universal behavior of critical dynamics far from equilibrium bo zheng physics department, zhejiang...
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Universal Behavior of
Critical Dynamics far from Equilibrium
Bo ZHENG
Physics Department, Zhejiang University
P. R. China
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Contents
I IntroductionII Short-time dynamic scalingIII Applications * second order transitions * Kosterlitz-Thouless transitions * disordered systems, spin glasses * weak first-order phase transitionsIV Deterministic dynamicsV Concluding remarks
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I Introduction
Many-body Systems
It is difficult to solve the Eqs. of motion
Statistical Physics
Equilibrium Ensemble theory
Non-equilibrium e .g. Langevin equations Monte Carlo dynamics
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,
/ i j ii j i
H kT K s s h s
Ising model
1is
/
{ }
/
{ }
C
C
0 T T
( ) T<T
1 1 1
{
i
i
H kTi id d
i s i
H kT
s
M s s eL Z L
Z e
M
M
Tc TTcTcT /)(
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Features of second order transitions
* Scaling form
It represents self-similarity are critical exponents
* Universality
Scaling functions and critical exponentsdepend only on symmetry and spatial dim.
Physical origin divergent correlation length, fluctuations
lawpowerbMbM 1
,
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Dynamics
Dynamic scaling form
z : the dynamic exponent .. Landau PRB36(87)567
PRB43(91)6006
Condition: t sufficiently largeOrigin : both correlation length and correlating time are divergent
mequilibriutSS ii 0
1,, btbMbtM Z
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II Short-time dynamic scaling
Is there universal scaling behavior inthe short-time regime? Recent answer: YES
Theory: renormalization group methods Janssen, Z.Phys. B73(89)539
Experiments: spin glasses “phase ordering” of KT systems …Simulation: Ising model, Stauffer (92), Ito (93)
Important:
* macroscopic short time * macroscopic initial conditions
mictt
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010 0( , , ) ( , , )xv Z vM t m b M b t b b m
: a “new” critical exponentOrigin: divergent correlating time
Dynamic process far from equilibrium e.g. t = 0 , T =
t > 0 , T = Tc
Langevin dynamics
Monte Carlo dynamics
Dynamic scaling form
0x
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Self-similarity in time direction, Ising model
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ztbm /10 small, ,0
z(xθt
mtMtmtM zxz
/)/ , ~
,1,
0
000
In most cases, Initial increase of the magnetization! Janssen (89) Zheng (98)
3D Ising model
0
.06 .04 .02 .00
.1014(5) .1035(4) .1059(20) .108(2)
0m
Grassberger (95), 0.104(3)
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3D Ising Model
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0 ,0 0 m
Auto-correlation
zdiid
ttSSL
tA / ~ 01
Even if contributes to dynamic behavior
Second moment
,0 0 m
C
iid
ttSL
tM
2
22 ~
1
Zdc 2
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3D Ising Model
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3D Ising Model
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Summary
* Short-time dynamic scaling a new exponent
* Scaling form ==> ==> static exponents Zheng, IJMPB (98), review
Li,Schuelke,Zheng, Phys.Rev.Lett. (95), Zheng, PRL(96)
Conceptually interesting and important Dynamic approach does not suffer from critical slowing down Compared with cluster algorithms, Wang-Landau … it applies to local dynamics
Z ,
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III Applications
* second order transitions
e.g. 2D Ising, 3D Ising, 2D Potts, … non-equilibrium kinetic models chaotic mappings 2D SU(2) lattice gauge theory, 3D SU(2) … Chiral degree of freedom of FFXY model … Ashkin-Teller model Parisi-Kardar-Zhang Eq. for growing interface
…
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2D FFXY model jji
iij SSKkTH
/,
-K K
Order parameter: Project of the spin configuration
on the ground state
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Initial state: ground state,
z / ~ | 1 /
02
2
dzd Lt
M
MU L finite
10 m
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2D FFXY model, Chiral degree of freedom
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2D FFXY model, Chiral degree of freedom
Initial state: random, smallm 0
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Tc Z 2β/ν ν
93 .454(3) .38(2) .80(5)
94 .454(2) .22(2) .813(5)
95 .452(1) 1
96 .451(1) .898(3)
OURS
(98)
.4545(2) .202(3) 2.17(4) .261(5) .81(2)
Ising .191(3) 2.165(10) .25 1
2D FFXY model, Chiral degree of freedom Luo,Schuelke,Zheng, PRL (98)
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* Kosterlitz-thouless transitions
e.g. 2D Clock model, 2D XY model, 2D FFXY model, … 2D Josephson junction array,… 2D Hard Disk model,…
Logarithmic corrections to the scaling Bray PRL(00)
Auto-correlation
zdttttA / 0 )]/ln(/[ ~
Second moment CttttM
02 )]/ln(/[ ~
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2D XY model, random initial state, Ying,Zheng et al PRE(01)
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* disordered systems
e.g. random-bond, random-field Ising model,… spin glasses
3D Spin glasses
Challenge: Scaling doesn’t hold for standard order parameter
Pseudo-magnetization: Project of the spin configuration on the ground state
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3D spin glasses, Luo,schulke,Zheng (99)
initial increase of the Pseudo-magnetization
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* weak first-order phase transitions
How to distinguish weak first order phase transitions from second order or KT phase transtions?
Non-equilibrium dynamic approach:
for a 2nd order transition: at Kc (~ 1/Tc) there is power law behavior
for a weak 1st order transition: at Kc there is NO power law behavior
However, there exist pseudo critical points!!
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disordered metastable state K* > Kc
M(0)=0
ordered metastable state K** < Kc
M(0)=1
For a 2nd order transition, K* = K**
2D q-state Potts model
zvdttM /)/2( )2( ~ )(
z / ~ )( ttM
ij
jiKH ,
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2D 7-state Potts model, heat-bath algorithm
M(0)=0
K*
Kc
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2D 7-state Potts model, heat-bath algorithm
M(0)=1
K**
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2D Potts models Schulke, Zheng, PRE(2000)
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IV Deterministic dynamics
Now it is NOT ‘statistical physics’
theory, isolated4
iiiiii mH
42222
!4
1
2
1
2
1
2
1
32
!3
12 iiiiii m
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22 tttttt iiii
Time discretization
Iteration up to very long time is difficultBehavior of the ordered initial state is not clear
Random initial state
01
00,,1,, mttMtmtM zxzz
ztFtmm 100 ~ ) (small
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Z
.176(7) 2.148(20) .24(3) .95(5)
Ising .191(1) 2.165(10) .25 1
4
2
Zheng, Trimper, Schulz, PRL (99)
Violation of the Lorentz invariance
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V Concluding remarks
* There exists universal scaling behavior in critical dynamics far from equilibrium -- initial conditions, systematic description
damage spreading glass dynamics phase ordering growing interface …
* Short-time dynamic approach to the equilibrium state predicting the future …