growth driven dynamics in mean-field models of interacting spins
TRANSCRIPT
Growth driven dynamics in mean-field models of interacting spins
Richard G. Morris* and Tim Rogers†
* Theoretical Physics, The University of Warwick, Coventry, UK
† Mathematics, The University of Bath, Bath, UK
Growth
How do physicists model growth?
kinetics is connected with the dynamical evolution of the system from the initialdisordered state to the ®nal equilibrium state.
Part of the fascination of the ®eld, and the reason why it remains a challengemore than three decades after the ®rst theoretical papers appeared, is that, in thethermodynamic limit, ®nal equilibrium is never achieved! This is because the longestrelaxation time diverges with the system size in the ordered phase, re¯ecting thebroken ergodicity. Instead, a network of domains of the equilibrium phasesdevelops, and the typical length scale associated with these domains increases withtime t. This situation is illustrated in ®gure 2, which shows a Monte Carlo simulationof a two-dimensional Ising model, quenched from TI ˆ 1 to TF ˆ 0. Inspection ofthe time sequence may persuade the reader that domain growth is a scalingphenomenon; the domain patterns at later times look statistically similar to thoseat earlier times, apart from a global change of scale. This `dynamic scalinghypothesis’ will be formalized below.
For pedagogical reasons, we have introduced domain growth in the context ofthe Ising model and shall continue to use magnetic language for simplicity. A relatedphenomenon that has been studied for many decades, however, by metallurgists, is
A. J. Bray484
Figure 2. Monte Carlo simulation of domain growth in the d ˆ 2 Ising model at T ˆ 0(taken from Kissner [8]). The system size is 256 £ 256, and the snapshots correspondto 5, 15, 60 and 200 Monte Carlo steps per spin after a quench from T ˆ 1.
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Kissner J. G., Ph. D., The University of Manchester (1992)
What if growth cannot be separated from relaxation?
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What if growth cannot be separated from relaxation?
Mean-field models
Mean-field modelsNon-growing
• Spins:
Mean-field models
�1, . . . ,�N 2 {�1,+1}
Non-growing
• Spins:
• Characterised by one variable:
Mean-field models
�1, . . . ,�N 2 {�1,+1}
Non-growing
x =1
N
NX
i=1
�i
• Spins:
• Characterised by one variable:
• Rate of flipping individuals spin is independent of the system size.
Mean-field models
�1, . . . ,�N 2 {�1,+1}
Non-growing
x =1
N
NX
i=1
�i
• Spins:
• Characterised by one variable:
• Rate of flipping individuals spin is independent of the system size.
• Stochastic evolution of jump size
Mean-field models
�1, . . . ,�N 2 {�1,+1}
O
✓1
N
◆
Non-growing
x =1
N
NX
i=1
�i
Mean-field modelsGrowing
• Spins:
Mean-field modelsGrowing
�1, . . . , �N(t) 2 {�1,+1}
• Spins:
• Characterised by two variables: or
Mean-field modelsGrowing
�1, . . . , �N(t) 2 {�1,+1}
(x, s) (u, v)
• Spins:
• Characterised by two variables: or
• Rate of flipping individuals spin is still independent of the system size.
Mean-field modelsGrowing
�1, . . . , �N(t) 2 {�1,+1}
(x, s) (u, v)
• Spins:
• Characterised by two variables: or
• Rate of flipping individuals spin is still independent of the system size.
• Asymptotics now defined in terms of i.e., a lower bound on
Mean-field modelsGrowing
�1, . . . , �N(t) 2 {�1,+1}
N0 ⌘ N (t = 0)N
(x, s) (u, v)
Mean-field modelsGrowth mechanism
Mean-field modelsGrowth mechanism
• Spins added from a reservoir at a rate: �
Mean-field modelsGrowth mechanism
• Spins added from a reservoir at a rate:
• Reservoir can be magnetised:
�
gu, gv
Mean-field modelsGrowth mechanism
• Spins added from a reservoir at a rate:
• Reservoir can be magnetised:
• Rate of flipping individual spins:
�
gu, gv
fu, fv
Mean-field modelsGrowth mechanism
• Spins added from a reservoir at a rate:
• Reservoir can be magnetised:
• Rate of flipping individual spins:
• Both and may depend on
�
gu, gv
fu, fv
gu, gv xfu, fv
Mean-field modelsGrowth mechanism
• Spins added from a reservoir at a rate:
• Reservoir can be magnetised:
• Rate of flipping individual spins:
• Both and may depend on
• Effects of growth are subordinate to flips but still macroscopic…
�
gu, gv
fu, fv
gu, gv xfu, fv
� ⇠ N↵, ↵ 2 (�1,+1)
Mean-field modelsFormal description
Mean-field modelsFormal description
d
dtP (u, v, t) =
N0
⇣E�1/N0u E+1/N0
v � 1⌘vfv
+N0
⇣E+1/N0u E�1/N0
v � 1⌘ufu
+⇣E�1/N0u � 1
⌘�gu
+⇣E�1/N0v � 1
⌘�gv
�P (u, v, t)
Mean-field modelsFormal description
E�1/N0u = 1� 1
N0
@
@u+
1
2N20
@2
@u2+O
�1/N3
0
�
Mean-field modelsFormal description
@
@tP (u, v, t) =�
✓@
@u� @
@v
◆vfvP (u, v, t)
+
✓@
@u� @
@v
◆ufuP (u, v, t)
� 1
N0
✓@
@ugu +
@
@vgv
◆�P (u, v, t)
+1
2N0
✓@
@u� @
@v
◆2
(vfv + ufu)P (u, v, t)
+O�1/N3
0
�
Mean-field modelsFormal description
du
dt= (vfv � ufu) +
�
N0gu +
rufu + vfv
N0⌘(t),
dv
dt= (ufu � vfv) +
�
N0gv �
rufu + vfv
N0⌘(t)
Mean-field modelsFormal description
ds
dt=
�
sN0
dx
dt=(1� x)fv � (1 + x)fu
+�
sN0(gu � gv � x)
+
s
2(1� x)fv + (1 + x)fu
sN0⌘(t) .
The Voter model
The Voter modelNon-growing
• “Pick a neighbouring spin and copy it…”
The Voter modelNon-growing
• “Pick a neighbouring spin and copy it…”
The Voter modelNon-growing
fu = (1� x)/2, fv = (1 + x)/2
• “Pick a neighbouring spin and copy it…”
The Voter modelNon-growing
dx
d⌧=
p2(1� x
2) ⌘(⌧)
fu = (1� x)/2, fv = (1 + x)/2
The Voter modelNon-growing
dx
d⌧=
p2(1� x
2) ⌘(⌧)
t
x
0 50 100 150 200
−0.2
0
0.2
τ
x
0 0.5 1 1.5 2
−0.2
0
0.2
The Voter modelNon-growing
dx
d⌧=
p2(1� x
2) ⌘(⌧)
t
x
0 50 100 150 200
−0.2
0
0.2
τ
x
0 0.5 1 1.5 2
−0.2
0
0.2
=) P1(x) = [�(x� 1) + �(x+ 1)] /2
The Voter modelConstant growth
• Choose
The Voter modelConstant growth
� = 1
• Choose
• Zero net reservoir-magnetisation:
The Voter modelConstant growth
� = 1
gu = gv = 1/2
• Choose
• Zero net reservoir-magnetisation:
The Voter modelConstant growth
� = 1
gu = gv = 1/2
ds
dt=
�
sN0=) s = 1 + t/N0
dx
dt= � x
sN0+
s2(1� x
2)
sN0⌘(t)
t
x
0 50 100 150 200
−0.2
0
0.2
τ
x
0 0.5 1 1.5 2
−0.2
0
0.2
The Voter modelConstant growth
ds
dt=
�
sN0=) s = 1 + t/N0
dx
dt= � x
sN0+
s2(1� x
2)
sN0⌘(t)
The Voter modelConstant growth
ds
dt=
�
sN0=) s = 1 + t/N0
dx
dt= � x
sN0+
s2(1� x
2)
sN0⌘(t)
d⌧
dt=
1
sN0=) dx
d⌧= �x+
p2(1� x
2)⌘(⌧)
The Voter modelConstant growth
ds
dt=
�
sN0=) s = 1 + t/N0
dx
dt= � x
sN0+
s2(1� x
2)
sN0⌘(t)
d⌧
dt=
1
sN0=) dx
d⌧= �x+
p2(1� x
2)⌘(⌧)
• “Noise-induced bi-stability!”
The Voter modelConstant growth cont’d
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
P (x, t) =✓3
�sin�1(x), (N0 + t)�4
�
⇡
p1� x
2
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
P1(x) = ⇡
�1(1� x
2)�1/2
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
P (T ) =1
2⇡(N0 + T )✓01
✓0,
N0
N0 + T
◆P1(x) = ⇡
�1(1� x
2)�1/2
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
P1(x) = ⇡
�1(1� x
2)�1/2
P (T ) ⇠ T�5/4
• Solve
The Voter modelConstant growth cont’d
dx/d⌧ = �x+p
2(1� x
2)⌘(⌧)
P (T ) ⇠ T�5/4
T
P (T)
100
101
102
103
104
105
10−8
10−6
10−4
10−2
100
The Glauber-Ising model
The Glauber-Ising modelNon-growing
The Glauber-Ising modelNon-growing
P ({�}) = e��H[{�}]/Z
The Glauber-Ising modelNon-growing
Z =X
x
e��H[{�}]
P ({�}) = e��H[{�}]/Z
The Glauber-Ising modelNon-growing
Z =X
x
e��H[{�}]
P ({�}) = e��H[{�}]/Z
H [{�}] = � J
N
NX
hi,ji
�i�j � hNX
i=1
�i
The Glauber-Ising modelNon-growing
P (x) = e
��H(x)/Z
Z =X
x
e��H(x)
H (x) = �J
2
�Nx
2 � 1��Nhx
The Glauber-Ising modelNon-growing
P (x) = e
��H(x)/Z
Z =X
x
e��H(x)
H (x) = �J
2
�Nx
2 � 1��Nhx
fu/d =1
2[1⌥ tanh� (Jx+ h)]
The Glauber-Ising modelNon-growing
The Glauber-Ising modelNon-growing
dx
dt= ��0(x) +
rD(x)
N
⌘(t)
The Glauber-Ising modelNon-growing
dx
dt= ��0(x) +
rD(x)
N
⌘(t)
�(x) =
1
2
x
2 � 1
�J
log cosh� (Jx+ h)
D(x) = 1� x tanh�(Jx+ h)
The Glauber-Ising modelGrowing
The Glauber-Ising modelGrowing
• Rescaling of time no-longer works due to deterministic “drift” terms:
The Glauber-Ising modelGrowing
• Rescaling of time no-longer works due to deterministic “drift” terms:
dx
dt= tanh� (Jx+ h)� x+O (1/N0)
The Glauber-Ising modelGrowing
• Rescaling of time no-longer works due to deterministic “drift” terms:
• Separation of of timescales implies instantaneous size-dependent escape rate:
dx
dt= tanh� (Jx+ h)� x+O (1/N0)
The Glauber-Ising modelGrowing
• Rescaling of time no-longer works due to deterministic “drift” terms:
• Separation of of timescales implies instantaneous size-dependent escape rate:
dx
dt= tanh� (Jx+ h)� x+O (1/N0)
(s) =
s�
00(x0)|�00
(x1)|D(x0)
4⇡
2D(x1)
exp
⇢�s
Zx1
x0
�
0(⇠)
D(⇠)
d⇠
�
The Glauber-Ising modelGrowing cont’d
The Glauber-Ising modelGrowing cont’d
• Survivor function:
The Glauber-Ising modelGrowing cont’d
• Survivor function:
P (T � t) = exp
⇢�Z t
0[s(t0)] dt0
�
The Glauber-Ising modelGrowing cont’d
• Survivor function:
• Constant growth:
P (T � t) = exp
⇢�Z t
0[s(t0)] dt0
�
The Glauber-Ising modelGrowing cont’d
• Survivor function:
• Constant growth:
P (T � t) = exp
⇢�Z t
0[s(t0)] dt0
�
P (T � t) = exp
⇢AN0
B�eB
⇣1� eB�t/N0
⌘�
A = (0), B = 0(0)/(0) < 0
The Glauber-Ising modelGrowing cont’d
• Survivor function:
• Constant growth:
P (T � t) = exp
⇢�Z t
0[s(t0)] dt0
�
A = (0), B = 0(0)/(0) < 0
P (T � 1) > 0
The Glauber-Ising modelGrowing cont’d
t
P(T
>t)
0 1 2 3 4 5 6 7 8 9 10
x 104
0
0.2
0.4
0.6
0.8
1
…back to the voter model
…back to the voter model
…back to the voter model
• What if …?� = N↵
…back to the voter model
• What if …?
• Rescale time:
� = N↵
⌧ =1
↵
n
N↵0 �
⇥
(1� ↵) t+N↵�10
⇤
↵↵�1
o
…back to the voter model
• What if …?
• Rescale time:
� = N↵
⌧ =1
↵
n
N↵0 �
⇥
(1� ↵) t+N↵�10
⇤
↵↵�1
o
⌧⇤ = limt!1
⌧ =N↵
0
↵
‘Freezing’ in the voter model
‘Freezing’ in the voter model
• Growing disc: � =pN
‘Freezing’ in the voter model
• Growing disc:
• Replication:
� =pN
gu = u = (1 + x)/2, gv = v = (1� x)/2
‘Freezing’ in the voter model
• Growing disc:
• Replication:
� =pN
gu = u = (1 + x)/2, gv = v = (1� x)/2
⌧⇤ =p
N0/2
dx
d⌧=
p2 (1� x
2)⌘(⌧)
‘Freezing’ in the voter model
t
x
0 50 100 150 200
−0.2
0
0.2
τ
x
0 0.5 1 1.5 2
−0.2
0
0.2
x
CDF(x)
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
−0.05 0 0.050.45
0.5
0.55
‘Freezing’ in the voter model
Wrap-up
Wrap-up
Wrap-up
• Growth affects the dynamics of even the simplest spin-models.
Wrap-up
• Growth affects the dynamics of even the simplest spin-models.
• Seems pretty cool… absorbing/meta-stable ‘switch’.
Wrap-up
• Growth affects the dynamics of even the simplest spin-models.
• Seems pretty cool… absorbing/meta-stable ‘switch’.
• Going forward: what can we say about spatial models?
Wrap-up
• Growth affects the dynamics of even the simplest spin-models.
• Seems pretty cool… absorbing/meta-stable ‘switch’.
• Going forward: what can we say about spatial models?
• What do you think / can you help?