SU2 Yang-Mills eos with fluctuating Temperature
description
Transcript of SU2 Yang-Mills eos with fluctuating Temperature
![Page 1: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/1.jpg)
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
![Page 2: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/2.jpg)
Entropy formulas, distributions
![Page 3: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/3.jpg)
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
![Page 4: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/4.jpg)
Example: Gibbs-Boltzmann
![Page 5: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/5.jpg)
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
![Page 6: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/6.jpg)
Compisition in small steps:
axyyxyxa
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
asymptotic
rule
![Page 7: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/7.jpg)
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
![Page 8: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/8.jpg)
Superstatistics
a. Kinetic simulation (NEBE)
b. Monte Carlo simulation
c. Superstatistics: effective partition
function
![Page 9: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/9.jpg)
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
![Page 10: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/10.jpg)
max: 1 – 1/c, mean: 1, spread: 1 / √ c
Gamma distribution
![Page 11: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/11.jpg)
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
![Page 12: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/12.jpg)
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
![Page 13: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/13.jpg)
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:
![Page 14: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/14.jpg)
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 =
![Page 15: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/15.jpg)
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
![Page 16: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/16.jpg)
![Page 17: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/17.jpg)
Test of Gamma deviates
![Page 18: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/18.jpg)
Lattice asymmetry
![Page 19: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/19.jpg)
Asymmetry parameter for c = 5.5
![Page 20: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/20.jpg)
![Page 21: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/21.jpg)
Euler-Gamma random deviates statistics
![Page 22: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/22.jpg)
Equipartition of action
![Page 23: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/23.jpg)
Compare action equipartition
![Page 24: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/24.jpg)
Electric / Magnetic ratio
![Page 25: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/25.jpg)
Random deviate spacing per link update
![Page 26: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/26.jpg)
Action difference at c = 1024
![Page 27: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/27.jpg)
Action difference at several c
![Page 28: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/28.jpg)
Zsol
t Sch
ram
, Deb
rece
n
![Page 29: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/29.jpg)
Ideal Tsallis-Bose gas
For c = 5.5 we have 1 / a = 4.5 and e ≈ 4 e_0
![Page 30: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/30.jpg)
Action sum at c = 1024
![Page 31: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/31.jpg)
Action sum at several c-s
![Page 32: SU2 Yang-Mills eos with fluctuating Temperature](https://reader035.fdocuments.in/reader035/viewer/2022062520/56815749550346895dc4ec7f/html5/thumbnails/32.jpg)
Composition rule entropy
Power-law not exponential
Superstatistics
Tsallis-Bose id.gas eos
SU2 YM Monte Carlo eos