Transcript Slide 1
Lecture 11: Ising model
Outline:
• equilibrium theory
•d=1
• mean field theory, phase transition
• critical phenomena
• kinetics (Glauber model)
• critical dynamics
• continuum description: Landau-Ginzburg model
•Langevin dynamics
“spins”
binary variables Si = ±1 (or 0/1)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
i
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
i
(low energy is “favorable”)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
(chain)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
and d=∞ ( all-to-all connectivity)
(chain)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj:
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj: ferromagnetism
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E Jij Si S j hi Si
ij
i
12 Jij Si S j hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj: ferromagnetism
hi > 0: favour Si = +1.
equilibrium stat mech
Gibbs distribution P[S]
1
exp(E)
Z
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
1
Z exp
2 J ij Si S j hi Si
{S}
ij
i
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
partition
1
Z exp
2 J ij Si S j hi Si
function
{S}
ij
i
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
partition
1
Z exp
2 J ij Si S j hi Si
function
{S}
ij
i
free energy:
F T log Z
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
partition
1
Z exp
2 J ij Si S j hi Si
function
{S}
ij
i
the original Ising model:
nearest-neighbor interactions, J > 0, d = 1,
hi = 0
free energy:
F T log Z
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
partition
1
Z exp
2 J ij Si S j hi Si
function
{S}
ij
i
free energy:
F T log Z
the original Ising model:
nearest-neighbor interactions, J > 0, d = 1,
hi = 0 Z expJ Si Si1
i
{S}
equilibrium stat mech
1
1
1
Gibbs distribution P[S] exp(E) exp
J S S hi Si
2 ij i j
Z
Z
ij
i
partition
1
Z exp
2 J ij Si S j hi Si
function
{S}
ij
i
free energy:
F T log Z
the original Ising model:
nearest-neighbor interactions, J > 0, d = 1,
hi = 0 Z expJ Si Si1
i
{S}
can also have “3-spin interactions”, etc:
E 16 KijkSi S j Sk 12 Jij Si S j hi Si
ijk
ij
i
solving 1-d Ising model by decimation
Z expJ Si Si1 e JS 1 S2 e JS 2 S3 e JS 3S4 e JS 4 S5 L
i
S1S2
{S}
S3
S4
S5
solving 1-d Ising model by decimation
Z expJ Si Si1 e JS 1 S2 e JS 2 S3 e JS 3S4 e JS 4 S5 L
i
S1S2
{S}
S3
S4
S5
e JS i Si 1 1 JSi Si1 12 (J) 2 (Si Si1 ) 2 3!1 (J) 3 (Si Si1) 3
cosh(J) Si Si1 sinh(J)
cosh(J)1 Si Si1 tanh(J)
solving 1-d Ising model by decimation
Z expJ Si Si1 e JS 1 S2 e JS 2 S3 e JS 3S4 e JS 4 S5 L
i
S1S2
{S}
S3
S4
S5
e JS i Si 1 1 JSi Si1 12 (J) 2 (Si Si1 ) 2 3!1 (J) 3 (Si Si1) 3
cosh(J) Si Si1 sinh(J)
cosh(J)1 Si Si1 tanh(J)
Z coshN J 1 S1S2 tanh(J)1 S2S3 tanh(J)1 S3S4 tanh(J)L
S1 S2
S3
S4
solving 1-d Ising model by decimation
Z expJ Si Si1 e JS 1 S2 e JS 2 S3 e JS 3S4 e JS 4 S5 L
i
S1S2
{S}
S3
S4
S5
e JS i Si 1 1 JSi Si1 12 (J) 2 (Si Si1 ) 2 3!1 (J) 3 (Si Si1) 3
cosh(J) Si Si1 sinh(J)
cosh(J)1 Si Si1 tanh(J)
Z coshN J 1 S1S2 tanh(J)1 S2S3 tanh(J)1 S3S4 tanh(J)L
S1 S2
S3
S4
sum on every other spin:
(1 S
2
S
tanh
J)(1
S
S
tanh
J)
2(1
S
S
tanh
J)
i1 i
i i1
i1 i1
Si
solving 1-d Ising model by decimation
Z expJ Si Si1 e JS 1 S2 e JS 2 S3 e JS 3S4 e JS 4 S5 L
i
S1S2
{S}
S3
S4
S5
e JS i Si 1 1 JSi Si1 12 (J) 2 (Si Si1 ) 2 3!1 (J) 3 (Si Si1) 3
cosh(J) Si Si1 sinh(J)
cosh(J)1 Si Si1 tanh(J)
Z coshN J 1 S1S2 tanh(J)1 S2S3 tanh(J)1 S3S4 tanh(J)L
S1 S2
S3
S4
sum on every other spin:
(1 S
2
S
tanh
J)(1
S
S
tanh
J)
2(1
S
S
tanh
J)
i1 i
i i1
i1 i1
Si
But this is an interaction J’ between Si-1 and Si+1 with tanhJ tanh2 J
correlation function
Repeat (“renormalization group”):
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
2exp(JM ) 2exp(JM )
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
tanhJM
2exp(JM ) 2exp(J M )
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
Si Sin expn /
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
Si Sin expn / => correlation length
1
log tanh J
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
Si Sin expn / => correlation length
low T:
tanh J 1 2exp( 2J)
1
log tanh J
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
Si Sin expn / => correlation length
low T:
tanhJ 1 2exp(2J) 12 exp(2J)
1
log tanh J
correlation function
Repeat (“renormalization group”):
m
after m steps, tanhJm (tanhJ)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z expJM S1SN
S1SN
2exp(JM ) 2exp(JM )
N
tanhJM tanhJ
2exp(JM ) 2exp(JM )
Si Sin expn / => correlation length
low T:
1
log tanh J
tanhJ 1 2exp(2J) 12 exp(2J)
Correlation length grows toward ∞ at low T, but no ordering
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
J
Z exp
2N Si S j h Si
{S}
ij
i
d=∞
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
J
J
Z exp
2N Si S j h Si
exp2N Si h Si
{S}
ij
i
{S}
i
i
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
J
J
Z exp
2N Si S j h Si
exp2N Si h Si
{S}
ij
i
{S}
i
i
Z dmNm Si exp12 NJm2 Nhm
{S}
i
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
J
J
Z exp
2N Si S j h Si
exp2N Si h Si
{S}
ij
i
{S}
i
i
Z dmNm Si exp12 NJm2 Nhm
{S}
i
dy
Z dm
expyNm Si exp12 NJm 2 Nhm
2i
{S}
i
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
J
J
Z exp
2N Si S j h Si
exp2N Si h Si
{S}
ij
i
{S}
i
i
Z dmNm Si exp12 NJm2 Nhm
{S}
i
dy
Z dm
expyNm Si exp12 NJm 2 Nhm
2i
{S}
i
dy
Z dm
exp Nym log(2coshy) 12 Jm2 hm
2i
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
J
J
Z exp
2N Si S j h Si
exp2N Si h Si
{S}
ij
i
{S}
i
i
Z dmNm Si exp12 NJm2 Nhm
{S}
i
dy
Z dm
expyNm Si exp12 NJm 2 Nhm
2i
{S}
i
dy
Z dm
exp Nym log(2coshy) 12 Jm2 hm
2i
exponent prop to N => can use saddle point to evaluate Z
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
ym log(2coshy) 12 Jm 2 hm 0 y (Jm h)
m
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
ym log(2coshy) 12 Jm 2 hm 0 y (Jm h)
m
put them together:
m tanh (Jm h)
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
ym log(2coshy) 12 Jm 2 hm 0 y (Jm h)
m
put them together:
h = 0: m tanh Jm
m tanh (Jm h)
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
ym log(2coshy) 12 Jm 2 hm 0 y (Jm h)
m
put them together:
m tanh (Jm h)
h = 0: m tanh Jm has 2 solutions (i.e 2 saddle points)
for βJ > 1, i.e., T < J
saddle point equations
Z
dm
dy
exp Nym log(2coshy) 12 Jm2 hm
2i
ym log(2coshy) 12 Jm 2 hm 0 m tanhy
y
ym log(2coshy) 12 Jm 2 hm 0 y (Jm h)
m
put them together:
m tanh (Jm h)
h = 0: m tanh Jm has 2 solutions (i.e 2 saddle points)
for βJ > 1, i.e., T < J
solution m: spontaneous magnetization
intuition: heuristic approach
from
1
1
P[S] exp
J S S h Si
2 ij i j
Z
ij
i
intuition: heuristic approach
from
1
1
P[S] exp
J S S h Si
2 ij i j
Z
ij
i
total field on Si is
Hi h Jij S j
j
intuition: heuristic approach
from
1
1
P[S] exp
J S S h Si
2 ij i j
Z
ij
i
total field on Si is Hi h Jij S j
j
replace it by its mean: H h Jij S j h Jm
j
intuition: heuristic approach
from
1
1
P[S] exp
J S S h Si
2 ij i j
Z
ij
i
total field on Si is Hi h Jij S j
j
replace it by its mean: H h Jij S j h Jm
j
and calculate m as the average S of a single spin in field H:
intuition: heuristic approach
from
1
1
P[S] exp
J S S h Si
2 ij i j
Z
ij
i
total field on Si is Hi h Jij S j
j
replace it by its mean: H h Jij S j h Jm
j
and calculate m as the average S of a single spin in field H:
e H e H
m S H
tanh (h Jm)
H
e
e
broken symmetry
T > Tc
T = Tc
T < Tc
Near Tc
expand
m tanh(Jm h) Jm h 13 (Jm)3
Near Tc
expand
h = 0:
m tanh(Jm h) Jm h 13 (Jm)3
(J 1)m 13 (Jm)3
Near Tc
expand
h = 0:
m tanh(Jm h) Jm h 13 (Jm)3
(J 1)m 13 (Jm) 3 m
3(Tc T)
, Tc J
Tc
Near Tc
m tanh(Jm h) Jm h 13 (Jm)3
expand
(J 1)m 13 (Jm) 3 m
h = 0:
1st order in h:
m Jm h
3(Tc T)
, Tc J
Tc
Near Tc
m tanh(Jm h) Jm h 13 (Jm)3
expand
(J 1)m 13 (Jm) 3 m
h = 0:
1st order in h:
m Jm h m
h
T Tc
3(Tc T)
, Tc J
Tc
Near Tc
m tanh(Jm h) Jm h 13 (Jm)3
expand
(J 1)m 13 (Jm) 3 m
h = 0:
1st order in h:
m Jm h m
3(Tc T)
, Tc J
Tc
h
1
, i.e.,
T Tc
T Tc
Near Tc
m tanh(Jm h) Jm h 13 (Jm)3
expand
(J 1)m 13 (Jm) 3 m
h = 0:
1st order in h:
T = Tc:
h 13 (Jm)3 m h
m Jm h m
1
3
3(Tc T)
, Tc J
Tc
h
1
, i.e.,
T Tc
T Tc
Near Tc
m tanh(Jm h) Jm h 13 (Jm)3
expand
(J 1)m 13 (Jm) 3 m
h = 0:
1st order in h:
T = Tc:
m Jm h m
h 13 (Jm)3 m h
mean field critical behaviour
1
3
3(Tc T)
, Tc J
Tc
h
1
, i.e.,
T Tc
T Tc
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m (T Tc ) , (T Tc ) , m h
1
12 , 1, 3
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m (T Tc ) , (T Tc ) , m h
1
12 , 1, 3
d = 2: Onsager exact solution:
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m (T Tc ) , (T Tc ) , m h
1
12 , 1, 3
d = 2: Onsager exact solution:
Tc 2.269J, 18 , 47 , 15
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
74
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
75
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
76
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
77
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
Jij = Jji
78
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
P(hi )
Jij = Jji
exphi
1
exphi exphi 1 exp(m2hi )
79
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
P(hi )
Jij = Jji
exphi
1
12 1 tanh(hi )
exphi exphi 1 exp(m2hi )
80
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
Jij = Jji
exphi
1
12 1 tanh(hi )
exphi exphi 1 exp(m2hi )
(equilibration of Si, given
current values of other S’s)
P(hi )
81
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
Jij = Jji
exphi
1
12 1 tanh(hi )
exphi exphi 1 exp(m2hi )
(equilibration of Si, given
current values of other S’s)
P(hi )
82
master equation for Glauber model
0
dP({S},t) 1
2 1 Si tanhhi (t)P(S1 L Si L SN )
dt
i
12 1 Si tanhhi (t)P(S1 L Si L SN )
i
master equation for Glauber model
0
dP({S},t) 1
2 1 Si tanhhi (t)P(S1 L Si L SN )
dt
i
12 1 Si tanhhi (t)P(S1 L Si L SN )
i
magnetization
Si (t) Si P({S},t)
{S}
master equation for Glauber model
0
dP({S},t) 1
2 1 Si tanhhi (t)P(S1 L Si L SN )
dt
i
12 1 Si tanhhi (t)P(S1 L Si L SN )
i
magnetization
Si (t) Si P({S},t)
{S}
time evolution:
d Si
dP({S},t)
0
0 Si
dt
dt
{S}
master equation for Glauber model
0
dP({S},t) 1
2 1 Si tanhhi (t)P(S1 L Si L SN )
dt
i
12 1 Si tanhhi (t)P(S1 L Si L SN )
i
magnetization
Si (t) Si P({S},t)
{S}
time evolution:
d Si
dP({S},t)
0
0 Si
dt
dt
{S}
12 Si tanhhi (t)P(S1 L Si L SN )
i
12 Si tanhhi (t)P(S1 L Si L SN )
i
master equation for Glauber model
0
dP({S},t) 1
2 1 Si tanhhi (t)P(S1 L Si L SN )
dt
i
12 1 Si tanhhi (t)P(S1 L Si L SN )
i
magnetization
Si (t) Si P({S},t)
{S}
time evolution:
d Si
dP({S},t)
0
0 Si
dt
dt
{S}
1
2
S tanhh (t)P(S L
i
i
i
1
Si L SN )
12 Si tanhhi (t)P(S1 L Si L SN )
i
Si tanhhi (t)
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation:
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation:
S j (t) S j (t) m(t)
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation: S j (t) S j (t) m(t)
dm
0
m tanh (Jm h)
dt
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation: S j (t) S j (t) m(t)
dm
0
m tanh (Jm h)
dt
t -> ∞: recover equilibrium result
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation: S j (t) S j (t) m(t)
dm
0
m tanh (Jm h)
dt
t -> ∞: recover equilibrium result
dm
T > Tc, h = 0: 0
m(1 J)
dt
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation: S j (t) S j (t) m(t)
dm
0
m tanh (Jm h)
dt
t -> ∞: recover equilibrium result
T T
dm
c
T > Tc, h = 0: 0
m(1 J) m m(0)exp
t
dt
T 0
mean field dynamics
d Si
0
Si (t) tanh
Jij S j (t) h
dt
j
mean field approximation: S j (t) S j (t) m(t)
dm
0
m tanh (Jm h)
dt
t -> ∞: recover equilibrium result
T T
dm
c
T > Tc, h = 0: 0
m(1 J) m m(0)exp
t
dt
T 0
“critical slowing down” T
0
T Tc
below Tc
0
dm
m tanh (Jm h)
dt
below Tc
0
dm
m tanh (Jm h)
dt
expand around
m0 tanh( Jm 0 )
below Tc
0
dm
m tanh (Jm h)
dt
expand around
m0 tanh(Jm0 )
dm
0
m sech2 (Jm0 ) Jm
dt
below Tc
0
dm
m tanh (Jm h)
dt
expand around
m0 tanh(Jm0 )
dm
0
m sech2 (Jm0 ) Jm
dt
m (1 m02 )Jm
below Tc
0
dm
m tanh (Jm h)
dt
m0 tanh(Jm0 )
expand around
dm
0
m sech2 (Jm0 ) Jm
dt
m (1 m02 )Jm
just below Tc: 0
dm
1 (J)(1 m02 )m
dt
below Tc
0
dm
m tanh (Jm h)
dt
m0 tanh(Jm0 )
expand around
dm
0
m sech2 (Jm0 ) Jm
dt
m (1 m02 )Jm
just below Tc: 0
dm
1 (J)(1 m02 )m
dt
1 (J)(1 3(Tc T) /Tc )m
below Tc
0
dm
m tanh (Jm h)
dt
m0 tanh(Jm0 )
expand around
dm
0
m sech2 (Jm0 ) Jm
dt
m (1 m02 )Jm
just below Tc: 0
dm
1 (J)(1 m02 )m
dt
1 (J)(1 3(Tc T) /Tc )m
2(Tc T)m
below Tc
0
dm
m tanh (Jm h)
dt
m0 tanh(Jm0 )
expand around
dm
0
m sech2 (Jm0 ) Jm
dt
m (1 m02 )Jm
just below Tc: 0
dm
1 (J)(1 m02 )m
dt
1 (J)(1 3(Tc T) /Tc )m
2(Tc T)m
correlation time
Tc
0
T
T
c
1
2
beyond MFT
still have critical slowing down, different critical exponents
Landau-Ginzburg model
continuous-valued spins in continuous space:
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
functional
integral:
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
functional
integral:
D
lim
d
x i 1 xi 0
i
i
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
(free) energy
functional:
E[] 12
functional
integral:
d xr (x)
d
2
0
D
1
2
lim
d
x i 1 xi 0
i
i
u0 4 (x) ((x))2
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
(free) energy
functional:
E[] 12
functional
integral:
d xr (x)
d
“potential” V() 12 r0 2 14 u04
2
0
D
1
2
lim
d
x i 1 xi 0
i
i
u0 4 (x) ((x))2
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
(free) energy
functional:
E[] 12
functional
integral:
d xr (x)
d
“potential” V() 12 r0 2 14 u04
2
0
D
1
2
lim
d
x i 1 xi 0
i
i
u0 4 (x) ((x))2
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
(free) energy
functional:
E[] 12
functional
integral:
d xr (x)
d
2
0
D
1
2
lim
d
x i 1 xi 0
i
i
u0 4 (x) ((x))2
2
“potential” V() 12 r0 2 14 u04 “bending energy” (x)
Landau-Ginzburg model
continuous-valued spins in continuous space: Si (x)
Z expE({S})
{S}
Z
D expE[ ]
(free) energy
functional:
E[] 12
functional
integral:
d xr (x)
d
2
0
D
1
2
d
lim
x i 1 xi 0
i
i
u0 4 (x) ((x))2
2
“potential” V() 12 r0 2 14 u04 “bending energy” (x)
continuum limit of
Si1Si 1
i
1
2
Si1 Si
2
i
mean field theory
no fluctuations: ϕ uniform
mean field theory
no fluctuations: ϕ uniform
dV
0
d
mean field theory
no fluctuations: ϕ uniform
dV
0 r0 0 u0 03 0
d
mean field theory
no fluctuations: ϕ uniform
dV
0 r0 0 u0 03 0
d
solutions:
0 0
0 0, r0 u0
mean field theory
no fluctuations: ϕ uniform
dV
0 r0 0 u0 03 0
d
solutions:
0 0
0 0, r0 u0
take r0 prop to T - Tc
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
E[] 12
d xr (x) ((x))
d
2
0
2
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
E[] 12
d xr (x) ((x))
d
Fourier transform:
dd p
2
2
2
1
E[ ] 2
r
(
p)
p
(
p)
(2
)d 0
2
0
2
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
Fourier transform:
dd p
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
0
2
(2
)d
2
0
r0 ( p) p2 ( p)
p
2
2
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
Fourier transform:
dd p
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
0
2
(2
)d
1
P[] expE[]
Z
2
0
r0 ( p) p2 ( p)
p
2
2
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
Z
Z p
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
product of independent Gaussians, one for each pair (p,-p)
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
product of independent Gaussians, one for each pair (p,-p)
1
2
( p)
r0 p 2
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
product of independent Gaussians, one for each pair (p,-p)
1
1
2
(x)
(y)
exp r0 x y
( p)
2
4 x y
r0 p
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
product of independent Gaussians, one for each pair (p,-p)
1
1
2
(x)
(y)
exp r0 x y
( p)
2
4 x y
r0 p
1
1
correlation length
r0
T Tc
fluctuations around MFT
E[] 12
r0 > 0: ignore ϕ4 term:
d xr (x) ((x))
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ] 2
r
(
p)
p
(
p)
r
(
p)
p
(
p)
0
2
(2
)d 0
p
1
1
2
P[ ] expE[ ] exp 12 (r0 p2 ) ( p)
(T = 1)
Z
Z p
product of independent Gaussians, one for each pair (p,-p)
1
1
2
(x)
(y)
exp r0 x y
( p)
2
4 x y
r0 p
1
1
correlation length
diverges at Tc
r0
T Tc
Langevin dynamics
(x)
E
(x,t),
t
(x)
(x,t)( x , t ) 2T d (x x ) (t t )
Langevin dynamics
(x)
E
(x,t),
t
(x)
(x,t)( x , t ) 2T d (x x ) (t t )
Fourier transform :
( p)
E
*
( p,t),
t
( p)
( p,t)( p, t ) 2T (t t )
Langevin dynamics
(x)
E
(x,t),
t
(x)
(x,t)( x , t ) 2T d (x x ) (t t )
Fourier transform :
( p)
E
*
( p,t),
t
( p)
E[ ]
1
2
r0 ( p) p 2 ( p)
p
2
( p,t)( p, t ) 2T (t t )
2
1
4
u0
( p) ( p) ( p) ( p p p)
p, p , p
Langevin dynamics
(x)
E
(x,t),
t
(x)
(x,t)( x , t ) 2T d (x x ) (t t )
Fourier transform :
( p)
E
*
( p,t),
t
( p)
E[ ]
1
2
r0 ( p) p 2 ( p)
p
2
( p,t)( p, t ) 2T (t t )
2
1
4
u0
( p) ( p) ( p) ( p p p)
p, p , p
( p)
r0 p2 ( p) u0 ( p) ( p) ( p p p) ( p,t)
t
p , p
Langevin dynamics
(x)
E
(x,t),
t
(x)
(x,t)( x , t ) 2T d (x x ) (t t )
Fourier transform :
( p)
E
*
( p,t),
t
( p)
E[ ]
1
2
r0 ( p) p 2 ( p)
2
p
( p,t)( p, t ) 2T (t t )
2
1
4
u0
( p) ( p) ( p) ( p p p)
p, p , p
( p)
r0 p2 ( p) u0 ( p) ( p) ( p p p) ( p,t)
t
p , p
or, back in real space,
(x)
r0 2 (x) u0 3 (x) (x,t)
t
small fluctuations above Tc
ignoring u0:
( p)
r0 p2 (p) ( p,t)
t
small fluctuations above Tc
( p)
r0 p2 (p) ( p,t)
t
We have solved this before:
ignoring u0:
small fluctuations above Tc
( p)
r0 p2 (p) ( p,t)
t
We have solved this before:
ignoring u0:
( p,t)
t
exp (r0 p 2 )(t t )( p, t )dt
small fluctuations above Tc
( p)
r0 p2 (p) ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) exp (r0 p 2 )(t t )( p, t )dt
average over noise (we did this before, too)
t
small fluctuations above Tc
( p)
r0 p2 (p) ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) exp (r0 p 2 )(t t )( p, t )dt
average over noise (we did this before, too)
t
( p,t) ( p, t )
T
2
exp
(r
p
) t t
0
2
r0 p
small fluctuations above Tc
( p)
r0 p2 (p) ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) exp (r0 p 2 )(t t )( p, t )dt
average over noise (we did this before, too)
t
( p,t) ( p, t )
T
2
exp
(r
p
) t t
0
2
r0 p
critical slowing down (long correlation time) of fluctuations
near Tc (small r0) for small p (long wavelengths)