTE)

and Zsolt Schram (DTP ATOMKI Debrecen)

1. Superstatistics: Euler-Gamma T

2. Monte Carlo with rnd. spacing

3. Ideal gas limit, effective action

4. Numerical results for SU2Non-Perturbative Methods in Quantum Field Theory, 10-12. 03. 2010 Hévíz, Hungary

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

ényi Rln1

1)(

Tsallis )(1

)1(1

),1ln(1

)(

1)0,(,),(

11

/

2

q

nona

aqa

non

a

eqa

fq

SL

ffa

S

aEZ

faxa

xL

axxhaxyyxyxh

Compisition in small steps:

axyyxyxa

ezL

axa

xL

axxGxh

xyGyxyxh

az

c

c

),(

1)(

)1ln(1

)(

1)0(1)0,(

)(),(

1

2

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

Superstatistics

a. Kinetic simulation (NEBE)

b. Monte Carlo simulation

c. Superstatistics: effective partition

function

Canonical distribution: POWER – LAW TAILED

f exp( - U / T ) = ( 1 + E / cT )

-(c+1)

Interpretations: fluctuating temperature, energy imbalance, multiplicative + additive noise,

. . .

( 1 + x / c ) = dt t e e -t -xt/c1(c+1)

-(c+1)

This equals to Gamma distributed Gibbs factors:

c

q = 1 + 1 / c

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

Gamma distribution

Fluctuating spacing

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 asymmetry parametert s

Action: S(t,U) = a(U) t + b(U) / t

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 ∞

cc

(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:

effc c

(c) - ln

• large a,b finite c: 2 ab • large a,b,c and a-b (a+b): a + b

( )ba+c( )

(c+v)/2

2K (2 b(a+c) )c+vS =

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

Test of Gamma deviates

Lattice 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

Zsol

t Sch

ram

, Deb

rece

n

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

Composition rule entropy

Power-law not exponential

Superstatistics

Tsallis-Bose id.gas eos

SU2 YM Monte Carlo eos