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Large deviation for the Ginzburg-Landau ∇φ interface model with aconservation law
Takao NISHIKAWA
March 29, 2008
Model
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Microscopic interface
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Interface φ = {φ(x) ∈ R; x ∈ Zd}
Zd( )Rd
Á( )x
x
φ(x): the height at posision x
Energy of miscroscopic interface
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Energy of the microscopic interface φ = {φ(x) ∈ R; x ∈ Zd}
H(φ) =12
∑x,y∈Zd,|x−y|=1
V (φ(x) − φ(y))
(V : R → R is C2, symm., strictly convex)
Dynamics without conservation laws
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Langevin eq.
dφ0t (x) = − ∂H
∂φ(x)(φ0
t )dt +√
2dwt(x), x ∈ ΓN (1)
• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s
• ∂H
∂φ(x)=
∑y:|x−y|=1
V ′(φ(x) − φ(y))
Dynamics without conservation laws
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Langevin eq.
dφ0t (x) = − ∂H
∂φ(x)(φ0
t )dt +√
2dwt(x), x ∈ ΓN (1)
• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s
• ∂H
∂φ(x)=
∑y:|x−y|=1
V ′(φ(x) − φ(y))
Dynamics without conservation laws
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Langevin eq.
dφ0t (x) = − ∂H
∂φ(x)(φ0
t )dt +√
2dwt(x), x ∈ ΓN (1)
• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s
• ∂H
∂φ(x)=
∑y:|x−y|=1
V ′(φ(x) − φ(y))
⇐⇒ Glauber dynamics
Dynamics with a conservation law (1)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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dφt(x) = Δ{
∂H
∂φ(·)(φt)}
(x)dt +√
2dwt(x), x ∈ ΓN (2)
• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure
E[ws(x)wt(y)] = Δ(x, y)s ∧ t
• Δ = ΔΓN: (discrete) Laplacian on ΓN
Δf(x) =∑
y∈ΓN ,|x−y|=1
(f(y) − f(x)), x ∈ ΓN
Dynamics with a conservation law (1)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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dφt(x) = Δ{
∂H
∂φ(·)(φt)}
(x)dt +√
2dwt(x), x ∈ ΓN (2)
• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure
E[ws(x)wt(y)] = Δ(x, y)s ∧ t
• Δ = ΔΓN: (discrete) Laplacian on ΓN
Δf(x) =∑
y∈ΓN ,|x−y|=1
(f(y) − f(x)), x ∈ ΓN
Dynamics with a conservation law (1)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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dφt(x) = Δ{
∂H
∂φ(·)(φt)}
(x)dt +√
2dwt(x), x ∈ ΓN (2)
• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure
E[ws(x)wt(y)] = Δ(x, y)s ∧ t
• Δ = ΔΓN: (discrete) Laplacian on ΓN
Δf(x) =∑
y∈ΓN ,|x−y|=1
(f(y) − f(x)), x ∈ ΓN
Dynamics with a conservation law (2)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Remark 1. By Ito’s formula, it is easy to see∑x∈ΓN
φt(x) ≡∑
x∈ΓN
φ0(x) (= const.), t ≥ 0, (3)
that is, the total sum of the height variable (� number of particle) isconserved by this time evolution.
Dynamics with a conservation law (2)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Remark 1. By Ito’s formula, it is easy to see∑x∈ΓN
φt(x) ≡∑
x∈ΓN
φ0(x) (= const.), t ≥ 0, (3)
that is, the total sum of the height variable (� number of particle) isconserved by this time evolution.
⇐⇒ Kawasaki dynamics
Macroscopic dynamics (scaling)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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1. dynamics without conservation lawshN,0(t, θ) (t ∈ [0, t], θ ∈ [0, 1)d =: T
d)
hN,0(t, x/N) = N−1φ0N2t(x), x ∈ ΓN
(N−1 in space, N2 in time)
2. dynamics with a conservation lawhN (t, θ) (t ∈ [0, t], θ ∈ T
d)
hN (t, x/N) = N−1φN4t(x), x ∈ ΓN
(N−1 in space, N4 in time)
Macroscopic dynamics (scaling)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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1. dynamics without conservation lawshN,0(t, θ) (t ∈ [0, t], θ ∈ [0, 1)d =: T
d)
hN,0(t, x/N) = N−1φ0N2t(x), x ∈ ΓN
(N−1 in space, N2 in time)
2. dynamics with a conservation lawhN (t, θ) (t ∈ [0, t], θ ∈ T
d)
hN (t, x/N) = N−1φN4t(x), x ∈ ΓN
(N−1 in space, N4 in time)
Hydrodynamic scaling limit (LLN)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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1. dynamics without conservation laws ([FS97],[N03])
hN,0 −→ h0 :∂h0
∂t= div∇σ(∇h0),
where σ : Rd → R is surface tension introduced via
thermodynamic limit
2. dynamics with a conservation law ([N02])
hN −→ h :∂h
∂t= −Δ div∇σ(∇h)
Hydrodynamic scaling limit (LLN)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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1. dynamics without conservation laws ([FS97],[N03])
hN,0 −→ h0 :∂h0
∂t= div∇σ(∇h0),
where σ : Rd → R is surface tension introduced via
thermodynamic limit
2. dynamics with a conservation law ([N02])
hN −→ h :∂h
∂t= −Δ div∇σ(∇h)
Hydrodynamic scaling limit (LLN)
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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1. dynamics without conservation laws ([FS97],[N03])
hN,0 −→ h0 :∂h0
∂t= div∇σ(∇h0),
where σ : Rd → R is surface tension introduced via
thermodynamic limit
2. dynamics with a conservation law ([N02])
hN −→ h :∂h
∂t= −Δ div∇σ(∇h)
Does Large deviation principle hold?
LDP for the non-conservative model
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Theorem 1 ([FN01]). hN,0 satisfies the large deviation principle inC([0, T ], L2
w(Td)) with the speed Nd. The rate functional has thefollowing expression if h is smooth enough:
I(h) =∫ T
0
∥∥∥∥ d
dth(t) − div∇σ(∇h(t))
∥∥∥∥2
L2
dt
LDP for the non-conservative model
Model
• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model
Main result
Rough sketch of theproof
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Theorem 1 ([FN01]). hN,0 satisfies the large deviation principle inC([0, T ], L2
w(Td)) with the speed Nd. The rate functional has thefollowing expression if h is smooth enough:
I(h) =∫ T
0
∥∥∥∥ d
dth(t) − div∇σ(∇h(t))
∥∥∥∥2
L2
dt
LDP for the dynamics with conservation law?
Main result
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Assumptions
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Assumption 1. We assume the following:
• d ≤ 3 (technical)
• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:
◦ 〈hN (0)〉 :=∫
Td
h(0, θ) dθ ≡ 0,
◦ ∃h0 ∈ H−1(Td) s.t. limN→∞
‖hN (0) − h0‖H−1 = 0,
◦ supN≥1
‖hN (0)‖H1 < ∞
Assumptions
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Assumption 1. We assume the following:
• d ≤ 3 (technical)
• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:
◦ 〈hN (0)〉 :=∫
Td
h(0, θ) dθ ≡ 0,
◦ ∃h0 ∈ H−1(Td) s.t. limN→∞
‖hN (0) − h0‖H−1 = 0,
◦ supN≥1
‖hN (0)‖H1 < ∞
Assumptions
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Assumption 1. We assume the following:
• d ≤ 3 (technical)
• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:
◦ 〈hN (0)〉 :=∫
Td
h(0, θ) dθ ≡ 0,
◦ ∃h0 ∈ H−1(Td) s.t. limN→∞
‖hN (0) − h0‖H−1 = 0,
◦ supN≥1
‖hN (0)‖H1 < ∞
Large deviation principle
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Theorem 2. hN satisfying large deviation principle inC([0, T ], H−1
w (Td)) with the speed Nd and the rate functionalI(h). The rate functional I(h) has the following expression if h issmooth:
I(h) =∫ T
0
∥∥∥∥ d
dth(t) − Δ div∇σ(∇h(t))
∥∥∥∥2
H−1
dt
The rate functional I(h)
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Remark 2. Exactly saying, the rate functional is expressed asfollows:
I(h) = supJ∈C∞([0,T ]×Td),〈J(t)〉=0
I(h; J)
I(h; J) =∫
Td
J(T, θ)h(T, θ) dθ −∫
Td
J(0, θ)h0(θ) dθ
−∫ T
0dt
∫Td
∂J
∂t(t, θ)h(t, θ) dθ
+∫ T
0dt
∫Td
(∇ΔJ(t, θ),∇σ(∇h(t, θ)))Rd dθ
+∫ T
0dt
∫Td
J(t, θ)ΔJ(t, θ) dθ
The rate functional I(h)
Model
Main result
• Assumptions• Large deviationprinciple• The rate functionalI(h)
Rough sketch of theproof
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Remark 2. Exactly saying, the rate functional is expressed asfollows:
I(h) = supJ∈C∞([0,T ]×Td),〈J(t)〉=0
I(h; J)
I(h; J) =∫
Td
J(T, θ)h(T, θ) dθ −∫
Td
J(0, θ)h0(θ) dθ
−∫ T
0dt
∫Td
∂J
∂t(t, θ)h(t, θ) dθ
+∫ T
0dt
∫Td
(∇ΔJ(t, θ),∇σ(∇h(t, θ)))Rd dθ
+∫ T
0dt
∫Td
J(t, θ)ΔJ(t, θ) dθ
Rough sketch of the proof
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
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Rough sketch of the proof
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
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The proof is by H−1-method in
• Landim-Yau, Commun. Pure Appl. Math. (’95)
• Funaki-N., Probab. Theory Relat. Fields (’01)
Rough sketch of the proof
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
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The proof is by H−1-method in
• Landim-Yau, Commun. Pure Appl. Math. (’95)
• Funaki-N., Probab. Theory Relat. Fields (’01)
Rough sketch of the proof
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
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The proof is by H−1-method in
• Landim-Yau, Commun. Pure Appl. Math. (’95)
• Funaki-N., Probab. Theory Relat. Fields (’01)
Notations (1)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
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• (Zd)∗: all oriented bonds in Zd, i.e.
(Zd)∗ = {(x, y) ∈ Zd × Z
d; |x − y| = 1}
• Γ∗N : all oriented bonds in ΓN
• X : all η ∈ R(Z
d)∗ satisfying the following conditions:
1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).
2. For every closed loops C∑b∈C
η(b) = 0
holds.
Notations (1)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
17 / 28
• (Zd)∗: all oriented bonds in Zd, i.e.
(Zd)∗ = {(x, y) ∈ Zd × Z
d; |x − y| = 1}
• Γ∗N : all oriented bonds in ΓN
• X : all η ∈ R(Z
d)∗ satisfying the following conditions:
1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).
2. For every closed loops C∑b∈C
η(b) = 0
holds.
Notations (1)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
17 / 28
• (Zd)∗: all oriented bonds in Zd, i.e.
(Zd)∗ = {(x, y) ∈ Zd × Z
d; |x − y| = 1}
• Γ∗N : all oriented bonds in ΓN
• X : all η ∈ R(Z
d)∗ satisfying the following conditions:
1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).
2. For every closed loops C∑b∈C
η(b) = 0
holds.
Notations (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
18 / 28
• ∇: discrete gradient
∇φ(b) = φ(x) − φ(y), b = (x, y)
• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X
• F (u): expected value of F : RX → R with respect to μu
F (u) = Eμu [F (η)]
• Λl = {x ∈ Zd; max |xi| ≤ l}
• τy : spatical shift by y ∈ Zd
• AvΛ F (η): sample mean of F on Λ � ΓN , that is,
Notations (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
18 / 28
• ∇: discrete gradient
∇φ(b) = φ(x) − φ(y), b = (x, y)
• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X
• F (u): expected value of F : RX → R with respect to μu
F (u) = Eμu [F (η)]
• Λl = {x ∈ Zd; max |xi| ≤ l}
• τy : spatical shift by y ∈ Zd
• AvΛ F (η): sample mean of F on Λ � ΓN , that is,
Notations (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
18 / 28
• ∇: discrete gradient
∇φ(b) = φ(x) − φ(y), b = (x, y)
• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X
• F (u): expected value of F : RX → R with respect to μu
F (u) = Eμu [F (η)]
• Λl = {x ∈ Zd; max |xi| ≤ l}
• τy : spatical shift by y ∈ Zd
• AvΛ F (η): sample mean of F on Λ � ΓN , that is,
Notations (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
18 / 28
• ∇: discrete gradient
∇φ(b) = φ(x) − φ(y), b = (x, y)
• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X
• F (u): expected value of F : RX → R with respect to μu
F (u) = Eμu [F (η)]
• Λl = {x ∈ Zd; max |xi| ≤ l}
• τy : spatical shift by y ∈ Zd
• AvΛ F (η): sample mean of F on Λ � ΓN , that is,
Notations (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
18 / 28
• ∇: discrete gradient
∇φ(b) = φ(x) − φ(y), b = (x, y)
• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X
• F (u): expected value of F : RX → R with respect to μu
F (u) = Eμu [F (η)]
• Λl = {x ∈ Zd; max |xi| ≤ l}
• τy : spatical shift by y ∈ Zd
• AvΛ F (η): sample mean of F on Λ � ΓN , that is,
Notations (3)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
19 / 28
• ηlx: sample mean of η ∈ X on Λl + x, i.e.,
ηlx = (2l + 1)−d
∑y∈Λl+x
d∑i=1
η(ei + y)ei ∈ Rd, (4)
where ei is i-th unit vector in Rd
• WFl : the difference between the sample mean and the
ensemble mean
WFl (η) = AvΛl
F (η) − F (ηl0), η ∈ X (5)
Notations (3)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
19 / 28
• ηlx: sample mean of η ∈ X on Λl + x, i.e.,
ηlx = (2l + 1)−d
∑y∈Λl+x
d∑i=1
η(ei + y)ei ∈ Rd, (4)
where ei is i-th unit vector in Rd
• WFl : the difference between the sample mean and the
ensemble mean
WFl (η) = AvΛl
F (η) − F (ηl0), η ∈ X (5)
Exponential a-priori estimate
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
20 / 28
Proposition 3 (A priori estimate). There exists a constant κ > 0such that for all T ≥ 0, 0 < β ≤ 1, N ≥ 1 and all initial datum
E
[exp
{βe−κT
∫ T
0
(∑b∈Γ∗
N
V ′(∇φNs (b))∇φN
s (b)
+ Nd‖hN (s)‖2H−1
)ds
}]
≤ exp{βNd‖hN‖2H−1 + 2βNd/κ}.
(6)
holds.
Superexponential local equilibrium
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
21 / 28
Theorem 4. For all α > 0 we have
limε↓0
limN→∞
N−d log E
⎡⎣exp
⎛⎝α
∫ T
0
∑x∈ΓN
∣∣τxWFNε(∇φN
t )∣∣ dt
⎞⎠⎤⎦ ≤ 0.
Superexponential local equilibrium
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
21 / 28
Theorem 4. For all α > 0 we have
limε↓0
limN→∞
N−d log E
⎡⎣exp
⎛⎝α
∫ T
0
∑x∈ΓN
∣∣τxWFNε(∇φN
t )∣∣ dt
⎞⎠⎤⎦ ≤ 0.
This theorem is the core of the proof of LDP
Superexponential 1-block estimate
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
22 / 28
Theorem 5. For all α > 0 we have
liml→∞
limN→∞
N−d log E
⎡⎣exp
⎛⎝α
∫ T
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
⎞⎠⎤⎦ ≤ 0.
Proof of Theorem 5 (1)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
23 / 28
Since we have
EφN
[exp
(α
∫ T
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
)]
≤(
supφN
EφN
[exp
(α
∫ N−dT
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
)])Nd
by Markov property, it is sufficient to show that
liml→∞
limN→∞
log supφN
EφN
[exp
(α
∫ N−dT
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
)]≤ 0.
Proof of Theorem 5 (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
24 / 28
Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have
liml→∞
limN→∞
supφN
EφN
⎡⎣∫ N−dT
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
⎤⎦ ≤ 0,
we obtain the conclusion.
Proof of Theorem 5 (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
24 / 28
Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have
liml→∞
limN→∞
supφN
EφN
⎡⎣∫ N−dT
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
⎤⎦ ≤ 0,
we obtain the conclusion.
Proof of Theorem 5 (2)
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
24 / 28
Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have
liml→∞
limN→∞
supφN
EφN
⎡⎣∫ N−dT
0
∑x∈ΓN
∣∣τxWFl (∇φN
t )∣∣ dt
⎤⎦ ≤ 0,
we obtain the conclusion.
This estimate is the ordinary 1-block estimateif d ≤ 3, and therefore we can prove it by astandard way!
Superexponential 2-blocks estimate
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
25 / 28
Theorem 6. For every α > 0 we have
liml→∞
limε↓0
limN→∞
N−d log EφN[exp
(α
∫ T
0
∑x∈ΓN
∣∣∣W f,Nl,ε (∇φN
t )∣∣∣ dt
)]≤ 0.
Once we have following theorem, we can obtain the above.
Theorem 7. For every α > 0 we have
liml→∞
limε↓0
limN→∞
N−d log EφN⎡⎣exp
⎛⎝α |ΛNε|−1
∫ T
0
∑x∈ΓN
∑y∈ΛNε�Λ2l
∣∣ηlx − ηl
x+y
∣∣2 dt
⎞⎠⎤⎦ ≤ 0.
Superexponential 2-blocks estimate
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
25 / 28
Theorem 6. For every α > 0 we have
liml→∞
limε↓0
limN→∞
N−d log EφN[exp
(α
∫ T
0
∑x∈ΓN
∣∣∣W f,Nl,ε (∇φN
t )∣∣∣ dt
)]≤ 0.
Once we have following theorem, we can obtain the above.
Theorem 7. For every α > 0 we have
liml→∞
limε↓0
limN→∞
N−d log EφN⎡⎣exp
⎛⎝α |ΛNε|−1
∫ T
0
∑x∈ΓN
∑y∈ΛNε�Λ2l
∣∣ηlx − ηl
x+y
∣∣2 dt
⎞⎠⎤⎦ ≤ 0.
Proof of Theorem 7
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
26 / 28
By a similar manner to [FN01], it is sufficient to show the following:
liml→∞
limε↓0
limN→∞
N−d log EφN
[exp
⎛⎝α |ΛNε|−1
∫ T
0
∑y∈ΛNε
⎛⎝∑
b∈Γ∗N
Ψy(b;∇φNt ) − 4Nd
⎞⎠ dt
⎞⎠] ≤ 0
(7)
• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))
} {η(b + y) − η(b)}• Thermodynamic Identity:
d∑i=1
E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1
Proof of Theorem 7
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
26 / 28
By a similar manner to [FN01], it is sufficient to show the following:
liml→∞
limε↓0
limN→∞
N−d log EφN
[exp
⎛⎝α |ΛNε|−1
∫ T
0
∑y∈ΛNε
⎛⎝∑
b∈Γ∗N
Ψy(b;∇φNt ) − 4Nd
⎞⎠ dt
⎞⎠] ≤ 0
(7)
• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))
} {η(b + y) − η(b)}• Thermodynamic Identity:
d∑i=1
E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1
Proof of Theorem 7
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
26 / 28
By a similar manner to [FN01], it is sufficient to show the following:
liml→∞
limε↓0
limN→∞
N−d log EφN
[exp
⎛⎝α |ΛNε|−1
∫ T
0
∑y∈ΛNε
⎛⎝∑
b∈Γ∗N
Ψy(b;∇φNt ) − 4Nd
⎞⎠ dt
⎞⎠] ≤ 0
(7)
• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))
} {η(b + y) − η(b)}• Thermodynamic Identity:
d∑i=1
E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1
Remaining Tasks
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
27 / 28
• LDP in the case of the dimension d ≥ 4
To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)
• LDP in the case of Dirichlet-type boundary condition
Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate
Remaining Tasks
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
27 / 28
• LDP in the case of the dimension d ≥ 4
To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)
• LDP in the case of Dirichlet-type boundary condition
Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate
Remaining Tasks
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
27 / 28
• LDP in the case of the dimension d ≥ 4
To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)
• LDP in the case of Dirichlet-type boundary condition
Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate
Remaining Tasks
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
27 / 28
• LDP in the case of the dimension d ≥ 4
To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)
• LDP in the case of Dirichlet-type boundary condition
Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate
Model
Main result
Rough sketch of theproof• Rough sketch of theproof
• Notations (1)
• Notations (2)
• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate
• Proof of Theorem 7
• Remaining Tasks
•
28 / 28
Thank you for your attention!