Exit time in presence of multiple metastable
states
Emilio N.M. Cirillo
Dipartimento Scienze di Base e Applicate per l’Ingegneria
Sapienza Università di Roma
Joint work with
F.R. Nardi, Department of Mathematics and Computer Science, Eindhoven University of
Technology, Eindhoven, The Netherlands
C. Spitoni, Department of Mathematics, Utrecht, The Netherlands
E. Olivieri, Dipartimento di Matematica, Roma Tor Vergata, Italy
Warwick – January 2016
Summary: first part
Metastable states in Statistical Mechanics models
Statistical Mechanics lattice models
Definition of metastable states
Properties of metastable states
Metastable states in the Blume–Capel model
Probabilistic Cellular Automata
Statistical Mechanics Lattice models
Λ = finite square with periodic boundary conditions
Λ
qσ(i)
σ(i) ∈ {1, . . . , k} spin variable associated with site i
Ω = {1, . . . , k}Λ configuration space
σ ∈ Ω configuration
Part 1 Metastable states in Statistical Mechanics models Statistical Mechanics lattice models
page 3/34
Statistical Mechanics Lattice models
Λ = finite square with periodic boundary conditions
Λ
qσ(i)
σ(i) ∈ {1, . . . , k} spin variable associated with site i
Ω = {1, . . . , k}Λ configuration space
σ ∈ Ω configuration
Equilibrium
• H(σ) = Hamiltonian
• µ(σ) = Gibbs measure =
e −H(σ)/T
Z (T )
• Z (T ) = partition function
Part 1 Metastable states in Statistical Mechanics models Statistical Mechanics lattice models
page 3/34
Metropolis dynamics: main features
The dynamics is a discrete time dynamics σ0 , σ1 , . . . , σt , . . . such that
• transitions increasing the energy are inhibited at T small. Consider
two configurations σ and η differing at a single site
H(η)
H(σ)
p(σ, η) ∼ 1
p(σ, η) ∝ e −[H(η)−H(σ)]/T
R
H(η)
H(σ)
• single spin–flip dynamics
• detailed balance (reversibility):
p(σ, η)e −H(σ)/T = e −H(η)/T p(η, σ)
• detailed balance ⇒ the Gibbs measure µ(σ) is stationary
Part 1 Metastable states in Statistical Mechanics models Statistical Mechanics lattice models
page 4/34
Metastable state definition
[Manzo, Nardi, Olivieri, Scoppola JSP 2004]
Height of a path ω = ω1 , . . . , ωn
qω2
Φω = max H(ωi )
qω3 ω
q 5 ω
q 7
6
ω4 q
ω6 q
ω8
i=1,...,n
Φω − H(ω1 )
qω1
Part 1 Metastable states in Statistical Mechanics models Definition of metastable states
?
page 5/34
Metastable state definition
[Manzo, Nardi, Olivieri, Scoppola JSP 2004]
Height of a path ω = ω1 , . . . , ωn
qω2
Φω = max H(ωi )
qω3 ω
q 5 ω
q 7
6
ω4 q
ω6 q
ω8
i=1,...,n
Φω − H(ω1 )
qω1
?
Communication height Φ(A, A0 ) between A, A0 ⊂ Ω
Φ(A, A0 ) = min 0 Φω
ω:A→A
Part 1 Metastable states in Statistical Mechanics models Definition of metastable states
page 5/34
Metastable state definition
[Manzo, Nardi, Olivieri, Scoppola JSP 2004]
Height of a path ω = ω1 , . . . , ωn
qω2
Φω = max H(ωi )
qω3 ω
q 5 ω
q 7
6
ω4 q
ω6 q
ω8
i=1,...,n
Φω − H(ω1 )
qω1
?
Communication height Φ(A, A0 ) between A, A0 ⊂ Ω
Φ(A, A0 ) = min 0 Φω
ω:A→A
Stability level of σ ∈ Ω
Vσ = Φ(σ, {states at energy smaller than σ}) − H(σ)
σ
V6
σ
?
Part 1 Metastable states in Statistical Mechanics models Definition of metastable states
page 5/34
Metastable state definition
Let Ωs be the set of the absolute minima of the Hamiltonian.
Define the maximal stability level
Γm = max s Vσ > 0
σ∈Ω\Ω
The set of metastable states is Ωm = {η ∈ Ω \ Ωs : Vη = Γm }.
The set of critical droplets Pc is the set of configurations where the
optimal paths from Ωm to Ωs attain the maximal energy level.
Pc
6
Γm
Ωm
?
Ωs
Part 1 Metastable states in Statistical Mechanics models Definition of metastable states
page 6/34
Metastable state properties
(Olivieri, Scoppola, Ben Arous, Cerf, Catoni,
Trouvé, Manzo, Nardi, Bovier, Eckhoff, Gayrard, Klein, den Hollander, Beltrán, Landim,
Slowick, Bianchi, Gaudillière, Sohier, C., ...)
Let σ ∈ Ωm
• for any ε > 0 we have lim Pσ (e (Γm −ε)/T < τΩs < e (Γm +ε)/T ) = 1
T →0
• lim T log Eσ (τΩs ) = Γm
T →0
• lim Pσ (τPc < τΩs ) = 1
T →0
Under suitable hypothesis on the structure of the set Ωm ∪ Ωs you can
compute the constant k > 0 such that
Eσ (τΩs ) =
1 Γm /T
e
[1 + o(1)]
k
Note that k is somehow related to the cardinality of the set of critical
droplets (entropy effect).
Part 1 Metastable states in Statistical Mechanics models Properties of metastable states
page 7/34
Comments
(on the general results)
• Not sharp estimates on exit time have been proven first in the case
of Metropolis dynamics and recently generalized also to not
reversible dynamics.
• General results on sharp estimates on exit time are valid under
hypotheses that exclude cases when multiple metastable states are
present.
The case we are interested to:
Ωm
Ωm
Ωs
• In any case finding out the set of metastable states in a concrete
model is often a very difficult task.
Part 1 Metastable states in Statistical Mechanics models Properties of metastable states
page 8/34
Summary: second part
Metastable states in Statistical Mechanics models
Metastable states in the Blume–Capel model
The Blume–Capel model
Metastability in presence of a single metastable state
Metastability in presence of multiple metastable states
Sketch of the proof
Sharp estimates on the exit time
Probabilistic Cellular Automata
Blume–Capel model
Λ
q
σ(i)
r r
• Λ = finite square with periodic boundary conditions
• σ(i) ∈ {−1, 0, +1} spin variable associated with site i
• h ∈ R magnetic field and λ ∈ R chemical potential
X
X
X
[σ(i)]2 − h
σ(i)
• H(σ) =
[σ(i) − σ(j)]2 − λ
hiji
Part 2 Metastable states in the Blume–Capel model The Blume–Capel model
i
i
page 10/34
Blume–Capel model
Λ
q
• Λ = finite square with periodic boundary conditions
σ(i)
r r
• σ(i) ∈ {−1, 0, +1} spin variable associated with site i
• h ∈ R magnetic field and λ ∈ R chemical potential
X
X
X
[σ(i)]2 − h
σ(i)
• H(σ) =
[σ(i) − σ(j)]2 − λ
i
hiji
i
Ground states: H(u) = −(h + λ)|Λ|, H(0) = 0, and H(d) = (h − λ)|Λ|
h
λ>h>0
@@ u 6
0
u
@
-λ
d
h>|λ|
not interesting
0
interesting
d
d
0
u
Part 2 Metastable states in the Blume–Capel model The Blume–Capel model
u
page 10/34
Blume–Capel model
Λ
q
• Λ = finite square with periodic boundary conditions
σ(i)
r r
• σ(i) ∈ {−1, 0, +1} spin variable associated with site i
• h ∈ R magnetic field and λ ∈ R chemical potential
X
X
X
[σ(i)]2 − h
σ(i)
• H(σ) =
[σ(i) − σ(j)]2 − λ
i
hiji
i
Ground states: H(u) = −(h + λ)|Λ|, H(0) = 0, and H(d) = (h − λ)|Λ|
h
λ>h>0
@@ u 6
0
u
@
-λ
d
h>|λ|
not interesting
0
interesting
d
d
0
u
u
• the candidates d and 0 are metastable states? Can they coexist?
• suppose d is metastable, does 0 have a role in the path from d to u?
Part 2 Metastable states in the Blume–Capel model The Blume–Capel model
page 10/34
Metropolis dynamics
Let σt the configuration at time t:
• chose at random with uniform probability 1/|Λ| a lattice site and call
it i;
• chose with probability 1/2 one of the two values in
{−1, 0, +1} \ {σt (i)}
and call it s;
• flip the spin σt (i) to s with probability 1 if the energy decreases and
with probabilty
exp{−∆H/T }
if the energy increases (∆H > 0).
Part 2 Metastable states in the Blume–Capel model The Blume–Capel model
page 11/34
Monte Carlo sequences:
• = −1 • = 0 • = +1
Parameters: Λ = 100 × 100, h = 0.1, λ = 0.2, T = 1.25
Parameters: Λ = 100 × 100, h = 0.1, λ = 0.02, T = 0.909
In both cases d is the unique metastable state: the transition 0 → u is
much faster than the transition d → 0.
Part 2 Metastable states in the Blume–Capel model Metastability in presence of a single metastable state
page 12/34
Rigorous results
h=2λ
h
6 h=λ
λ
• Ωm = {d}
• Pc =
0 d
u
with `c =
• Γm = H(Pc ) − H(d) ∼
• Energy landscape:
2−h+λ
h
8
h
(does not depend on λ)
d
0
u
Part 2 Metastable states in the Blume–Capel model Metastability in presence of a single metastable state
page 13/34
Rigorous results
h=2λ
h
6 h=λ
λ
• Ωm = {d}
• Pc =
d
0
with `c =
• Γm = H(Pc ) − H(d) ∼
• Energy landscape:
2
h−λ
4
h−λ
d
0
u
Part 2 Metastable states in the Blume–Capel model Metastability in presence of a single metastable state
page 14/34
Zero chemical potential Blume–Capel model
Hamiltonian H(σ) =
P
hiji [σ(i)
− σ(j)]2 − h
P
i
σ(i)
Ground states:
h
@ u6
@
0
@ ud
H(u) = −h|Λ|, H(0) = 0, H(d) = h|Λ|
λ
Guess:
d
d
0
0
d
u
u
0
u
2λ>h>λ>0
h>2λ>0
h>λ=0
Critical droplet: `c = b2/hc + 1
d
Γm = H( d
0
d
0
d)
− H(d) = H( 0
u
0)
− H(0) ∼
4
h
0
Part 2 Metastable states in the Blume–Capel model Metastability in presence of multiple metastable states
page 15/34
Monte Carlo sequences:
• = −1 • = 0 • = +1
Parameters: Λ = 100 × 100, h = 0.1, λ = 0.02, T = 0.909
Parameters: Λ = 100 × 100, h = 0.1, λ = 0, T = 0.909
Result to be proven: d and 0 are both metastable: the transitions 0 → u
and d → 0 take approximatively the same time.
Part 2 Metastable states in the Blume–Capel model Metastability in presence of multiple metastable states
page 16/34
Rigorous results
We prove the model dependent results:
1. Ωs = {u}
d
2. Γm = maxσ∈Ω\Ωs Vσ = H( d
0
d)
− H(d) ≡ Γ
d
3. Ωm = {η ∈ Ω \ Ωs : Vη = Γm } = {d, 0}
4. Pc =
0
d
(critical droplet between d and 0)
5. Qc =
u
0
(critical droplet between 0 and u)
Then we get that for any σ ∈ Ωm
• for any ε > 0 we have lim Pσ (e (Γ−ε)/T < τu < e (Γ+ε)/T ) = 1
T →0
• lim T log Eσ (τu ) = Γ
T →0
• lim Pd (τPc < τu ) = 1
T →0
and
lim P0 (τQc < τu ) = 1
T →0
Part 2 Metastable states in the Blume–Capel model Metastability in presence of multiple metastable states
page 17/34
Lemma
(finding out the metastable states)
Assume A ⊂ Ω \ Ωs and a ∈ R are such that
Φ(η, Ωs ) − H(η) = a
for any η ∈ A
and
Φ(σ, Ωs ) − H(σ) < a
for any σ ∈ Ω \ (A ∪ Ωs ) (recurrence)
Then
Γm = a and Ωm = A
where (recall) Γm = max s Vσ .
σ∈Ω\Ω
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 18/34
Proof of some of the model dependent ingredients
To prove the model dependent inputs
d
Γm = maxσ∈Ω\Ωs Vσ = H( d
0
d)
− H(d) ≡ Γ
d
and
Ωm = {η ∈ Ω \ Ωs : Vη = Γm } = {d, 0}
we have to prove the following:
• Φ(d, u) − H(d) = Γ
• Φ(0, u) − H(0) = Γ
• Φ(σ, u) − H(σ) < Γ for all σ ∈ Ω \ {d, 0, u} (recurrence)
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 19/34
Proof of some of the model dependent ingredients
To prove the model dependent inputs
d
Γm = maxσ∈Ω\Ωs Vσ = H( d
0
d)
− H(d) ≡ Γ
d
and
Ωm = {η ∈ Ω \ Ωs : Vη = Γm } = {d, 0}
we have to prove the following:
• Φ(d, u) − H(d) = Γ
• Φ(0, u) − H(0) = Γ
• Φ(σ, u) − H(σ) < Γ for all σ ∈ Ω \ {d, 0, u} (recurrence)
Recurrence is not very difficult but terribly boring. In the sequel I sketch
the proof of the first of the three conditions listed above. The second one
is similar.
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 19/34
Minmax: upper bound
Find a path connecting d to u attaining o
⇒ Φ(d, u) ≤ H(Pc )
its highest energy level at Pc
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 20/34
Minmax: upper bound
Find a path connecting d to u attaining o
⇒ Φ(d, u) ≤ H(Pc )
its highest energy level at Pc
Pc
000
00
00
0
00
00
0
00
00
0
qq
q
q
q 00 00 00
00
q
q 00 00
00
q 00
q 2−h
000
q
q
q
q
−h
q
2−h
q q q
q
q
`c
−h
`c −1
`c −1
`c
`c
4−h
d q
Then the path goes down to 0 and the from 0 to u in a similar fashion a
plus droplet is nucleated inside the sea of zeros.
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 20/34
Minmax: lower bound
Prove that all the paths connecting d to u
attain an energy level greater than
or equal to H(Pc )
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
)
⇒ Φ(d, u) ≥ H(Pc )
page 21/34
Minmax: lower bound
Prove that all the paths connecting d to u
attain an energy level greater than
or equal to H(Pc )
Strategy (serial dynamics): if there exists Ω̄ ⊂ Ω
such that
)
⇒ Φ(d, u) ≥ H(Pc )
su
• Pc ∈ Ω̄
• all the paths connecting d to u necessarily
pass through Ω̄
Ω
s
Pc
Ω̄
sd
• min H(σ) = H(Pc )
σ∈Ω̄
It than follows that all the paths connecting d to u attain an energy level
greater than or equal to H(Pc ).
Remark: with this strategy you do not get the model dependent input 4,
namely, you do not prove that the maximum along the path is necessarily
attained at Pc . To prove that a deeper investigation is needed.
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 21/34
Minmax: lower bound
In two state spin systems (e.g. Ising) life is easier: you have to count the
flipped spins. Here you have to count the minus spins that are not
flipped:
Ω̄ = set of configurations having |Λ| − [(`c − 1)`c + 1] minus spins
Then
• Pc = d
d
0
d∈
Ω̄ ⇐ by definition
d
• paths from d to u pass through Ω̄ ⇐ single spin–flip dynamics
(great simplification due to the continuity of the dynamics)
In order to complete the proof of the lower bound one has to show that
min H(σ) = H(Pc )
σ∈Ω̄
The analogous of such a result, although not trivial, is less difficult in two
state spin systems. In our case the proof is much more delicate due to
the presence of three state.
Part 2 Metastable states in the Blume–Capel model Sketch of the proof
page 22/34
Sharp estimate
Consider the Ising model with h > 0 small (Bovier and Manzo 2002):
Pc =
d
d
u
Ed (τu )
3
=
Γ
/T
m
T →0 e
4(2`c − 1)|Λ|
lim
u
For the Blume–Caple model with λ = 0 we expect (same critical
droplets):
d
lim
T →0
0
Ed (τ{u,0} )
E0 (τu )
3
= lim Γ /T =
T →0 e m
4(2`c − 1)|Λ|
e Γm /T
u
What can be said about Ed (τu )?
Part 2 Metastable states in the Blume–Capel model Sharp estimates on the exit time
page 23/34
Sharp estimate
Since it can be proven that
lim Pd [τu < τ0 ] = 0
T →0
We expect that the time for the transition d → u is the sum of the time
for the transitions d → 0 and 0 → u.
Part 2 Metastable states in the Blume–Capel model Sharp estimates on the exit time
page 24/34
Sharp estimate
Since it can be proven that
lim Pd [τu < τ0 ] = 0
T →0
We expect that the time for the transition d → u is the sum of the time
for the transitions d → 0 and 0 → u.
Theorem (Landim, Lemire 8 days ago on arXiv)
For the zero chemical potential Blume–Capel model for h small we have
that
lim
T →0
E0 (τu )
3
=
Γ
/T
m
4(2`c − 1)|Λ|
e
and
lim
T →0
Ed (τu )
3
=2×
Γ
/T
m
4(2`c − 1)|Λ|
e
Part 2 Metastable states in the Blume–Capel model Sharp estimates on the exit time
page 24/34
Sharp estimate: numerical check
Prefactor = (averaged exit time from d to u)/ exp{Γm /T }
Parameters: Λ = 60 × 60, h = 0.8, T = 0.4
Colors for λ: • 0, • 0.001, • 0.01, • 0.02, • 0.04, • 0.06,
Part 2 Metastable states in the Blume–Capel model Sharp estimates on the exit time
page 25/34
Sharp estimate: numerical check
Prefactor = (averaged exit time from d to u)/ exp{Γm /T }
Parameters: Λ = 60 × 60, h = 0.8, T = 0.27027
Colors for λ: • 0, • 0.01, • 0.02
Part 2 Metastable states in the Blume–Capel model Sharp estimates on the exit time
page 26/34
Summary: third part
Metastable states in Statistical Mechanics models
Metastable states in the Blume–Capel model
Probabilistic Cellular Automata
Definition and examples
Results
Probabilistic Cellular Automata
• Λ = finite square with periodic boundary conditions
• σ(i) ∈ {1, . . . , k} state variable associated with site i
q0
Λ
I
qi
• Ω = {1, . . . , k}Λ state space, σ ∈ Ω state
• I ⊂ Λ finite
Part 3 Probabilistic Cellular Automata Definition and examples
page 28/34
Probabilistic Cellular Automata
• Λ = finite square with periodic boundary conditions
• σ(i) ∈ {1, . . . , k} state variable associated with site i
q0
Λ
I
qi
• Ω = {1, . . . , k}Λ state space, σ ∈ Ω state
• I ⊂ Λ finite
• fσ : {1, . . . , k} → [0, 1] is a probability distribution depending on the
state σ restricted to I
• Θi : Ω → Ω shifts a configuration so that the site i is mapped to the
origin 0
• Probabilistic Cellular Automaton = Markov chain σ0 , σ1 , . . . , σt , . . .
on Ω with transition matrix
Y
p(σ, η) =
fΘi σ (η(i))
∀σ, η ∈ Ω
i∈Λ
Remark: parallel and local character of the evolution; all sites updated at
time t looking at the state at time t − 1.
Part 3 Probabilistic Cellular Automata Definition and examples
page 28/34
Reversible Probabilistic Cellular Automata
Assume Ω = {−1, +1}Λ , I symmetrical with respect to the origin, and
fσ (s) =
h 1 X
io
1n
1 + s tanh
σ(j) + h
for all s ∈ {−1, +1}
2
T
j∈I
where T > 0 and h ∈ R are called temperature and magnetic field.
Part 3 Probabilistic Cellular Automata Definition and examples
page 29/34
Reversible Probabilistic Cellular Automata
Assume Ω = {−1, +1}Λ , I symmetrical with respect to the origin, and
fσ (s) =
h 1 X
io
1n
1 + s tanh
σ(j) + h
for all s ∈ {−1, +1}
2
T
j∈I
where T > 0 and h ∈ R are called temperature and magnetic field.
Reversibility [Grinstein et. al. PRL 1985, Kozlov, Vasiljev Ad. Prob. 1980]:
G (σ) = −h
X
i∈Λ
σ(i) − T
X
i∈Λ
log cosh
h1 X
T
σ(j) + h
i
j∈i+I
• detailed balance: p(σ, η)e −G (σ)/T = e −G (η)/T p(η, σ)
• detailed balance ⇒ the measure exp{−G (σ)/T }/Z (T ) is stationary
Remark (compare to Metropolis): G depends on T ; parallel rule.
Part 3 Probabilistic Cellular Automata Definition and examples
page 29/34
Two examples
Nearest neighbor model: I = four nearest neighbors of the origin
I
q
q
0
q
q
Cross model: I = four nearest neighbors of the origin plus the origin
I
q
q q0 q
q
Part 3 Probabilistic Cellular Automata Definition and examples
page 30/34
The cross PCA
I
q
q q0 q
q
Consider the cross PCA model with positive and small magnetic field h > 0.
Result:
+++++
+++++
+++++
+++++
+++++
u
−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
d
6
Γ
Ωs = {u} and Ωm = {d}
d
?
u
Part 3 Probabilistic Cellular Automata Definition and examples
page 31/34
The cross PCA
I
q
q q0 q
q
Consider the cross PCA model with positive and small magnetic field h > 0.
Result:
+++++
+++++
+++++
+++++
+++++
u
−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
d
6
Γ
Ωs = {u} and Ωm = {d}
?
d
Critical droplet:
and
u
++++
++++
+++++
++++
++++
q
6
?
H(d)
Γ
++++
+++++
+++++
++++
++++
p
2
`6
c =b h c+1
?
T →0
Γ = H(q) + ∆(q, p) − H(d) ∼
Part 3 Probabilistic Cellular Automata Definition and examples
16
h
page 31/34
The nearest neighbor PCA
q
I
q
0
q
q
Consider the nearest neighbor PCA model with a positive and
small magnetic field h > 0.
Result: flip–flopping metastable state
+++++
+++++
+++++
+++++
+++++
u
−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
d
+ −+ −+
−+ −+ −
+ −+ −+
−+ −+ −
+ −+ −+
c
Part 3 Probabilistic Cellular Automata Results
6
Γ
6
s
m
Ω = {u}, Ω = {d,c}
Γ
d
?
c
?
u
page 32/34
The nearest neighbor PCA
q
I
q
0
q
q
Consider the nearest neighbor PCA model with a positive and
small magnetic field h > 0.
Result: flip–flopping metastable state
+++++
+++++
+++++
+++++
+++++
u
−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
d
+ −+ −+
−+ −+ −
+ −+ −+
−+ −+ −
+ −+ −+
c
6
Γ
6
s
m
Ω = {u}, Ω = {d,c}
Γ
d
?
c
?
u
Critical droplet in the sea of minuses:
+ +
−−−−
−−−−
+ +
−−−−
+
+
−−−−
+ +
−−−−
+
+
q
6
?
H(d)
Γ
−−−−
+ +
−−−−
+
+ +
−−−−
+ +
−−−−
+
+
−−−−
+ +
p
2
`6
c =b h c+1
?
and
T →0
Γ = H(q) + ∆(q, p) − H(d) ∼
Part 3 Probabilistic Cellular Automata Results
8
h
page 32/34
Tuning the self–interaction
PCA nearest neighbor model
I
q
q
0
q
q
κ=0
PCA cross model
q
q q0 q
q
κ=1
Let I be the set of the four nearest neighbors of the origin. Let
fσ (s) =
h1
io
X
1n
1 + s tanh
κσ(0) +
σ(j) + h
2
T
j∈I
for σ ∈ Ω, s ∈ {−1, +1} e κ ∈ [0, 1].
The parameter κ tunes the self–interaction: for κ = 0, 1 we get the
nearest neighbor and the cross PCA models.
Part 3 Probabilistic Cellular Automata Results
page 33/34
Close bibliography
[C., Olivieri JSP 1996] “Metastability and nucleation for the Blume-Capel model.
Different mechanisms of transition.”
[C., Nardi JSP 2003] “Metastability for the Ising model with a parallel dynamics.”
[C., Nardi JSP 2013] “Relaxation height in energy landscapes: an application to
multiple metastable states.”
[C., Nardi, Spitoni AIMI III 2010] “Competitive nucleation in metastable systems.”
[Landim, Lemire 2016] “Metastability of the two–dimensional Blume–Capel model
with zero chemical potential and small magnetic field.”
[Manzo, Nardi, Olivieri, Scoppola JSP 2004] “On the essential features of
metastability: tunnelling time and critical configurations.”
Part 3 Probabilistic Cellular Automata Results
page 34/34
Bibliography
[Bigelis, C., Lebowitz, Speer PRE 1999] “Critical droplets in metastable probabilistic
cellular automata.”
[Bovier et. al. CMP 2002] “Metastability and low lying spectra in reversible Markov
chains.”
[Cassandro et. al. JSP 1984] “Metastable behavior of stochastic dynamics: A
pathwise approach.”
[C., Lebowitz JSP 1998] “Metastability in the two–dimensional Ising model with free
boundary conditions.”
[C., Nardi, Spitoni PRE 2008] “Competitive nucleation in reversible Probabilistic
Cellular Automata.”
[C., Nardi, Spitoni JSP 2008] “Metastability for reversible probabilistic cellular
automata with self–interaction.”
[Dai Pra, Louis, Roelly ESAIM Probab. Statist. 2002] “Stationary measures and
phase transition for a class of probabilistic cellular automata.”
[Derrida 1990] “Dynamical phase transition in spin model and automata,”
Fundamental problem in Statistical Mechanics VII, H. van Beijeren, Editor,
Elsevier Science Publisher B.V..
Part 3 Probabilistic Cellular Automata Results
page 35/34
Bibliography
[Grinstein et. al. PRL 1985] “Statistical Mechanics of Probabilistic Cellular
Automata.”
[Isakov CMP 1984] “Nonanalytic features of the first order phase transition in the
Ising model.”
[Kozlov, Vasiljev Ad. Prob. 1980] “Reversible Markov chain with local interactions,”
in “Multicomponent random system.”
[Neves, Schonmann CMP 1991] “Critical Droplets and Metastability for a Glauber
Dynamics at Very Low Temperatures.”
[Olivieri, Scoppola JSP 1995] “Markov chains with exponentially small transition
probabilities: First exit problem from a general domain. I. The reversible case,”
[Trouvé Annales IHP section B 1996] “Rough large deviation estimates for the
optimal convergence speed exponent of generalized simulated annealing
algorithms.”
Part 3 Probabilistic Cellular Automata Results
page 36/34
Avanzi
Part 3 Probabilistic Cellular Automata Results
page 37/34
Proof of the general Lemma
Assume for simplicity A = {η} (single metastable state case).
Let σ 6= η and σ 6∈ Ωs , then H(σ) > H(Ωs ) implies
Vσ ≤ Φ(σ, Ωs ) − H(σ) < a
Since H(η) > H(Ωs ) we have
Vη ≤ Φ(η, Ωs ) − H(η) = a
By absurdity assume Vη < a:
• there exists σ 6= η such that H(σ) < H(η) and Φ(η, σ) − H(η) < a
• there exists a path ω 1 : η → σ such that Φω1 − H(η) < a
• if σ ∈ Ωs we get a contraddiction
• if σ 6∈ Ωs , there exists a path ω 2 : σ → Ωs such that
Φω2 − H(σ) < a,
• since Φω1 ω2 − H(η) < a we get a contraddiction
Conclusions:
Vη = a, Vσ < a for σ 6= η and σ 6∈ Ωs ⇒ Γm = a
Part 3 Probabilistic Cellular Automata Results
page 38/34
Lemma
Assume 0 < h < 1, b2/hc not integer, and |Λ| ≥ 49/h4 .
Pick σ ∈ Ω̄ and let Nσ be the collection of sites i in Λ such that
σ(i) 6= −1.
Then
1. Nσ is not a nearest neighbor connected subset of Λ winding around
the torous (recall we assumed periodic boundary conditions)
2. if σ ∈ {set of minima of Ω̄} then σ(i) = 0 for all i ∈ Nσ
3. Pc ∈ {set of minima of Ω̄}
4. min H(σ) = H(Pc )
σ∈Ω̄
Remark: 4 follows from 3 trivially. We have to prove 1 – 3.
Part 3 Probabilistic Cellular Automata Results
page 39/34
Item 1
Pick σ ∈ Ω̄ and let Nσ be the collection of sites i in Λ such that
σ(i) 6= −1. Then Nσ does not wind around the torous.
Proof. Since h < 1
|Nσ | = `c (`c − 1) + 1 ≤
2
h
+1
2
h
+1≤
7
h2
The statement follows since we assumed that |Λ| is finite but large
enough with respect to 1/h. More precisely we assumed |Λ| ≥ 49/h4 .
Part 3 Probabilistic Cellular Automata Results
page 40/34
Item 2
Pick σ ∈ Ω̄ and let Nσ be the collection of sites i in Λ such that
σ(i) 6= −1. If σ ∈ {set of minima of Ω̄} then σ(i) = 0 for all i ∈ Nσ .
Proof. Pick σ ∈ Ω̄ and recall H(σ) =
X
hiji
[σ(i) − σ(j)]2 − h
X
σ(i).
i
−−−−−−−
0
− − − − − − − Let σ ∈ Ω̄ obtained by flipping + → 0 in σ. Then
− 0 +−+ 0 −
H(σ 0 ) − H(σ) = rh−3`−m ≤ rh − (` + m)
−++−+ 0 −
−+−−−−−
− − 0 0 0 − − The + unit squares form a polyomino whose perimeter
− − − − − − − is equal to ` + m and whose area is r .
By general polyomino properties we have that (` + m)2 ≥ 16r . Hence
√
Then
H(σ 0 ) − H(σ) ≤ rh − r
√
• H(σ 0 ) − H(σ) ≤ 0 ⇐= r < 4/h
√
• r < 4/h ⇐= r ≤ `c (`c − 1) + 1 < 7/h2
Part 3 Probabilistic Cellular Automata Results
page 41/34
Item 3
Pick σ ∈ Ω̄ and let Nσ be the collection of sites i in Λ such that
σ(i) 6= −1. Then Pc ∈ set of minima of Ω̄.
Proof. Pick σ ∈ {set of minima of Ω̄}.
− − − − − − − Recall H(σ) = X[σ(i) − σ(j)]2 − h X σ(i), then
−−−−−−−
i
hiji
−0 0−0 0−
−0 0−0 0−
H(σ) − H(u) = perimeter of the polyomino
− 0 −−−−−
−− 0 0 0 −−
−h[`c (`c − 1) + 1]
−−−−−−−
Hence the configurations in the set of minima of Ω̄ are such that the
zeros form a polyomino of area `c (`c − 1) + 1 and having minimal
perimeter.
From general results on polyominoes we have that the configuration Pc is
an example of such (minimal perimeter) configurations.
Remark: the configuration Pc is not the unique perimeter minimizer =⇒
we do not have the escape mechanism statement for free!!!!
Part 3 Probabilistic Cellular Automata Results
page 42/34