Métastabilité
dans des équations de Ginzburg–Landau
avec bruit blanc spatiotemporel
Nils Berglund
MAPMO, Université d’Orléans
CNRS, UMR 6628 et Fédération Denis Poisson
www.univ-orleans.fr/mapmo/membres/berglund
Collaborateurs:
Florent Barret, Ecole Polytechnique, Palaiseau
Bastien Fernandez, CPT, CNRS, Marseille
Barbara Gentz, Université de Bielefeld
Lausanne, 26 juin 2009
Metastability in physics
Examples:
• Supercooled liquid
• Supersaturated gas
• Wrongly magnetised ferromagnet
1
Metastability in physics
Examples:
• Supercooled liquid
• Supersaturated gas
• Wrongly magnetised ferromagnet
. Near first-order phase transition
. Nucleation implies crossing energy barrier
Free energy
Order parameter
1-a
Metastability in stochastic lattice models
. Lattice: Λ ⊂⊂ Z d
. Configuration space: X = S Λ, S finite set (e.g. {−1, 1})
. Hamiltonian: H : X → R (e.g. Ising or lattice gas)
. Gibbs measure: µβ (x) = e−βH(x) /Zβ
. Dynamics: Markov chain with invariant measure µβ
(e.g. Metropolis: Glauber or Kawasaki)
2
Metastability in stochastic lattice models
. Lattice: Λ ⊂⊂ Z d
. Configuration space: X = S Λ, S finite set (e.g. {−1, 1})
. Hamiltonian: H : X → R (e.g. Ising or lattice gas)
. Gibbs measure: µβ (x) = e−βH(x) /Zβ
. Dynamics: Markov chain with invariant measure µβ
(e.g. Metropolis: Glauber or Kawasaki)
Results (for β 1) on
• Transition time between + and −
or empty and full configuration
• Transition path
• Shape of critical droplet
. Frank den Hollander, Metastability under stochastic dynamics, Stochastic
Process. Appl. 114 (2004), 1–26.
. Enzo Olivieri and Maria Eulália Vares, Large deviations and metastability,
Cambridge University Press, Cambridge, 2005.
2-a
Reversible diffusion
dxt = −∇V (xt) dt +
√
2ε dWt
. V : R d → R : potential, growing at infinity
. Wt: d-dim Brownian motion on (Ω, F , P)
Reversible w.r.t.
invariant measure:
e−V (x)/ε
µε(dx) =
dx
Zε
(detailed balance)
3
Reversible diffusion
dxt = −∇V (xt) dt +
√
2ε dWt
. V : R d → R : potential, growing at infinity
. Wt: d-dim Brownian motion on (Ω, F , P)
Reversible w.r.t.
invariant measure:
Mont
Puget
e−V (x)/ε
µε(dx) =
dx
Zε
(detailed balance)
Col de Sugiton
z
Luminy
x
Calanque de Sugiton
y
τyx: first-hitting time of small ball Bε(y), starting in x
“Eyring–Kramers law” (Eyring 1935, Kramers 1940)
• Dim 1: E[τyx] ' √ 00 2π 00
e[V (z)−V (x)]/ε
V (x)|V
r (z)|
• Dim>
2: E[τyx] ' |λ 2π
1 (z)|
|det(∇2 V (z))| [V (z)−V (x)]/ε
e
det(∇2 V (x))
3-a
Towards a proof of Kramers’ law
• Large deviations (Wentzell & Freidlin 1969):
lim ε log(E[τyx]) = V (z) − V (x)
ε→0
• Analytic (Helffer, Sjöstrand 85, Miclo 95, Mathieu 95, Kolokoltsov 96,. . . ):
low-lying spectrum of generator
• Potential theory/variational (Bovier, Eckhoff, Gayrard, Klein 2004):
E[τyx] = |λ 2π
1 (z)|
r
h
i
|det(∇2 V (z))| [V (z)−V (x)]/ε
1/2
1/2
e
1 + O(ε
|log ε|
)
det(∇2 V (x))
and similar asymptotics for eigenvalues of generator
• Witten complex (Helffer, Klein, Nier 2004):
full asymptotic expansion of prefactor
• Distribution of τyx (Day 1983, Bovier et al 2005):
lim P
ε→0
o
x
x
τy > tE[τy ] = e−t
n
4
Ginzburg–Landau equation
∂tu(x, t) = ∂xxu(x, t) + u(x, t) − u(x, t)3 + noise
x ∈ [0, L], u(x, t) ∈ R represents e.g. magnetisation
• Periodic b.c.
• Neumann b.c. ∂xu(0, t) = ∂xu(L, t) = 0
Noise: weak, white in space and time
5
Ginzburg–Landau equation
∂tu(x, t) = ∂xxu(x, t) + u(x, t) − u(x, t)3 + noise
x ∈ [0, L], u(x, t) ∈ R represents e.g. magnetisation
• Periodic b.c.
• Neumann b.c. ∂xu(0, t) = ∂xu(L, t) = 0
Noise: weak, white in space and time
Deterministic system is gradient
δV
3
∂xxu + u − u = −
δu
Z L
1 0
1
1
u (x)2 − u(x)2 + u(x)4 dx
V [u] =
2
4
0 2
→ min
5-a
Stationary solutions
u00(x) = −u(x) + u(x)3 = −
d
dx
"
#
6
Stationary solutions
u00(x) = −u(x) + u(x)3 = −
d
dx
"
#
• u±(x) ≡ ±1: global minima of V , stable
• u0(x) ≡ 0: unstable
• Neumann b.c:
q for k =
1, 2, . . . , if L > πk,
√
kx
2m
√
+ K(m), m
2k m + 1 K(m) = L
uk,±(x) = ± m+1 sn
m+1
6-a
Stationary solutions
u00(x) = −u(x) + u(x)3 = −
d
dx
"
#
• u±(x) ≡ ±1: global minima of V , stable
• u0(x) ≡ 0: unstable
• Neumann b.c:
q for k =
1, 2, . . . , if L > πk,
√
kx
2m
√
+ K(m), m
2k m + 1 K(m) = L
uk,±(x) = ± m+1 sn
m+1
u+
u1,+
u2,+
u3,+
u4,+
0
π
2π
3π
L
u0
u4,−
u3,−
u2,−
u1,−
u−
6-b
Stability of stationary solutions
2
d
Linearisation at u(x): ∂tϕ = A[u]ϕ, A[u] = dx2 + 1 − 3u(x)2
7
Stability of stationary solutions
2
d
Linearisation at u(x): ∂tϕ = A[u]ϕ, A[u] = dx2 + 1 − 3u(x)2
• u±(x) ≡ ±1: eigenvalues −(2 + (πk/L)2), k ∈ N
• u0(x) ≡ 0: eigenvalues 1 − (πk/L)2, k ∈ N
Number of positive eigenvalues:
u+ ≡ 1
u1,+
u2,+
0
1
2
u3,+
3
u4,+
0
π
2π
1
3π
2
L
3
4
u0 ≡ 0
u4,−
1
0
2
3
u3,−
u2,−
u1,−
u− ≡ −1
7-a
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u3 )
Ẅtx: space–time white noise (formal derivative of Brownian sheet)
8
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u3 )
Ẅtx: space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution:
1. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos(kπx/L) = e−(kπ/L) t cos(kπx/L)
8-a
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u3 )
Ẅtx: space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution:
1. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos(kπx/L) = e−(kπ/L) t cos(kπx/L)
√
2. u̇t = ∆ut + 2ε Ẅtx
Z t
√
⇒
ut = e∆t u0 + 2ε
e∆(t−s) Ẇx(ds)
|0
wt(x) =
XZ t
k
0
{z
=: wt (x)
}
2
(k)
e−(kπ/L) (t−s) dWs cos(kπx/L)
8-b
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u3 )
Ẅtx: space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution:
1. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos(kπx/L) = e−(kπ/L) t cos(kπx/L)
√
2. u̇t = ∆ut + 2ε Ẅtx
Z t
√
⇒
ut = e∆t u0 + 2ε
e∆(t−s) Ẇx(ds)
|0
wt(x) =
XZ t
k
0
{z
=: wt (x)
}
2
(k)
e−(kπ/L) (t−s) dWs cos(kπx/L)
√
⇒
2ε Ẅtx + f (ut)
Z t
√
ut = e∆t u0 + 2ε wt +
e∆(t−s) f (us) ds =: Ft[u]
⇒
Existence and a.s. uniqueness (Faris & Jona-Lasinio 1982)
3. u̇t = ∆ut +
0
8-c
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
Fourier variables: ut(x) =
∞
X
(∆ ≡ ∂xx , f (u) = u − u3 )
zk (t) ei πkx/L
k=−∞
⇒
X
dzk = −λk zk dt −
√
(k)
zk1 zk2 zk3 dt + 2ε dWt
k1 +k2 +k3 =k
where λk = −1 + (πk/L)2
9
Ginzburg–Landau equation with noise
√
u̇t(x) = ∆ut(x) + f (ut(x)) + 2ε Ẅtx
Fourier variables: ut(x) =
∞
X
(∆ ≡ ∂xx , f (u) = u − u3 )
zk (t) ei πkx/L
k=−∞
⇒
X
dzk = −λk zk dt −
√
(k)
zk1 zk2 zk3 dt + 2ε dWt
k1 +k2 +k3 =k
where λk = −1 + (πk/L)2
Energy functional:
∞
X
1
1 X
1
2
V [u] =
λk |zk | +
zk1 zk2 zk3 zk4
L
2 k=−∞
4 k +k +k +k =0
1
2
3
4
9-a
The question
How long does the system take to get from u−(x) ≡ −1
to (a neighbourhood of) u+(x) ≡ 1?
Metastability: Time of order econst /ε
(rate of order e−const /ε)
10
The question
How long does the system take to get from u−(x) ≡ −1
to (a neighbourhood of) u+(x) ≡ 1?
Metastability: Time of order econst /ε
(rate of order e−const /ε)
We seek constants ∆W (activation energy), Γ0 and α such that
the random transition time τ satisfies
i
−1
α
E[τ ] = Γ0 + O(ε ) e∆W/ε
h
10-a
The question
How long does the system take to get from u−(x) ≡ −1
to (a neighbourhood of) u+(x) ≡ 1?
Metastability: Time of order econst /ε
(rate of order e−const /ε)
We seek constants ∆W (activation energy), Γ0 and α such that
the random transition time τ satisfies
i
−1
α
E[τ ] = Γ0 + O(ε ) e∆W/ε
h
Large deviations (Faris & Jona-Lasinio 1982)
. L 6 π : ∆W = V [u0] − V [u−] = L
4
h
i
(1−m)(3m+5)
1
. L > π : ∆W = V [u1,±]−V [u−] = √
8 E(m)−
K(m)
1+m
3 1+m
10-b
Formal computation for Ginzburg–Landau (R.S. Maier, D. Stein, 01)
Case L < π:
v
u
2 V [u ])
|λ1(u0)| u
det(∇
−
t
Kramers: Γ0 '
2π
|det(∇2V [u0])|
Eigenvalues at V [u−] ≡ −1: µk = 2 + (πk/L)2
Eigenvalues at V [u0] ≡ 0: λk = −1 + (πk/L)2
11
Formal computation for Ginzburg–Landau (R.S. Maier, D. Stein, 01)
Case L < π:
v
u
2 V [u ])
|λ1(u0)| u
det(∇
−
t
Kramers: Γ0 '
2π
|det(∇2V [u0])|
Eigenvalues at V [u−] ≡ −1: µk = 2 + (πk/L)2
Eigenvalues at V [u0] ≡ 0: λk = −1 + (πk/L)2
s
v
√
u Y
∞
|λ0| u
µk
1
sinh( 2L)
t
Thus formally Γ0 '
= 3/4
2π k=0 |λk |
sin L
2
π
Case L > π:
Spectral determinant computed by Gelfand’s method
11-a
Formal computation for Ginzburg–Landau (R.S. Maier, D. Stein, 01)
Case L < π:
v
u
2 V [u ])
|λ1(u0)| u
det(∇
−
t
Kramers: Γ0 '
2π
|det(∇2V [u0])|
Eigenvalues at V [u−] ≡ −1: µk = 2 + (πk/L)2
Eigenvalues at V [u0] ≡ 0: λk = −1 + (πk/L)2
s
v
√
u Y
∞
|λ0| u
µk
1
sinh( 2L)
t
Thus formally Γ0 '
= 3/4
2π k=0 |λk |
sin L
2
π
Case L > π:
Spectral determinant computed by Gelfand’s method
Problems:
1. What happens when L → π−? (bifurcation)
2. Is the formal computation correct in infinite dimension?
11-b
Potential theory
Consider first Brownian motion Wtx = x + Wt
x = inf{t > 0 : W x ∈ A}, A ⊂ R d
Let τA
t
12
Potential theory
Consider first Brownian motion Wtx = x + Wt
x = inf{t > 0 : W x ∈ A}, A ⊂ R d
Let τA
t
x ] satisfies
Fact 1: wA(x) = E[τA
∆wA(x) = 1
wA(x) = 0
GAc (x, y) Green’s function ⇒ wA(x) =
x ∈ Ac
x∈A
Z
Ac
GAc (x, y) dy
12-a
Potential theory
Consider first Brownian motion Wtx = x + Wt
x = inf{t > 0 : W x ∈ A}, A ⊂ R d
Let τA
t
x ] satisfies
Fact 1: wA(x) = E[τA
x ∈ Ac
∆wA(x) = 1
x∈A
wA(x) = 0
GAc (x, y) Green’s function ⇒ wA(x) =
y
Z
Ac
GAc (x, y) dy
A
12-b
Potential theory
x < τ x ] satisfies
Fact 2: hA,B (x) = P[τA
B
∆hA,B (x) = 0
⇒ hA,B (x) =
Z
∂A
x ∈ (A ∪ B)c
hA,B (x) = 1
x∈A
hA,B (x) = 0
x∈B
GB c (x, y) ρA,B (dy)
13
Potential theory
x < τ x ] satisfies
Fact 2: hA,B (x) = P[τA
B
x ∈ (A ∪ B)c
∆hA,B (x) = 0
⇒ hA,B (x) =
Z
∂A
hA,B (x) = 1
x∈A
hA,B (x) = 0
x∈B
GB c (x, y) ρA,B (dy)
ρA,B : “surface charge density” on ∂A
−
−
−
−
−
+ + ++
+
+
+ A
+
+
+ ++
+
−
− −−
−
−
−
B
−
−
−
− − −−
1V
13-a
Potential theory
Capacity: capA(B) =
Z
∂A
ρA,B (dy)
14
Potential theory
Capacity: capA(B) =
Z
∂A
ρA,B (dy)
Key observation: let C = Bε(x), then (using G(y, z) = G(z, y))
Z
ZAc
Ac
hC,A(y) dy =
hC,A(y) dy =
Z
Z
ZAc ∂C
∂C
GAc (y, z)ρC,A(dz) dy
wA(z)ρC,A(dz) ' wA(x) capC (A)
14-a
Potential theory
Capacity: capA(B) =
Z
∂A
ρA,B (dy)
Key observation: let C = Bε(x), then (using G(y, z) = G(z, y))
Z
ZAc
Ac
hC,A(y) dy =
hC,A(y) dy =
Z
Z
ZAc ∂C
∂C
GAc (y, z)ρC,A(dz) dy
wA(z)ρC,A(dz) ' wA(x) capC (A)
Z
⇒
x ] = w (x) '
E[τA
A
Ac
hBε(x),A(y) dy
capBε(x)(A)
14-b
Potential theory
Capacity: capA(B) =
Z
∂A
ρA,B (dy)
Key observation: let C = Bε(x), then (using G(y, z) = G(z, y))
Z
ZAc
Ac
hC,A(y) dy =
hC,A(y) dy =
Z
Z
ZAc ∂C
∂C
GAc (y, z)ρC,A(dz) dy
wA(z)ρC,A(dz) ' wA(x) capC (A)
Z
⇒
x ] = w (x) '
E[τA
A
Ac
hBε(x),A(y) dy
capBε(x)(A)
Variational representation: Dirichlet form
capA(B) =
Z
(A∪B)c
k∇hA,B (x)k2 dx =
inf
Z
h∈HA,B (A∪B)c
k∇h(x)k2 dx
(HA,B : set of sufficiently smooth functions satisfying b.c.)
14-c
Potential theory
General case: dxt = −∇V (xt) dt +
√
2ε dWt
Generator: ∆ 7→ ε∆ − ∇V · ∇
Then
Z
x ] = w (x) '
E[τA
A
where
capA(B) = ε
inf
Z
h∈HA,B (A∪B)c
Ac
hBε(x),A(y) e−V (y)/ε dy
capBε(x)(A)
k∇h(x)k2 e−V (x)/ε dx
15
Potential theory
General case: dxt = −∇V (xt) dt +
√
2ε dWt
Generator: ∆ 7→ ε∆ − ∇V · ∇
Then
Z
x ] = w (x) '
E[τA
A
where
capA(B) = ε
inf
Z
h∈HA,B (A∪B)c
Ac
hBε(x),A(y) e−V (y)/ε dy
capBε(x)(A)
k∇h(x)k2 e−V (x)/ε dx
Rough a priori bounds on h show that if x potential minimum,
Z
(2πε)d/2 e−V (x)/ε
−V
(y)/ε
hBε(x),A(y) e
dy ' q
c
A
det(∇2V (x))
15-a
Estimation of capacity
Truncated energy functional: retain only modes with k 6 d
d
X
2 + ...
1 V [u] = − 1 z 2 + u (z ) + 1
λ
|z
|
1
1
k k
L
2 0
2 k=2
2 + 3 z4
u1(z1) = 1
λ
z
1
1
2
8 1
16
Estimation of capacity
Truncated energy functional: retain only modes with k 6 d
d
X
2 + ...
1 V [u] = − 1 z 2 + u (z ) + 1
λ
|z
|
1
1
k k
L
2 0
2 k=2
2 + 3 z4
u1(z1) = 1
λ
z
1
1
2
8 1
Theorem: For all L < π,
Z ∞
e−u1(z1)/ε dz1 Y
d u
i
u 2πε h
−∞
t
√
1 + R(ε)
capBε(u−)(Bε(u+)) = ε
λj
2πε
j=2
v
where R(ε) = O((ε|log ε|)1/4) is uniform in d.
16-a
Estimation of capacity
Truncated energy functional: retain only modes with k 6 d
d
X
2 + ...
1 V [u] = − 1 z 2 + u (z ) + 1
λ
|z
|
1
1
k k
L
2 0
2 k=2
2 + 3 z4
u1(z1) = 1
λ
z
1
1
2
8 1
Theorem: For all L < π,
Z ∞
e−u1(z1)/ε dz1 Y
d u
i
u 2πε h
−∞
t
√
1 + R(ε)
capBε(u−)(Bε(u+)) = ε
λj
2πε
j=2
v
where R(ε) = O((ε|log ε|)1/4) is uniform in d.
Corollary:
Expected first-passage time converges in the limit d → ∞
(Liu, 2003)
16-b
Sketch of proof
Upper bound:
cap = inf Φ(h) 6 Φ(h+)
Φ(h) = ε
h
q
Let δ =
Z
k∇h(z)k2 e−V (z)/ε dz
cε|log ε|, choose
h+(z) =



1



f (z0)
0
for
for
for
z0 < −δ
−δ < z0 < δ
z0 > δ
where εf 00(z0) + ∂z0 V (z0, 0)f 0(z0) = 0 with b.c. f (±δ) = 0, 1
17
Sketch of proof
Upper bound:
cap = inf Φ(h) 6 Φ(h+)
Φ(h) = ε
h
q
Let δ =
Z
k∇h(z)k2 e−V (z)/ε dz
cε|log ε|, choose
h+(z) =



1



f (z0)
0
for
for
for
z0 < −δ
−δ < z0 < δ
z0 > δ
where εf 00(z0) + ∂z0 V (z0, 0)f 0(z0) = 0 with b.c. f (±δ) = 0, 1
Lower bound:
Bound Dirichlet Φ form below by restricting domain, taking only
1st component of gradient and use for b.c. a priori bound on h
17-a
Computation of prefactor
L < π:
Γ0 =
1
23/4 π
r
√
sinh( 2L)
sin L
18
Computation of prefactor
L < π:
Γ0 =
r
1
23/4 π
√
r
sinh( 2L)
λ1
√
√ λ1
Ψ
+
sin L
λ1 + 3ε/4L
3ε/4L
where λ1 = −1 + (π/L)2 and
r
Ψ+(α) =
2
α(1+α) α2 /16
α
e
K
1/4
8π
16
18-a
Computation of prefactor
L < π:
Γ0 =
r
1
23/4 π
√
r
sinh( 2L)
λ1
√
√ λ1
Ψ
+
sin L
λ1 + 3ε/4L
3ε/4L
where λ1 = −1 + (π/L)2 and
r
Ψ+(α) =
2
α(1+α) α2 /16
α
e
K
1/4
8π
16
In particular,
q
√
Γ(1/4)
lim Γ0 =
sinh( 2π)ε−1/4
7 1/4
L→π−
2(3π )
18-b
Computation of prefactor
L < π:
Γ0 =
r
1
23/4 π
√
r
sinh( 2L)
λ1
√
√ λ1
Ψ
+
sin L
λ1 + 3ε/4L
3ε/4L
where λ1 = −1 + (π/L)2 and
r
Ψ+(α) =
2
α(1+α) α2 /16
α
e
K
1/4
8π
16
In particular,
q
√
Γ(1/4)
lim Γ0 =
sinh( 2π)ε−1/4
7 1/4
L→π−
2(3π )
12
11
(product of eigenvalues computed using
path-integral techniques, cf. Maier and Stein)
ε = 0.0003
9
8
Similar expression for L > π
Γ0
10
ε = 0.001
7
6
5
4
ε = 0.003
ε = 0.01
3
2
L
π
1
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7
18-c
References
• W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat
equation with noise, J. Phys. A 15, 3025–3055 (1982)
• R. S. Maier and D. L. Stein, Droplet nucleation and domain wall motion in
a bounded interval, Phys. Rev. Lett. 87, 270601-1 (2001)
• Anton Bovier, Michael Eckhoff, Véronique Gayrard and Markus Klein, Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and
exit times, J. Eur. Math. Soc. 6, 399–424 (2004)
• N. B. and Barbara Gentz, The Eyring–Kramers law for potentials with nonquadratic saddles, arXiv/0807.1681 (2008)
• N. B. and Barbara Gentz, Anomalous behavior of the Kramers rate at bifurcations in classical field theories, J. Phys. A: Math. Theor. 42, 052001
(2009)
• Florent Barret, N. B. and Barbara Gentz, in preparation
19