Metastability
in a system of interacting nonlinear diffusions
Nils Berglund
Centre de Physique Théorique - CNRS
Marseille Luminy
http://berglund.univ-tln.fr
Joint work with:
Bastien Fernandez, CPT, Marseille
Barbara Gentz, University of Bielefeld
Seminar über Stochastische Prozesse, Zürich, November 2006
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
Metastability in reversible diffusions
dxσ (t) = −∇V (xσ (t)) dt + σ dB(t)
. V : R d → R : potential, growing at infinity
. dB(t): d-dim Brownian motion on (Ω, F , P)
Invariant measure:
e−2V (x)/σ
µσ (x) =
Zσ
2
3
Metastability in reversible diffusions
dxσ (t) = −∇V (xσ (t)) dt + σ dB(t)
. V : R d → R : potential, growing at infinity
. dB(t): d-dim Brownian motion on (Ω, F , P)
Invariant measure:
e−2V (x)/σ
µσ (x) =
Zσ
2
Pizzo Centrale
Pizzo
Gotthard
Lucendro
Zuerich
Bellinzona
3-a
Metastability in reversible diffusions
dxσ (t) = −∇V (xσ (t)) dt + σ dB(t)
. V : R d → R : potential, growing at infinity
. dB(t): d-dim Brownian motion on (Ω, F , P)
Invariant measure:
e−2V (x)/σ
µσ (x) =
Zσ
2
Pizzo Centrale
Pizzo
Gotthard
Lucendro
Zuerich
Bellinzona
τ : transition time between potential wells (first-hitting time)
• Large deviations (Wentzell & Freidlin): limσ→0 σ 2 log(E{τ })
• Analytic (Miclo, Mathieu, Kolokoltsov): spectrum of generator
• Variational (Bovier el al): spectrum and distribution of τ
3-b
Metastability in reversible diffusions
. Stationary pts: S = {x : ∇V (x) = 0}
. Saddles of index k: Sk = {x ∈ S : Hess V (x) has k ev < 0}
. Graph G = (S0, E), x ↔ y if x, y ∈ unst. manif. of s ∈ S1
. xt ∼ markovian jump process on G
4
Metastability in reversible diffusions
. Stationary pts: S = {x : ∇V (x) = 0}
. Saddles of index k: Sk = {x ∈ S : Hess V (x) has k ev < 0}
. Graph G = (S0, E), x ↔ y if x, y ∈ unst. manif. of s ∈ S1
. xt ∼ markovian jump process on G
4-a
The model
• Lattice: Λ = Z /N Z , N > 2
5
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
dxi(t) =
5-a
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
4 − 1 x2
• Local bistable potential U (x) = 1
x
4
2
dxi(t) = f (xi(t)) dt
f (x) = −U 0(x) = x − x3
5-b
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
4 − 1 x2
• Local bistable potential U (x) = 1
x
4
2
• Coupling between sites: discretised Laplacian, intensity γ
i
γh
dxi(t) = f (xi(t)) dt + xi+1(t) − 2xi(t) + xi−1(t) dt
2
f (x) = −U 0(x) = x − x3
5-c
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
4 − 1 x2
• Local bistable potential U (x) = 1
x
4
2
• Coupling between sites: discretised Laplacian, intensity γ
• Independent white noise on each site
i
√
γh
dxi(t) = f (xi(t)) dt + xi+1(t) − 2xi(t) + xi−1(t) dt + N σ dBi(t)
2
f (x) = −U 0(x) = x − x3
5-d
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
4 − 1 x2
• Local bistable potential U (x) = 1
x
4
2
• Coupling between sites: discretised Laplacian, intensity γ
• Independent white noise on each site
i
√
γh
dxi(t) = f (xi(t)) dt + xi+1(t) − 2xi(t) + xi−1(t) dt + N σ dBi(t)
2
f (x) = −U 0(x) = x − x3
. Interacting diffusions (e.g. Dawson & Gärtner, Deuschel, Méléard, . . . )
5-e
The model
• Lattice: Λ = Z /N Z , N > 2
• i ∈ Λ 7→ xi ∈ R , configuration space X = R Λ
4 − 1 x2
• Local bistable potential U (x) = 1
x
4
2
• Coupling between sites: discretised Laplacian, intensity γ
• Independent white noise on each site
i
√
γh
dxi(t) = f (xi(t)) dt + xi+1(t) − 2xi(t) + xi−1(t) dt + N σ dBi(t)
2
f (x) = −U 0(x) = x − x3
. Interacting diffusions (e.g. Dawson & Gärtner, Deuschel, Méléard, . . . )
Gradient System: dxσ (t) = −∇Vγ (xσ (t)) dt +
√
N σ dB(t)
γ X
Potential: Vγ (x) =
U (xi) +
(xi+1 − xi)2
4 i∈Λ
i∈Λ
X
5-f
Weak coupling
. γ = 0: S = {−1, 0, 1}Λ, S0 = {−1, 1}Λ, G = hypercube.
6
Weak coupling
. γ = 0: S = {−1, 0, 1}Λ, S0 = {−1, 1}Λ, G = hypercube.
Theorem: ∀N , ∃γ ?(N ) > 0 s.t. points of each Sk (γ) continuous in
γ for 0 6 γ < γ ?(N )
q
√
√ 1 6 inf
1
?
?
3 + 2 3 − 3 = 0.2701 . . .
N >2 γ (N ) 6 γ (3) = 3
4
6-a
Weak coupling
. γ = 0: S = {−1, 0, 1}Λ, S0 = {−1, 1}Λ, G = hypercube.
Theorem: ∀N , ∃γ ?(N ) > 0 s.t. points of each Sk (γ) continuous in
γ for 0 6 γ < γ ?(N )
q
√
√ 1 6 inf
1
?
?
3 + 2 3 − 3 = 0.2701 . . .
N >2 γ (N ) 6 γ (3) = 3
4
. 0 < γ 1:
γ X ?
?
?
Vγ (x (γ)) = V0(x (0)) +
(xi+1(0) − x?i(0))2 + O(γ 2)
4 i∈Λ
−
−
−
−
−
−
−
−
1
4
1
4
+ 32 γ
+ 21 γ
V +
−
−
0
−
−
−
−
−
−
−
+
−
−
−
−
−
−
−
+
0
−
−
−
−
−
−
+
+
−
−
−
−
−
0
+
+
−
−
−
−
−
+
+
+
−
−
−
−
−
+
+
+
0
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
0
−
−
−
+
+
+
+
+
−
−
0
+
+
+
+
+
−
−
+
+
+
+
+
+
−
−
+
+
+
+
+
+
0
−
+
+
+
+
+
+
+
−
+
+
+
+
+
+
+
0
+
+
+
+
+
+
+
+
(1,1,1,...,1)
(1,1,1,...,−1)
(−1,1,1,...,−1)
N
4
2γ
0
time
Figure 1
(−1,−1,−1,...,−1)
6-b
Strong coupling: Synchronisation
Remarks: • I ± = ±(1, 1, . . . 1) ∈ S0 ∀γ
• O = (0, 0, . . . 0) ∈ S ∀γ
7
Strong coupling: Synchronisation
Remarks: • I ± = ±(1, 1,(a)
. . . 1) ∈I +S0 ∀γ
• O = (0, 0, . . . 0) ∈ S ∀γ
h
i
2
1
N 1 − O(N −2 )
Let γ1 =
= 2π
2
1 − cos(2π/N )
Theorem:
• S = {I −, I +, O} ⇔ γ > γ1
• S1 = {O} ⇔ γ > γ1
I−
(b)
I+
O
I−
Figure 1
7-a
Strong coupling: Synchronisation
Remarks: • I ± = ±(1, 1,(a)
. . . 1) ∈I +S0 ∀γ
• O = (0, 0, . . . 0) ∈ S ∀γ
h
i
2
1
N 1 − O(N −2 )
Let γ1 =
= 2π
2
1 − cos(2π/N )
Theorem:
• S = {I −, I +, O} ⇔ γ > γ1
• S1 = {O} ⇔ γ > γ1
(b)
I+
O
I−
I−
Figure 1
Proof:
 1−γ γ/2
γ/2 
γ/2 . . . . . .
3

ẋ = Ax − F (x), A =
. . . . . . γ/2 , Fi(x) = xi
γ/2
γ/2 1−γ
X
1
1 kx − Rxk2
Lyapunov function:
W (x) = 2
(xi − xi+1)2 = 2
i∈Λ
Rx = (x2, . . . , xN , x1)
dW (x)
d (x−Rx)i
=
hx−Rx,
dt
dt
6 hx−Rx, A(x−Rx)i 6 (1− γγ )kx−Rxk2
1
7-b
Strong coupling: Synchronisation
Remark: V (O) − V (I −) = V (O) − V (I +) = N/4
−
Corollary: ∀N , ∀γ > γ1(N ), ∀0 < r < R 6 1
2 , ∀x0 ∈ B(I , r):
• Let τ+ = τ hit(B(I +, r)). Then ∀δ > 0,
n
2
2o
x
(1/2−δ)/σ
(1/2+δ)/σ
lim P 0 e
6 τ+ 6 e
=1
σ→0
1
lim σ 2 log E x0 {τ+} =
σ→0
2
• Let τO = τ hit(B(O, r)),
and τ− = inf{t > τ exit(B(I −, R)) : xt ∈ B(I −, r)}. Then
n
o
x
lim P 0 τO < τ+ τ+ < τ− = 1
σ→0
R
I−
r
O
I+
8
Symmetry groups
Potential Vγ invariant by
• R(x1, . . . , xN ) = (x2, . . . , xN , x1)
• S(x1, . . . , xN ) = (xN , xN −1, . . . , x1)
• C(x1, . . . , xN ) = −(x1, . . . , xN )
⇒ Vγ invariant by group G = DN × Z 2 generated by R, S, C
9
Symmetry groups
Potential Vγ invariant by
• R(x1, . . . , xN ) = (x2, . . . , xN , x1)
• S(x1, . . . , xN ) = (xN , xN −1, . . . , x1)
• C(x1, . . . , xN ) = −(x1, . . . , xN )
⇒ Vγ invariant by group G = DN × Z 2 generated by R, S, C
G acts as group of transformations on X , S, Sk ∀k
• Orbit of x ∈ X : Ox = {gx : g ∈ G}
• Isotropy group of x ∈ X : Cx = {g ∈ G : gx = x} / G
• Fixed-point space of H / G: Fix(H) = {x ∈ X : hx = x ∀h ∈ H}
Properties:
|Cx||Ox| = |G|
Cgx = gCxg −1
Fix(gHg −1) = g Fix(H)
9-a
N =2
z?
Oz ?
Cz ?
Fix(Cz ? )
(0, 0)
(1, 1)
(1, −1)
(1, 0)
{(0, 0)}
{(1, 1), (−1, −1)}
{(1, −1), (−1, 1)}
{±(1, 0), ±(0, 1)}
G
D2 = {id, S}
{id, CS}
{id}
{(0, 0)}
{(x, x)}x∈R = D
{(x, −x)}x∈R
{(x, y)}x,y∈R = X
10
N =2
z?
Oz ?
Cz ?
Fix(Cz ? )
(0, 0)
(1, 1)
(1, −1)
(1, 0)
{(0, 0)}
{(1, 1), (−1, −1)}
{(1, −1), (−1, 1)}
{±(1, 0), ±(0, 1)}
G
D2 = {id, S}
{id, CS}
{id}
{(0, 0)}
{(x, x)}x∈R = D
{(x, −x)}x∈R
{(x, y)}x,y∈R = X
0
[×2]
(1, 1)
[×1]
(0, 0)
[×2]
[×4]
1/3
1/2
γ
(x, x)
(0, 0)
(x, y)
(1, 0)
Aa
I+
A
Aa
I−
O
A
(x, −x)
(1, −1)
I±
Aa
I+
A
I−
I+
O
I−
Figure 1
10-a
N =3
0
[×2]
(1, 1, 1)
[×1]
(0, 0, 0)
[×6]
(0, 0, 0)
(x, x, y)
(1, 1, 0)
(x, x, y)
O
A
∂a
(x, x, y)
∂b
I±
B
(x, −x, 0)
[×6] (1, −1, 0)
[×6]
γ
(x, x, x)
(0, 0, 1)
[×6] (1, 1, −1)
2/3
γ!
∂b
I+
I+
I+
A
A
O
∂a
I
−
I−
I−
Figure 1
11
N =4
0
1
3
γ!
γ̃1
[×2]
(1, 1, 1, 1)
[×1]
(0, 0, 0, 0)
[×2]
(1, −1, 1, −1)
[×4]
(1, 0, 1, 0)
[×8]
(1, 0, 1, −1)
[×8]
(1, −1, 0, 0)
[×8]
(0, 1, 0, 0)
[×4]
(1, 0, −1, 0)
2
5
1
2
2
3
1
γ̃2
γ
(x, x, x, x)
(0, 0, 0, 0)
[×4]
(1, 1, −1, −1)
[×8]
(1, 1, 0, 0)
[×16]
(1, 1, 0, −1)
[×8]
(1, 1, 1, −1)
[×8]
(1, 1, 1, 0)
(x, −x, x, −x)
I±
O
A(2)
(x, y, x, y)
(x, y, x, z)
(x, −x, y, −y)
(x, y, x, z)
B
(x, 0, −x, 0)
A
(x, x, −x, −x)
(x, x, y, y)
Aa
(x, y, z, t)
(x, y, x, z)
(x, y, x, z)
I
+
Aaα
∂a
I − ∂b
Aaα
∂a
∂b
Aaα
A
I+
Aa
I+
A
I−
I+
A
I−
Figure 1
I+
A
I−
O
I−
12
Desynchronisation
Theorem: ∀ even N , ∃δ(N ) > 0 s.t. for γ1 − δ(N ) < γ < γ1,
|S| = 2N + 3, and can be decomposed as
S0 = OI + = {I +, I −}
S1 = OA = {A, RA, . . . , RN −1A}
S2 = OB = {B, RB, . . . , RN −1B}
S3 = OO = {O}
with
q
γ
γ
2
2π
1
Aj (γ) = √
1 − γ sin N j − 2 + O 1 − γ
1
1
3
Vγ (A)
γ 2
γ 3
1
= −6 1 − γ
+ O (1 − γ )
N
1
1
13
Desynchronisation
Theorem: ∀ even N , ∃δ(N ) > 0 s.t. for γ1 − δ(N ) < γ < γ1,
|S| = 2N + 3, and can be decomposed as
S0 = OI + = {I +, I −}
S1 = OA = {A, RA, . . . , RN −1A}
S2 = OB = {B, RB, . . . , RN −1B}
S3 = OO = {O}
with
q
γ
γ
2
2π
1
Aj (γ) = √
1 − γ sin N j − 2 + O 1 − γ
1
1
3
Vγ (A)
γ 2
γ 3
1
= −6 1 − γ
+ O (1 − γ )
N
1
1
. N odd: similar result, |S| > 4N + 3
. Similar corollary τ , with τ0 7→ τ∪gA
. A and B have particular symmetries
13-a
Symmetries
N = 4L
xL
N = 4L + 2
xL+1
xL
x1
N = 2L + 1
x1
x1
x1
xL
0
A
−x1
−xL
−xL
xL
xL
x1
x1
x1
x1
0
0
0
0
−x1 −x1
−x1
−xL
−x1
−xL+1
xL
B
−x1 −xL
−x1 −x1
x1
−x1
−xL
xL
x1
x0
xL
x1
−xL
Figure 1
N
x
Fix(Cx)
4L
A
B
(x1, . . . , xL, xL, . . . , x1, −x1, . . . , −xL, −xL, . . . , −x1)
(x1, . . . , xL, . . . , x1, 0, −x1, . . . , −xL, . . . , −x1, 0)
4L + 2
A
B
(x1, . . . , xL+1, . . . , x1, −x1, . . . , −xL+1, . . . , −x1)
(x1, . . . , xL, xL . . . , x1, 0, −x1, . . . , −xL, −xL, . . . , −x1, 0)
2L + 1
A
B
(x1, . . . , xL, −xL, . . . , −x1, 0)
(x1, . . . , xL, xL, . . . , x1, x0)
1
14
Case N large
Let γe = γγ = γ(1 − cos(2π/N )),
1
1−cos(2π/N )
γeM = 1−cos(2πM/N )
= M12 + O N12
Theorem: ∀M > 1, ∃NM < ∞ s.t. for N > NM and γeM +1 < γe < γeM ,
S can be decomposed as
S0 = OI + = {I +, I −}
S2m−1 = OA(m)
S2m = OB (m)
S2M +1 = OO = {O}
(m)
with Aj (γe ) = a(m2γe ) sn
4 K(κ(m2 γ
e))
N
m = 1, . . . , M
m = 1, . . . , M ,
2e
m j−1
2 , κ(m γ )
+O M
N
1.1
1.0
and κ(γe ), a(γe ) implicitly defined by
2
γe = 4 K(κ(γe))π2(1+κ(γe)2)
2κ(γ
e
)2
2
a(γe ) = 1+κ(γe)2
0.9
0.8
a(!
γ)
0.7
0.6
κ(!
γ)
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 1
0.8
0.9
1.0
1.1
γ
!
15
Case N large: bifurcation diagram (N=4L)
0
[×2]
14L
[×1]
04L
[×2L]
(1L−1 0(−1)L−1 0)2
[×2L]
1L (−1)L 1L (−1)L
[×4L]
12L−1 0(−1)2L−1 0
[×4L]
12L (−1)2L
[×8L]
12L 0(−1)2L−2 0
[×16L]
γ
!2
1
γ
!
12L 0(−1)2L−1
Figure 1
16
V (A(2) )−V (I ± )
N
Potential difference:
V (A)−V (I ± )
H(γe ) =
N
h
i
E(κ)
2+κ2
1
1
= 4 − 3(1+κ2) 1+κ2 − 2 K(κ)
2
+ O κN
κ = κ(γe )
0.2
V (A) − V (I ± )
N
0.1
0.0
0.0 0.1 0.2 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
Figure 1
1
17
γ
!
V (A(2) )−V (I ± )
N
Potential difference:
V (A)−V (I ± )
H(γe ) =
N
h
i
E(κ)
2+κ2
1
1
= 4 − 3(1+κ2) 1+κ2 − 2 K(κ)
2
+ O κN
κ = κ(γe )
0.2
V (A) − V (I ± )
N
0.1
0.0
0.0 0.1 0.2 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
γ
!
Figure 1
Corollary: ∀0 < γe 6 1, ∃N0(γe ) s.t. ∀N > N0(γe ),
−
∀0 < r < R 6 1
2 , ∀x0 ∈ B(I , r):
• Let τ+ = τ hit(B(I +, r)). Then ∀δ > 0,
n
2
2o
x
(2H(
γ
e
)−δ)/σ
(2H(
γ
e
)+δ)/σ
lim P 0 e
6 τ+ 6 e
=1
σ→0
lim σ 2 log E x0 {τ+} = 2H(γe )
σ→0
S
hit
• Let τA = τ ( g∈G B(gA, r)),
and τ− = inf{t > τ exit(B(I −, R)) : xt ∈ B(I −, r)}. Then
n
o
x
0
lim P
τA < τ+ τ+ < τ− = 1
σ→0
1
17-a
Ideas of the proof
x∈S
⇔
⇔
h
i
γ
f (xn) + 2 xn+1 − 2xn + xn−1 = 0
(
1 ε2 f (x )
xn+1 = xn + εwnh − 2
n
i
1
wn+1 = wn − 2 ε f (xn) + f (xn+1)
r
2π
√ 1
'
ε= 2
γ
N γ
e
18
Ideas of the proof
x∈S
⇔
⇔
h
i
γ
f (xn) + 2 xn+1 − 2xn + xn−1 = 0
(
1 ε2 f (x )
xn+1 = xn + εwnh − 2
n
i
1
wn+1 = wn − 2 ε f (xn) + f (xn+1)
r
2π
√ 1
'
ε= 2
γ
N γ
e
. Area-preserving map
. Discretisation of ẍ = −f (x)
2 + w 2 ) − 1 x4
. Almost conserved quantity: C(x, w) = 1
(x
2
4
3
C(xn+1, wn+1) = C(xn, wn) + O(ε )
18-a
Ideas of the proof
x∈S
⇔
⇔
h
i
γ
f (xn) + 2 xn+1 − 2xn + xn−1 = 0
(
1 ε2 f (x )
xn+1 = xn + εwnh − 2
n
i
1
wn+1 = wn − 2 ε f (xn) + f (xn+1)
r
2π
√ 1
'
ε= 2
γ
N γ
e
. Area-preserving map
. Discretisation of ẍ = −f (x)
2 + w 2 ) − 1 x4
. Almost conserved quantity: C(x, w) = 1
(x
2
4
3
C(xn+1, wn+1) = C(xn, wn) + O(ε )
In action-angle variables (I, ψ):

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
I = h(C), and (ψ, C) 7→ (x, w) involves elliptic functions.
18-b
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Ω(I) monotonous in I ⇒ twist map.
19
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Ω(I) monotonous in I ⇒ twist map.
. “ε3 = 0”:

ψ = ψ + nεΩ(I )
n
0
0
In = I
0
(mod 2π)
Orbit of period N if N εΩ(I0) = 2πM , M ∈ {1, 2, . . . }.
ν = M/N : rotation number, j 7→ xj has 2M sign changes.
19-a
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Ω(I) monotonous in I ⇒ twist map.
. “ε3 = 0”:

ψ = ψ + nεΩ(I )
n
0
0
In = I
0
(mod 2π)
Orbit of period N if N εΩ(I0) = 2πM , M ∈ {1, 2, . . . }.
ν = M/N : rotation number, j 7→ xj has 2M sign changes.
. ε > 0: Poincaré–Birkhoff theorem: ∃ at least two periodic orbits
for each ν with 2πν/ε in range of Ω.
Problem: Show that there are only two orbits for each ν.
19-b
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Generating function: (ψn, ψn+1) 7→ G(ψn, ψn+1) such that
∂1G(ψn, ψn+1) = −In
∂2G(ψn, ψn+1) = In+1
20
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Generating function: (ψn, ψn+1) 7→ G(ψn, ψn+1) such that
∂1G(ψn, ψn+1) = −In
∂2G(ψn, ψn+1) = In+1
Property: Orbits of period N are stationary points of
GN (ψ1, . . . , ψN ) = G(ψ1, ψ2) + G(ψ2, ψ3) + · · · + G(ψN , ψ1 + 2πN ν)
20-a
Ideas of the proof

ψ
3
n+1 = ψn + εΩ(In) + ε f (ψn, In, ε)
3
I
n+1 = In + ε g(ψn, In, ε)
(mod 2π)
Generating function: (ψn, ψn+1) 7→ G(ψn, ψn+1) such that
∂1G(ψn, ψn+1) = −In
∂2G(ψn, ψn+1) = In+1
Property: Orbits of period N are stationary points of
GN (ψ1, . . . , ψN ) = G(ψ1, ψ2) + G(ψ2, ψ3) + · · · + G(ψN , ψ1 + 2πN ν)
In our case,
!
∞
X
!
ψ2 − ψ1
ψ2 − ψ1
3
b
, ε +2ε
, ε cos p(ψ1+ψ2)
G(ψ1, ψ2) = εG0
Gp
ε
ε
p=1
. N particles “connected by springs” in a periodic ext. potential.
. Stationary pts can be analysed by Fourier transf. for (ψ1, . . . , ψn).
20-b