Sin t

1
BUSSE-HEIKES MODEL
THEORETICAL MODEL
? ? ? ?potential dynamics
?
nonrelaxational potential flow
? ? 0
otherwise ? nonpotential
2
BUSSE-HEIKES MODEL
ZERO-DIMENSIONAL SYSTEMS
?
3
BUSSE-HEIKES MODEL
Stationary Solutions
4
BUSSE-HEIKES MODEL
The Case ? ? ?
Residual dynamics
Relaxational part
After t O(1) ? x(t) ? 1 (asymptotic dynamics)
5
BUSSE-HEIKES MODEL
Hamiltonian dynamics
E ? 0
6
BUSSE-HEIKES MODEL
? 0, ? 1.3
7
BUSSE-HEIKES MODEL
The case ? ? ?
? 0.1, ? 1.3
8
BUSSE-HEIKES MODEL
? ?1.3, ?0.01 ? ?3, ?0.01 ? ?1.3, ?0.1 ?
?3, ?0.1
lines with slope 6?/?
9
BUSSE-HEIKES MODEL
Busse-Heikes Model with Noise
White noise processes ??i(t)?j(t? )?2??(t-t?
)?ij
E(t) ? ?E? ? T(t) ? ?T? T(?E?)
10
BUSSE-HEIKES MODEL
? ?1.3, ?0.01 ? ?3, ?0.01 ? ?1.3, ?0.1 ?
?3, ?0.1
? 0.1, ? 1.3, ? 106
? 102 106
11
BUSSE-HEIKES MODEL
? small
? ?1.3, ?0.01 ? ?3, ?0.01 ? ?1.3, ?0.1 ?
?3, ?0.1
Saddle-point type integration ?E? ? (?/?)2, ? ?
0, ? small
? 102 106
12
BUSSE-HEIKES MODEL
ONE-DIMENSIONAL SYSTEMS
13
BUSSE-HEIKES MODEL
Dynamics of an Isolated Kink
v(?,?)h(?) ? O(?2)
Simulation
Perturbative approach
v
14
BUSSE-HEIKES MODEL
Multikink Configurations
?tL(t) ? 2v(?,?) ? exp(?1/2L(t))
L(t) t
L(t) log t
15
BUSSE-HEIKES MODEL
Domain Growth and Dynamical Scaling
L(t) log t crossover t
crossover
L(t) t
L(t) log t
? 3.5, ?103
16
BUSSE-HEIKES MODEL
TWO-DIMENSIONAL SYSTEMS
POTENTIAL
NONPOTENTIAL
?
nonpotential dynamics formation of vertex points
? 3, ? 2
? 3, ? 0
17
BUSSE-HEIKES MODEL
Spiral dynamics
?(?,?) ? vp1/2 ?03/2
18
BUSSE-HEIKES MODEL
dc ? 2.24/?0 ? ??1
Consequence coarsening will occur for system
sizes S ? dc
19
BUSSE-HEIKES MODEL
Vertex Motion
- Even number of vertices half and half
- Vertices disappear by pairs of opposite sence
of rotation
Periodic BC
- Correlated motions are observed
20
BUSSE-HEIKES MODEL
Null BC
Periodic BC
21
BUSSE-HEIKES MODEL
Domain Growth and Dynamical Scaling
? 0 (potential limit) ? L(t) ? t1/2 dynamical
scaling (3 fields)
? ? 0 (nonpotential limit) ? L(t) ? t1/2
dynamical scaling (2 fields)
22
BUSSE-HEIKES MODEL
Spatial-dependent Terms
23
BUSSE-HEIKES MODEL
? small
?2
(êj?)2
24
BUSSE-HEIKES MODEL
? large
?2
(êj?)2