SU2 Yang-Mills eos with fluctuating Temperature
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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
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