Metastability
in irreversible diffusion processes
and stochastic resonance
Nils Berglund
Centre de Physique Théorique - CNRS
Marseille Luminy
http://berglund.univ-tln.fr
Joint work with Barbara Gentz, WIAS, Berlin
Max-Planck-Institut Leipzig, May 2005
Stochastic resonance: typical example
h
i
3
dxt = xt − xt + A cos εt dt + σ dWt
|
{z
}
∂
= − V (xt, t)
∂x
Potential:
1 4 1 2
V (x, t) = x − x − Ax cos εt.
4
2
1
Sample paths {xt}t
ε = 0.001
A = 0, σ = 0.3
A = 0.24, σ = 0.2
A = 0.1, σ = 0.27
A = 0.35, σ = 0.2
Qualitative measures:
• Power spectrum, signal-to-noise ratio.
• Residence-time distribution: law of interwell transition times.
2
Theory of large deviations
[Freidlin, Wentzell]
dxt = f (xt) dt + σ dWt
 Z
T

1
kϕ̇t − f (ϕt)k2 dt
Rate function: I[0,T ](ϕ) = 2 0

+∞
for ϕ ∈ H1
otherwise
2
Probability of staying close to ϕ ∼ e−I(ϕ)/σ .
Large-deviation principle: for Γ set of paths,
n
o
2
P (xt)06t6T ∈ Γ ∼ e− inf Γ I/σ as σ → 0.
Meaning:
− inf◦ I 6 lim inf σ 2 log P (xt)t ∈ Γ
σ→0
Γ
6 lim sup σ 2 log P (xt)t ∈ Γ 6 − inf I
σ→0
Γ
3
Gradient case
(reversible)
H
dxt = −∇V (xt) dt + σ dWt
• I(ϕ) minimized on paths with ϕ̇t = −f (ϕt).
• Cost of leaving potential well
inf{I(ϕ) : ϕ0 = bottom, ∃t : ϕt 6∈ well} = 2H.
2
• Expected time to leave well: E(τ ) ∼ e2H/σ
[Eyring, Kramers, Freidlin, Wentzell, . . . ]
• Law of τ /E(τ ) → Exp(1) as σ → 0
[Day]
• Subexponential behaviour known, related to small
eigenvalues of generator of diffusion
[Bovier, Eckhoff, Gayrard, Klein], [Helffer, Klein, Nier]
4
Stochastic resonance: quasistatic regime
h
i
dxt = xt − x3
t + A cos εt dt + σ dWt
Take ε =
2π
,
T (σ)
T (σ) ∼ eλ/σ
(for simplicity)
2
xt crosses barrier whenever
T (σ) e2H(t)/σ
H(t) = instantaneous barrier height
2
φ(t, λ): deterministic function, switches to deeper well whenever
2H(t) < λ. (⇒ possibility of hysteresis.)
Theorem:
[Freidlin, 2000]
For S, p, δ > 0,
Z S p
lim P
xsT (σ) − φ(sT (σ), λ) ds > δ = 0.
σ→0
0
5
What about non-quasistatic regime?
h
i
3
dxt = xt − xt + A cos Ωyt dt + σ dWt
dyt = dt
Irreversible (degenerate) diffusion process.
Interwell transition → crossing unstable periodic orbit tracking the
saddle. Distribution of transition locations?
x
well
saddle
y
periodic orbit
well
6
Exit problem
dxt = f (xt) dt + σ dWt
x?
x0 ∈ D, open, bdd set, ∂D smooth.
First-exit time: τ = inf{t > 0 : xt ∈
/ D}.
First-exit location: xτ ∈ ∂D.
∂D
Case 1: D ⊂ basin of attraction of asympt. stable equil. point x?
n
o
?
?
V (x , y) = inf I(ϕ) : ϕ0 = x , ϕt = y .
t>0
V = inf V (x?, y).
y∈∂D
Quasipotential
2
V /σ
• E(τ ) ∼
ne
2
2o
(V
−δ)/σ
(V
+δ)/σ
• lim P e
6τ 6e
= 1.
σ→0
• If y 7→nV (x?, y) has non-degenerate
global minimum in x1 ∈ ∂D,
o
lim P kxτ − x1k > δ = 0.
[Freidlin, Wentzell]
σ→0
7
Exit problem
∂D
Case 2: ∂D is unstable periodic orbit (characteristic boundary)
• Quasipotential V (x?, y) = V is constant on ∂D.
⇒ no concentration of exit location?
2
• E(τ ) ∼ eV /σ still holds.
[Day]
• As σ → 0, density of xτ is translated along ∂D proportionally
to |log σ|: cycling.
[Day]
8
How to compute law of exit location
In
J1
δ
J2
T
J3
2T
Jn−1
(n − 1)T
3T
nT
If exit occurs in In = [y, y + ∆] ⊂ [nT, (n + 1)T ]:
Rate function has n minimizers of comparable value.
n
P 0,0 yτ ∈ In
o
n
X
n
n
o
o
0,0
δ,J
'
P ` yτ ∈ In P
yτ 0 ∈ J `
{z
}|
{z
}
`=1 |
P`
Qn−` (y)
P` ' const e−`ε exp − σV12 1 − e−2`λT
,
2
−V
/σ
ε=Te
Qk (y) ' C(y) e−2kλT exp − σV22 1 − c(y) e−2kλT
λ = Lyapunov exponent of unstable orbit.
9
Theorem:
∂D: unstable periodic orbit, Lyapunov exponent λ.
θ(y): “natural ” parametrization of boundary, θ(y+T ) = θ(y)+λT .
∀∆ > σ 1/2,
−1
P 0,θ (θ0)
n
o
θ(yτD ) ∈ [θ1, θ1 + ∆] =
Z θ +∆
1
θ1
p(θ|θ0) dθ [1 + O(σ 1/2)]
e−(θ−θ0)/λTK
1
PλT (θ − |log σ|)
p(θ|θ0) = ftrans(θ, θ0)
N
λTK
• ftrans(θ, θ0) grows from 0 to 1 for θ − θ0 of order |log σ|.
C V /σ 2
,
Kramers time.
• TK = TK(σ) = e
σ
∞
X
1 −2z
1 −2z
• PλT (x) =
A(x − kλT )
A(z) = e
exp − e
.
2
2
k=−∞
10
First-passage-time distributions
(a)
σ = 0.4, T = 2
(c)
σ = 0.5, T = 2
V = 0.5, λ = 1
(b)
σ = 0.4, T = 20
(d)
σ = 0.5, T = 5
11
Application to residence-time distribution
xt crosses xper (t) at time s.
τ : time of first crossing back after s.
s
τ
• First-passage-time density:
o
∂ s,xper (s)n
p(t|s) = P
τ <t .
∂t
• Asymptotic transition-phase density:
ψ(t) =
Z t
−∞
p(t|s)ψ(s − T /2) ds = ψ(t + T ).
• Residence-time distribution:
q(t) =
Z T
0
p(s + t|s)ψ(s − T /2) ds.
12
Computation of residence-time distribution
Without forcing
(A = 0):
p(t|s) ∼ exponential, ψ(t) uniform ⇒ q(t) ∼ exponential.
With forcing
(A σ 2):
time change t 7→ θ(t)/λ
1
e−(t−s)/TK
PλT (λt − |log σ|)
p(t|s) ' ftrans(t, s)
N
TK
h
i
1
ψ(s) ' PλT (λt − |log σ|) 1 + O(T /TK)
T
∞
e−t/TK λT X
1
q(t) ' f˜trans(t)
TK
2 k=−∞ cosh2(λ(t + T /2 − kT ))
13
Residence-time distributions
(a)
σ = 0.2, T = 2
V = 0.5, λ = 1
(b)
σ = 0.4, T = 10
References
• N. B., B. Gentz, On the noise-induced passage through an unstable periodic
orbit I: Two-level model, J. Statist. Phys. 114, 1577–1618 (2004)
•
, Universality of first-passage and residence-time distributions in nonadiabatic stochastic resonance, Europhys. Letters 70, 1–7 (2005)
• In preparation . . .
14