chaotic behavior and entropy production in yang-mills theory

1

Chaotic behavior and entropy production in Yang-Mills theory

---use of distribution functions in semiclassical approximation ----

Teiji Kunihiro (Kyoto U)

based on works in collaboration with

H Iida (Kyoto U) H Tsukiji A Ohnishi (YITP) TTTakahashi (Gunma CT)

B Muumlller (DukeBNL) A Schaumlfer (Regensburg)

YITP workshop on

ldquoNumerical approaches to the holographic principle quantum gravity and cosmologyrdquo

July 21-24 2015 YITP Kyoto Japan

Contents bull IntroductionMotivation bull Chaotic behavior of classical Yang-Mills system bull Chaoticity Lyapunov exponent KS entropy rate Boltzmann-Gibbs entropy in classicaldiscrete systems bull Toward coarse graining in quantum mechanical systems Wigner and Husimi function Husimi-Wehrl entropy

bull Simple quantum mechanical systems whose classical systems are chaotic

Numerical methods

bull Time evolution of H-W entropy of Y-M system

bull Summary

CGC Color glass condensate a QCD matter dominated by saturated gluons which can be treated as a classical field in a good approximation (L McLerran and R Venugopalan) Glasma the QCD matter just after the collision the soft part of which is created by CGC as the source and may be treated in the semiclassical aprox (C Lappi and L McLerran)

Huge entropy must be produced before the QGP formation Thermalization time ~ (05-20) fmc Fluc-induced Instability Entropy of chaotic YM classical field

4

Classical Yang-Mills Field Yang-Mills action CYM Hamiltonian in temporal gauge (A0=0)

TK BMueller A Ohnishi A Schaefer TT Takahashi AYamamoto PRD82 (2010)

5

Chaotic behavior of classical Yang-Mills fields TK BMueller A Ohnishi A Schaefer TT Takahashi AYamamoto PRD82 (2010)

Distance between the adjacent fields

One can see an exponential growth of the distance reminiscent of chaos

Essential role of Initial fluctuations for the creation of chaotic behavior H Iida TK B Muller AOhnishi ASchafer and TTTakahashi PRD 88 (2013)

Lyapunov exponents

How many are there positive Lyapunov exprsquos λ i gt0

7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM

Chaoticity in CYM T S Biro S G Matinyan B Muller Lect Notes Phys 56 (94) 1 S G Matinyan E B Prokhorenko G K Savvidy JETP Lett 44 (86) 138 NPB 298 (88) 414 B Muller A Trayanov PRL 68 (92) 3387 T S Biro C Gong B Muller PRD 52 (95)1260 C Gong PRD 49 (94) 2642

Exponential growth of distance from adjacent init cond Rapid spread of positive Lyapunov exponents

Kunihiro Muumlller AO Schaumlfer Takahashi Yamamoto (10) Iida Kunihiro BMuumlller AOhnishi Schaumlfer Takahashi (13)

Instability

spread exp growth

Typical Lyapunov spectrum

Sum of all Lyapunov exponent = 0 (Liouville theorem) 13 Positive 13 negative and 13 zero (or pure imag)

13 of DOF = gauge DOF

Iida Kunihiro BMuumlller AOhnishi Schaumlfer Takahashi (13)

total DOF=83x3x3x2 =9216

10

Coarse graining rarr Entropy

exp growth complexity of phase space dist

Chaoticity and Entropy

Kolmogorov-Sinai entropy rate (Krylov 1950 Kolmogorov 1959 Sinai1962)

Sensitivity to initial conditions

( ) (0)iti iX t e Xλδ δ= λ i gt0 positive Lyapunov exponent

Mixing and Information loss

= a partition of the phase space 1T minus backward time-evolution operator

Pesin theorem 0i

KS ihλ

λgt

= sum

Q

Y Pesin Russ Math Surv 32 (1977)55

What entropy

V Latora and MBaranger PRL (rsquo99) M Baranger V Latora and ARapisarda Chaos Soliton Fractals (2002)

A generalized Cat map

The slope of the linear rise coincides with the KS entropy hKS=248157096 069 calculated from the positive Lyapunov exponent

cell( ) ( ) log ( )i i

iS t p t p t= minussum

Coarse-grained Boltzmann-Gibbs entropy pi(t) The prob that the state of the system falls incide the cell ci of the phase space at t

248 069

Lyapunov exp ＝ｈKS

Chaos Entropy production

How about in Quantum Mechanics

An essential role of the coarse graining (averaging of orbits) Notice

How implement a coarse graining in Quantum Mechanics

We have seen for a map that

Which is true for other (cotinuous) classical sytems

13

Entropy production in quantum systems Entropy in quantum mech

Time evolution is unitary then the von Neumann entropy is const

Two ways of entropy production at the quantum level Entanglement entropy Partial trace over environment rarr Loss of info rarr entropy production Coarse grained entropy Coarse graining rarr entropy production Yes we can define it even in isolated systems such as HIC and early univ

Distribution function in Quantum Mechanics

The Wigner function

Caution It is a mere (Weyl) transformation of the density matrix a pure QM object and can be negative hence no ability of describing entropy production

The need of incorporation of coarse graining which inevitably enters through the observation process

12πInverse tr

Weyl-Wigner tr of op Ex

a product of oprsquos Moyal prod

Husimi function

We consider Gaussian smeared Wigner function which leads to Husimi function

Husimi function [Husimi(1940)]

More generally it is written in terms of a coherent state

Husimi function is semi-positive definite and is considered as a quantum distribution function

For the pure state

Coarse-grained within the amount consistent with the uncertainty principle of QM

Time evolution

Semi-classical approximation

With canonical EOM

Vanishes for HO

Gauss smearing

An alternative way (we do not take) Cf H-M Tsai B Muller PhysRev E85 (2012) 011110

17

Husimi Function A simple example with instability Inverted Harmonic Oscillator

exponential growth shrink Wigner function Husimi function

Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )

B Muller A Schaefer A Ohnishi and TK PTP 121(2008)555

ICQuantum dist

18

Husimi-Wehrl Entropy (1) Husimi-Wehrl entropy = Wehrl entropy using Husimi function Wehrl (78) Husimi (40) Anderson Halliwell (93) Kunihiro Muller Ohnishi Schafer (09)

Coarse grained entropy by minimum wave packet

Harmonic oscillator in equilibrium Min value SHW=1 (1 dim) from smearing Lieb (78) Wehrl (79)

Husimi-Wehrl = von Neumann at high T (T ℏω gtgt 1) Anderson Halliwell (93) Kunihiro Muller AO Schafer (09)

Husimi-Wehrl

von Neumann

19

Husimi-Wehrl Entropy (2) Inverted Harmonic Oscillator Many Harmonic amp Inverted Harmonic Oscillators

~ exp( λt ) λ=Lyapunov exp

independent of Δ

Classical unstable modes plays an essential role in entropy production at quantum level

bull The growth rate of the H-W entropy is given by the sum of the positive Lyapunov exponents (KS entropy) in the classical system bull Conversely KS entropy even gives the growth rate of the quantum entropy as given by H-W entropy

20

Husimi-Wehrl Entropy in Multi-Dimensions (1) Challenge Evolution of Husimi fn amp Multi-Dim integral Monte-Carlo + Semi-classical approx Tsukiji Iida Kunihiro Ohnishi Takahashi PTEP in press [arXiv150504698 [hep-ph]]

Two-step Monte-Carlo method Monte-Carlo integral + Liouville theorem [ fW(qpt)=fW(q0p0t=0) ] Test particle method Test particle evol + Monte-Carlo integral

21

Husimi-Wehrl Entropy in Multi-Dimensions (2)

Two-step Monte-Carlo integral Test particle method test particle evolution + MC integral

Outside MC rarr S Inside MC rarr fH

Tsukiji Iida Kunihiro Ohnishi Takahashi PTEP in press [arXiv150504698 [hep-ph]]

Liouville

22

ldquoYang-Millsrdquo Quantum Mechanics Yang-Mills quantum mechanics

Quartic interaction term rarr almost globally chaotic S G Matinyan G K Savvidy N G Ter-Arutunian Savvidy Sov Phys JETP 53 421 (1981) A Carnegie and I C Percival J Phys A Math Gen 17 801 (1984) P Dahlqvist and G Russberg Phys Rev Lett 65 2837 (1990)

23

Monte-Carlo + Semi-Classical Approx

Semi-Classical + MC methods reproduce mesh integral values of SHW Two-step MC results converge from above Test particle + MC results

converge from below

Time Time

SHW SHW Two-step MC Test particle + MC

Tsai-Muller

N (MC samples)

N(test particles)


H-M Tsai B Muller PhysRev E85 (2012) 011110 Test-particle method combined with moment applied directly to EOM of Husimi function up to ℏ2 corrections

Another QM system with a few degrees of freedom with a chaotic behavior in the classical limit Modified Quantum Y-M system

Poincare map

ε= 01 02

10

Integrable case

Evolution of Husimi-Wehrl entropy of Modified Quantum Y-M system

Two step MC Test-particle method

The two methods give the same results within the error bars

26

Husimi-Wehrl entropy of YM

Husimi-Wehrl entropy of CYM on the lattice

D=576 on 43 lattice for Nc=2 rarr 1152 dim integral average exponent ~ D (problem with large deviation )

Hartree approximation

Hartree approx gives error of 10-20 in HW entropy for 2d quantum mech

Tsukiji Iida Kunihiro Ohnishi Takahashi in progress

27

Check in the case of quantum mechanical systems

Product ansatz gives consistent results within 10 error bar

PA

HTsukiji HIida TK AOhnishi and TTTakahashi work in progress

28

Preliminary Results

Preliminary numerical results of SU(2) CYM on a 43 lattice Initial cond = min wave packet (gaussian) rarr SHW ~ 576 Entropy production is observed


Time (1 x 10-3 a)

29

Two step Monte-Carlo method

Test particle method

Many test particles

Lattice time Lattice time

Many Monte-Carlo samples Many

test particles


Summary 1 We have proposed to use Husimi function to describe isolated quantum systems so that an entropy (Husimi-Wehrl entropy) is defined 2 For quantum systems whose classical limit are chaotic or unstable the growth rate of their Husimi-Wehrl entropy is given by the KS entropy (the sum of positive Lyapunov exponents) in the classical system 3The classical Yang-Mills system shows chaotic behavior 4 To trigger the instability leading to the chaotic behavior the initial fluctuations as given by the initial quantum distribution is necessary 5 We have shown that the semiclassical approximation makes the numerical evaluation of the Husimi function and the H-W entropy feasible even for manybody systems including the QFT 6 We have shown that the entropy is created in the quantum Y-M theory which refelects the chaotic behavior in the classical limit

Future problems

bull Clarify the physical meaning of product ansatz

bull Calculate H-W entropy on a larger lattice

bull Case of expanding geometry

Back Ups

33

Conformal Property

No conformal anomaly in CYM rarr Any average quantity scales as εn4 (ε energy density n mass dim)

LLE temporally local ILE integral during exp growing period

Kunihiro Muumlller AO Schaumlfer Takahashi Yamamoto (10)

Sampling with Wigner function

t [2] (p+p q+p) Gauss (pq)

[3] (p q) Gauss (p+p-p q+q-q)

[1] (p(0) q(0)) from init dist

(p q)

(p(0) q(0)) rarr f(pq)

(p(0)q(0))

(pq)

Conjecture on entropy production at each stage

BMuller and A Schaefer Int J Mod Phys E20 2235 (2011)

RJ Fries et al arXiv 09065293

The separation of scales in the relativistic heavy-ion collisions

Liouville Boltzmann Fluid dyn Hamiltonian (Kinetic eq)

Slower dynamics

Navier-Stokes eq

（A reduction of the dynamical system)

Early thermalization Entropy production mechanism Possible origin Chaoticity of (classical) YM theory with the nonlinear coupling provided that some coarse graining is taken for granted But how coarse graining in QMQFT

Chaotic behavior and entropy production in Yang-Mills theory---use of distribution functions in semiclassical approximation ----

Contents

スライド番号 3

Classical Yang-Mills Field



Chaoticity of CYM






Entropy production in quantum systems


Husimi function


Husimi Function

Husimi-Wehrl Entropy (1)




ldquoYang-Millsrdquo Quantum Mechanics






Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





Contents bull IntroductionMotivation bull Chaotic behavior of classical Yang-Mills system bull Chaoticity Lyapunov exponent KS entropy rate Boltzmann-Gibbs entropy in classicaldiscrete systems bull Toward coarse graining in quantum mechanical systems Wigner and Husimi function Husimi-Wehrl entropy

bull Simple quantum mechanical systems whose classical systems are chaotic

Numerical methods

bull Time evolution of H-W entropy of Y-M system

bull Summary



4



5





Lyapunov exponents


7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM




Instability

spread exp growth






10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property







4



5





Lyapunov exponents


7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM




Instability

spread exp growth






10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





5





Lyapunov exponents


7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM




Instability

spread exp growth






10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property






Lyapunov exponents


7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM




Instability

spread exp growth






10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





7 Time Index

LEs (λ)

Const B background

Chaoticity of CYM




Instability

spread exp growth






10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property










10









Pesin theorem 0i

KS ihλ

λgt

= sum

Q


What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





What entropy







248 069








13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property











13





The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property






The Wigner function



12πInverse tr



Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





Husimi function





For the pure state


Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





Time evolution


With canonical EOM

Vanishes for HO

Gauss smearing


17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





17



Husimi

Wigner

t=0 t=2λ

growth

shrink

growth

finite

~ exp( λt )


ICQuantum dist

18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





18





Husimi-Wehrl

von Neumann

19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





19



independent of Δ



20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





20



21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





21





Liouville

22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





22



23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





23



converge from below

Time Time


Tsai-Muller

N (MC samples)

N(test particles)




Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property






Poincare map

ε= 01 02

10

Integrable case




26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property








26







27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





27



PA


28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





28

Preliminary Results



Time (1 x 10-3 a)

29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





29



Many test particles



test particles



Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property






Future problems




Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





Back Ups

33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





33

Conformal Property








(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property









(p q)


(p(0)q(0))

(pq)






Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property










Slower dynamics

Navier-Stokes eq




Contents





Chaoticity of CYM








Husimi function


Husimi Function











Preliminary Results


Summary

Future problems

Back Ups

Conformal Property





chaotic behavior and entropy production in yang-mills theory

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