Non-equilibrium macroscopic evolution of chain
of oscillators with conservative noise.
Stefano Olla
CEREMADE, Université Paris-Dauphine
Supported by ERC MALADY
Budapest, July 23, 2015
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Scalings
Happy Birthday Balint!
Let’s make an orgie of scalings.
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Scalings
A model with different macroscopic space-time scales
The same dynamics has different macroscopic behavior ( → 0) at
different space-time scaling:
▸
Ballistic (hyperbolic scale): εx, εt.
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Scalings
A model with different macroscopic space-time scales
The same dynamics has different macroscopic behavior ( → 0) at
different space-time scaling:
▸
Ballistic (hyperbolic scale): εx, εt.
▸
Diffusive scale: εx, ε2 t.
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Scalings
A model with different macroscopic space-time scales
The same dynamics has different macroscopic behavior ( → 0) at
different space-time scaling:
▸
Ballistic (hyperbolic scale): εx, εt.
▸
Diffusive scale: εx, ε2 t.
▸
Superdiffusive scale: εx, εα t, α < 2.
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Scalings
A model with different macroscopic space-time scales
The same dynamics has different macroscopic behavior ( → 0) at
different space-time scaling:
▸
Ballistic (hyperbolic scale): εx, εt.
▸
Diffusive scale: εx, ε2 t.
▸
Superdiffusive scale: εx, εα t, α < 2.
Mechanical equilibrium is reached on the hyperbolic scale, while
thermal equilibrium is reached in the larger superdiffusive or
diffusive scales.
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Scalings
Chain of oscillators with tension
Hamiltonian dynamics (FPU type):
ṙj (t) = pj (t) − pj−1 (t),
j = 1, . . . , N,
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt,
dpN (t) = (τ1 (t/N) − V (rN (t))) dt,
′
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Scalings
j = 1, . . . , N − 1,
Chain of oscillators with tension
Hamiltonian dynamics (FPU type):
ṙj (t) = pj (t) − pj−1 (t),
j = 1, . . . , N,
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt + γ noise,
j = 1, . . . , N − 1,
dpN (t) = (τ1 (t/N) − V (rN (t))) dt + γ noise,
′
we add a noise that conserve energy and momentum:
momentum exchange For each couple of nearest neighbor particle,
we randomly exchange momentum,
(pi , pi+1 ) → (pi+1 , pi ), with intensity 1.
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Scalings
Chain of oscillators with tension
Hamiltonian dynamics (FPU type):
ṙj (t) = pj (t) − pj−1 (t),
j = 1, . . . , N,
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt + γ noise,
j = 1, . . . , N − 1,
dpN (t) = (τ1 (t/N) − V (rN (t))) dt + γ noise,
′
we add a noise that conserve energy and momentum:
momentum exchange For each couple of nearest neighbor particle,
we randomly exchange momentum,
(pi , pi+1 ) → (pi+1 , pi ), with intensity 1.
diffusive exchange of momentum 3–particle continuous exchange
of momentum.
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Scalings
Chain of oscillators: infinite model
Hamiltonian dynamics (FPU type): j ∈ Z
ṙj (t) = pj (t) − pj−1 (t),
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt
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Scalings
Chain of oscillators: infinite model
Hamiltonian dynamics (FPU type): j ∈ Z
ṙj (t) = pj (t) − pj−1 (t),
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt + γ noise
we add a noise that conserve energy and momentum:
momentum exchange For each couple of nearest neighbor particle,
we randomly exchange momentum,
(pi , pi+1 ) → (pi+1 , pi ), with intensity γ.
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Scalings
Chain of oscillators: infinite model
Hamiltonian dynamics (FPU type): j ∈ Z
ṙj (t) = pj (t) − pj−1 (t),
dpj (t) = (V ′ (rj+1 (t)) − V ′ (rj (t))) dt + γ noise
we add a noise that conserve energy and momentum:
momentum exchange For each couple of nearest neighbor particle,
we randomly exchange momentum,
(pi , pi+1 ) → (pi+1 , pi ), with intensity γ.
diffusive exchange of momentum 3–particle continuous exchange
of momentum.
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Scalings
Gibbs measures and Thermodynamic Entropy
Ej =
pj2
2
+ V (rj )
energy of particle j
Gibbs measure at temperature β −1 , tension τ and momentum p
are:
dµβ,τ,p = ∏ e −β(EE j −ppj −τ rj )−G(β,τ,p) dpj drj
j
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Scalings
Gibbs measures and Thermodynamic Entropy
Ej =
pj2
2
+ V (rj )
energy of particle j
Gibbs measure at temperature β −1 , tension τ and momentum p
are:
dµβ,τ,p = ∏ e −β(EE j −ppj −τ rj )−G(β,τ,p) dpj drj
j
Thermodynamic entropy is
S(u, r ) = inf {−βτ r + βu − G(β, τ, 0)}
τ,β
β(u, r ) = ∂u S(u, r ),
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τ (u, r ) = −β −1 ∂r S(u, r ).
Scalings
Ergodicity (of the infinite system)
The stocastic perturbation of the dynamics is sufficient to give
ergodicity:
Theorem
(Fritz, Funaki, Lebowitz, PTRF 1994)
Assume that ν is translation invariant and stationary, with finite
entropy density, and furthemore that
ν(dp∣r )
is exchangeable.
Then ν is a convex combination of Gibbs measure dµβ,τ,p .
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Scalings
Ergodicity (of the infinite system)
The stocastic perturbation of the dynamics is sufficient to give
ergodicity:
Theorem
(Fritz, Funaki, Lebowitz, PTRF 1994)
Assume that ν is translation invariant and stationary, with finite
entropy density, and furthemore that
ν(dp∣r )
is exchangeable.
Then ν is a convex combination of Gibbs measure dµβ,τ,p .
▸
▸
ν(dp∣r ) maxwellian (Gallavotti-Verboven 1975)
ν(dp∣r ) convex combination of maxwellians (Olla, Varadhan,
Yau, 1993).
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Scalings
Hyperbolic Scaling: Euler equations
3 conserved quantities:
1
N ∑i G (i/N)ri (Nt)
πN (t)[G ] = N1 ∑i G (i/N)pi (Nt)
eN (t)[G ] = N1 ∑i G (i/N)Ei (Nt)
stretch RN (t)[G ] =
momentum
energy
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Scalings
Hyperbolic Scaling: Euler equations
3 conserved quantities:
1
N ∑i G (i/N)ri (Nt)
πN (t)[G ] = N1 ∑i G (i/N)pi (Nt)
eN (t)[G ] = N1 ∑i G (i/N)Ei (Nt)
stretch RN (t)[G ] =
momentum
energy
(RN (t), πN (t), eN (t)) Ð→ (r (x, t)dx, π(x, t)dx, e(x, t)dx)
∂t r = ∂x π
π(0, t) = 0,
∂t π = ∂x τ
∂t e = ∂x (τ π)
τ (r (1, t), U(1, t)) = τ1 (t)
U = e − π 2 /2 : internal energy. For smooth solutions this is proven
in:
▸
N. Even, S.O., ARMA (2014)
▸
S.O., SRS Varadhan, HT Yau, CMP (1993)
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Scalings
entropy evolution
∂t r = ∂x π
π(0, t) = 0,
U = e − π 2 /2, β =
∂S
∂U ,
∂t π = ∂x τ
∂t e = ∂x (τ π)
τ (r (1, t), U(1, t)) = τ1 (t)
τ = − β1 ∂S
∂r
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Scalings
entropy evolution
∂t r = ∂x π
π(0, t) = 0,
U = e − π 2 /2, β =
∂S
∂U ,
∂t π = ∂x τ
∂t e = ∂x (τ π)
τ (r (1, t), U(1, t)) = τ1 (t)
τ = − β1 ∂S
∂r
d
S(r (x, t), U(x, t)) = 0
dt
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for smooth solutions
Scalings
entropy evolution
∂t r = ∂x π
π(0, t) = 0,
U = e − π 2 /2, β =
∂S
∂U ,
∂t π = ∂x τ
∂t e = ∂x (τ π)
τ (r (1, t), U(1, t)) = τ1 (t)
τ = − β1 ∂S
∂r
d
S(r (x, t), U(x, t)) = 0
dt
for smooth solutions
When shocks will appear, we will have dissipation and
1
d
∫ S(r (x, t), U(x, t))dx > 0
dt 0
and (eventually) convergence to a mechanical equilibrium,
characterized by constant tension and momentum.
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Scalings
Mechanical Equilibrium
In particular starting with smooth initial profiles such that
p0 (x) = 0,
τ (u0 (x), r0 (x)) = τ0 = cost
p 2 (x)
u0 (x) = e0 (x) − 2 = e0 (x)
and non constant entropy profile S(u0 (x), ro (x))
are stationary for the Euler equations.
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Scalings
Mechanical Equilibrium
In particular starting with smooth initial profiles such that
p0 (x) = 0,
τ (u0 (x), r0 (x)) = τ0 = cost
p 2 (x)
u0 (x) = e0 (x) − 2 = e0 (x)
and non constant entropy profile S(u0 (x), ro (x))
are stationary for the Euler equations.
This means that they evolve towards thermal equilibrium at a
larger time scale:
▸ diffusive scaling (N 2 t, Nx), if thermal conductivity is finite,
with heat equation governing the evolution of the temperature
profile,
▸ superdiffusive scaling (N α t, Nx), 1 < α < 2, if thermal
conductivity is infinite.
If ∂r τ (u, r ) > 0 (acoustic chains), we expect superdiffusion of heat,
confirmed by many numerical experiments.
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Scalings
Energy Superdiffusion
Consider the Harmonic chain with noise conserving energy and
momentum.
V (r ) =
r2
,
2
τ (r , U) = r ,
S = 1 + log(πβ −1 )
1
1
e(x, 0) = r (x, 0)2 + π(x, 0)2 + β −1 (x, 0)
2
2
In the hyperbolic space-time scale limit we have
∂t r = ∂x π
∂t π = ∂x r
∂t e = ∂x (r π)
linear wave equation: no shocks! no dissipation!
Profiles of temperature and entropy do not change in time:
1
1
β −1 (x, t) = e(x, t) − r (x, t)2 − π(x, t)2 = β −1 (x, 0)
2
2
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Scalings
Energy Superdiffusion
Consider the Harmonic chain with noise conserving energy and
momentum.
V (r ) =
r2
,
2
τ (r , U) = r ,
S = 1 + log(πβ −1 )
1
1
e(x, 0) = r (x, 0)2 + π(x, 0)2 + β −1 (x, 0)
2
2
In the hyperbolic space-time scale limit we have
∂t r = ∂x π
∂t π = ∂x r
∂t e = ∂x (r π)
linear wave equation: no shocks! no dissipation!
Profiles of temperature and entropy do not change in time:
1
1
β −1 (x, t) = e(x, t) − r (x, t)2 − π(x, t)2 = β −1 (x, 0)
2
2
Since microscopically the noise made the dynamic ’ergodic’, the
convergence to equilibrium should happen in another space-time
scale.
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Scalings
Temperature profile
In the infinite space case, in the hyperbolic scale, after t → ∞, all
the phonon modes go to infinity, and the phonon energy
1
1
r (x, t)2 + π(x, t)2 Ð→ 0
t→∞
2
2
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Scalings
Temperature profile
In the infinite space case, in the hyperbolic scale, after t → ∞, all
the phonon modes go to infinity, and the phonon energy
1
1
r (x, t)2 + π(x, t)2 Ð→ 0
t→∞
2
2
e(x, t) Ð→ β −1 (x, 0)
t→∞
The initial temperature profile β −1 (x, 0) is constituted by the
initial profile of the variances of rx and px and the energy of the
high frequencies.
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Scalings
Temperature profile
In the infinite space case, in the hyperbolic scale, after t → ∞, all
the phonon modes go to infinity, and the phonon energy
1
1
r (x, t)2 + π(x, t)2 Ð→ 0
t→∞
2
2
e(x, t) Ð→ β −1 (x, 0)
t→∞
The initial temperature profile β −1 (x, 0) is constituted by the
initial profile of the variances of rx and px and the energy of the
high frequencies.
This temperature profile will start to evolve at a larger time scale,
supediffusive, shorter than the diffusive one.
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Scalings
What superdiffusion?
Assume V (r ) =
αr 2
2 ,
and centered initial conditions:
< px (0) >= 0,
< rx (0) >= 0
This is maintained at any positive time t.
Theorem (Jara-Komorowski-Olla, CMP 2015)
Assume
1
N
∑x ⟨Ei (0)⟩ < C , G ∈ C0 (R):
1
3/2
∑ G (i/N) ⟨Ei (N t)⟩ Ð→ ∫ G (x) T (t, x)dx
N→∞
N i
R
∂t T (x, t) = −ĉ∣∆x ∣3/2 T (x, t)
ĉ =
α3/4
29/4 (3γ)1/2
T (x, 0) = β −1 (x, 0).
In pinned chains, we prove usual diffusive behavior with heat
equation.
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Scalings
Phonon modes: diffusive scale
f±i (t) = pi (t) ± α1/2 (ri (t) ±
▸
3γ − 1
(ri+1 − ri ))
2
Ballistic movement on the hyperbolic scale:
1
±
1
±
1/2
∑ G (i/N) fi (N t) Ð→ ∫ G (x) f̄ (0, x ± α t)dx
N→∞
N i
R
▸
Then diffusive after re-centering
1
−1
1/2 1
±
2
±,d
∑ G (N i ∓ α N t) fi (N t) Ð→ ∫ G (x) f̄ (t, x)dx
N→∞
N i
R
3γ 2 ±,d
∂t f̄±,d =
∂ f̄
2 x
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Scalings
Conjectured result for FPUT dynamics
This behavior at large space-time scale (diffusive for phonon and
superdiffusive for heat mode) is also conjectured for the equilibrium
fluctuations dynamics for the Fermi-Pasta-Ulam-Tsingou model
V (r ) = r 2 + αr 3 + βr 4
Recent fluctuation hydrodynamics and mode coupling calculation
by Herbert Spohn (JSP 2014) for the deterministic FPU dynamics:
▸
▸
If V symmetric (α = 0 or β-FPU), and τ = 0: then diffusion
for phonons and ∣∆∣3/4 -superdiffusion for heat (temperature),
If V asymmetric (α-FPU), or symmetric but tension τ ≠ 0,
then phonon KPZ-superdiffuse, and ∣∆∣5/6 -superdiffusion for
heat.
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Scalings
Non-acoustic chains
These are tensionless chains, obtained by considering also
non-nearest neighbor interactions, for example:
⎤
⎡ p2
⎥
⎢ j
⎢
H = ∑ ⎢ + V (∆qj )⎥⎥ ,
2
⎥
j ⎢
⎦
⎣
∆qj = qj+1 + qj−1 − 2qj
Here the relevant balanced quantity is not the volume strain
rj = qj − qj−1 , but the curvature κj = ∆qj (Beam problem).
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Scalings
Non-acoustic chains
These are tensionless chains, obtained by considering also
non-nearest neighbor interactions, for example:
⎤
⎡ p2
⎥
⎢ j
⎢
H = ∑ ⎢ + V (∆qj )⎥⎥ ,
2
⎥
j ⎢
⎦
⎣
∆qj = qj+1 + qj−1 − 2qj
Here the relevant balanced quantity is not the volume strain
rj = qj − qj−1 , but the curvature κj = ∆qj (Beam problem).
For the harmonic dynamics with random moment exchange we
obtain:
▸
Thermal conductivity is finite! (This provides a first rigorous
counterexample for a famous conjecture).
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Scalings
Non-acoustic chains: heat equation
Starting with initial conditions with 0-curvature and a non-constant
temperature profile we have (T. Komorowski, S.O., 2015):
1
2
∑ G (i/N) ⟨Ei (N t)⟩ Ð→ ∫ G (x) T (t, y )dy
N→∞
N i
R
∂t T (y , t) = D∆y T (y , t)
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Scalings
Non-acoustic chains: heat equation
Starting with initial conditions with 0-curvature and a non-constant
temperature profile we have (T. Komorowski, S.O., 2015):
1
2
∑ G (i/N) ⟨Ei (N t)⟩ Ð→ ∫ G (x) T (t, y )dy
N→∞
N i
R
∂t T (y , t) = D∆y T (y , t)
With a non constant initial profiles of curvature and momentum,
everything evolves on the diffusive scale and the macroscopic
equations are:
∂t κ(y , t) = ∆y p(y , t),
∂t p(y , t) = C ∆y κ(y , t) + γ∆y p(y , t)
∂t T (y , t) = D∆y T (y , t) + ∆y κ2 (y , t)
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Scalings
Wave function and Wigner distribution
ry = qy − qy −1 , y ∈ Z:
q̇x (t) = px (t)
dpx (t) = ∆qx
dt + γ 1/2
∑
Yx+` px (t) ○ dwx+k (t),
`=−1,0,1
{wx (t)}x∈Z iid Wiener processes,
Yx = (px − px+1 )∂px−1 + (px+1 − px−1 )∂px + (px−1 − px )∂px+1
Wave function and Wigner distribution
ry = qy − qy −1 , y ∈ Z:
q̇x (t) = px (t)
dpx (t) = − (α ∗ q(t))x dt + γ 1/2
∑
Yx+` px (t) ○ dwx+k (t),
`=−1,0,1
αx symmetric, compact support or ∣αx ∣ ≤ Ce −∣x∣/C , α̂(k) > 0 and
▸ α̂(0) = 0 unpinned chain
▸ α̂′′ (0) > 0 acustic chain
Dispersion function:
ω(k) = α̂(k)1/2
(= 2∣ sin(πk)∣),
This implies
∫
∣k∣≤δ
ω ′ (k)2
dk = +∞
k2
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Scalings
k ∈ T.
Wave function
fˆ(k) = ∑ f (y )e −i2πyk
y ∈Z
Complex wave function in Fourier space:
ψ̂(t, k) ∶= ω(k)q̂(t, k) + i p̂(t, k)
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Scalings
Wave function
fˆ(k) = ∑ f (y )e −i2πyk
y ∈Z
Complex wave function in Fourier space:
ψ̂(t, k) ∶= ω(k)q̂(t, k) + i p̂(t, k)
d ψ̂(t, k) ∶= − iω(k)ψ̂(t, k)dt − γ
β̂(k)
[ψ̂(t, k) − ψ̂(t, −k)∗ ]dt
4
+ γ 1/2 i ∫ r (k, k ′ )[ψ̂(k − k ′ ) − ψ̂(−k + k ′ )∗ ]dw (t, k ′ )
T
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Scalings
Wave function
fˆ(k) = ∑ f (y )e −i2πyk
y ∈Z
Complex wave function in Fourier space:
ψ̂(t, k) ∶= ω(k)q̂(t, k) + i p̂(t, k)
d ψ̂(t, k) ∶= − iω(k)ψ̂(t, k)dt − γ
β̂(k)
[ψ̂(t, k) − ψ̂(t, −k)∗ ]dt
4
+ γ 1/2 i ∫ r (k, k ′ )[ψ̂(k − k ′ ) − ψ̂(−k + k ′ )∗ ]dw (t, k ′ )
T
< dw ∗ (t, k)dw (s, k ′ ) >= δ(k−k ′ )δ(t−s)dt dk,
< dw (t, k)dw (s, k ′ ) >= 0
r (k, k ′ ) = 2r12 (k)r1 (2(k − k ′ )) + 2r1 (2k)r12 (k − k ′ ),
β̂(k) =
8r12 (k) + 4r12 (2k)
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Scalings
∼ k
2
r1 (k) = sin(πk)
Wigner distribution
̂ (η, k, t) = ⟨ψ ∗ (k − η, t)ψ(k + η, t)⟩
W
W (y , k, t) = ∫
(T/)
̂ (η, k, t) dη
e i2πy η W
This is different from the energy < E[−1 y ] >, but can be proven that
W (y , k, t)− < E[−1 y ] (t) > Ð→ 0.
→0
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Scalings
Start with initial distribution such that sup < ∥φ(0)∥2 >≤ C , and
̂ (η, k, 0) dk Ð→ W0 (η)
W̄ (η, 0) = ∫ W
Theorem
̂ (η, k, −3/2 t) Ð→ e −ĉt∣η∣3/2 W0 (η)
W
→0
with ĉ =
α̂′′ (0)3/4
.
29/4 (3γ)1/2
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Scalings
previous results
▸
Basile, O. Spohn, ARMA 2009: kinetic limit/weak noise:
γ = γ ′ , δ = 1, convergence to a linear Boltzmann equation
̂(η, k, t) + iηω ′ (k)W (η, k, t) = γLW
̂(η, k, t)
∂t W
with the scattering operator
̂(η, k, t) = ∫ R(k, k ′ )(W
̂(η, k ′ , t) − W
̂(η, k, t))
LW
R(k, k ′ ) = R(k ′ , k) = r 2 (k, k − k ′ ) + r 2 (k, k + k ′ )
3
= ∑ Ri (k)R−i (k ′ ) for the continous noise
4 i=±1
9
= R(k)R(k) for the velocity exchange noise
4
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Scalings
previous results
▸
Basile, O. Spohn, ARMA 2009: kinetic limit/weak noise:
γ = γ ′ , δ = 1, convergence to a linear Boltzmann equation
̂(η, k, t) + iηω ′ (k)W (η, k, t) = γLW
̂(η, k, t)
∂t W
▸
Jara, Komorowski, O.; Basile, Bovier: invariance principle for
the corresponding Markov process:
λX (λ
−3/2
t) = λ ∫
λ−3/2 t
0
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ω ′ (K (s)) ds Ð→
λ→0
Scalings
3
− stable Levy
2
previous results
▸
Basile, O. Spohn, ARMA 2009: kinetic limit/weak noise:
γ = γ ′ , δ = 1, convergence to a linear Boltzmann equation
̂(η, k, t) + iηω ′ (k)W (η, k, t) = γLW
̂(η, k, t)
∂t W
▸
Jara, Komorowski, O.; Basile, Bovier: invariance principle for
the corresponding Markov process:
λX (λ
▸
−3/2
t) = λ ∫
λ−3/2 t
0
ω ′ (K (s)) ds Ð→
λ→0
3
− stable Levy
2
Mellet, Mishler, Mouhot, ARMA 2011: Related work by
analytic approach.
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Scalings
previous results
▸
Basile, O. Spohn, ARMA 2009: kinetic limit/weak noise:
γ = γ ′ , δ = 1, convergence to a linear Boltzmann equation
̂(η, k, t) + iηω ′ (k)W (η, k, t) = γLW
̂(η, k, t)
∂t W
▸
Jara, Komorowski, O.; Basile, Bovier: invariance principle for
the corresponding Markov process:
λX (λ
▸
▸
−3/2
t) = λ ∫
λ−3/2 t
0
ω ′ (K (s)) ds Ð→
λ→0
3
− stable Levy
2
Mellet, Mishler, Mouhot, ARMA 2011: Related work by
analytic approach.
Cedric Bernardin, Milton Jara, Patricia Goncalves, 2014: 2
conserved quantity model, equilibrium fluctuations.
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Scalings
Time evolution of the Wigner distribution
Anti-Wigner distribution:
̂ (η, k, t) = ⟨ψ(k − η, t)ψ(k + η, t)⟩
Z
for small , δ = 2/3,
̂ (η, k, −δ t) = −i−δ+1 ω ′ (k)η W
̂ + γ−δ LW
̂
∂t W
̂ (η, k, t) + Z
̂ (−η, k, t)]
− γ−δ L [Z
̂ + o()
̃W
− γ2−δ η 2 L
̂ (η, k, −δ t) = . . .
∂t Z
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Scalings
Time evolution of the Wigner distribution
Anti-Wigner distribution:
̂ (η, k, t) = ⟨ψ(k − η, t)ψ(k + η, t)⟩
Z
for small , δ = 2/3,
̂ (η, k, −δ t) = −i−δ+1 ω ′ (k)η W
̂ + γ−δ LW
̂
∂t W
̂ (η, k, t) + Z
̂ (−η, k, t)]
− γ−δ L [Z
̂ + o()
̃W
− γ2−δ η 2 L
̂ (η, k, −δ t) = . . .
∂t Z
▸
▸
̂ (η, k, −δ t) Ð→ 0.
Because of fast time fluctuations Z
̂ (η, k, −δ t) Ð→ W̄ (η, t),
By averaging for many collisions W
constant in k.
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Scalings
̂ = 0 and smaller terms in , we are left
Let’s drop Z
̂ (η, k, −δ t) = −i−δ+1 ω ′ (k)η W
̂ + γ−δ LW
̂
∂t W
̂ (η, k, t) = ∫ R(k, k ′ )(W
̂(η, k ′ , t) − W
̂ (η, k, t))dk ′
LW
R(k, k ′ ) =
3
′
∑ Ri (k)R−i (k )
4 i=±1
To simplify the argument assume the simpler scattering rate
R(k, k ′ ) = R(k)R(k ′ ),
∫ R(k)dk = 1,
R(k) ∼ k 2
̂ (η, k, t) = R(k) ∫ R(k ′ )W
̂ (η, k ′ , t)dk − R(k)W
̂ (η, k, t)
LW
̂ > −R(k)W
̂ (η, k, t)
≡ R(k) < R, W
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Scalings
̂ (η, k, −δ t)dt,
Taking Laplace Transform ŵ (η, k, λ) = ∫0 e −λt W
′
′
′
and denoting < R, ŵ >= ∫ R(k )ŵ (η, k , λ)dk
∞
δ λŵ + iω ′ (k)η ŵ − R(k) < R, ŵ > +R(k)ŵ = δ W0 (η, k)
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Scalings
̂ (η, k, −δ t)dt,
Taking Laplace Transform ŵ (η, k, λ) = ∫0 e −λt W
′
′
′
and denoting < R, ŵ >= ∫ R(k )ŵ (η, k , λ)dk
∞
δ λŵ + iω ′ (k)η ŵ − R(k) < R, ŵ > +R(k)ŵ = δ W0 (η, k)
rearranging
w (η, k, λ) =
δ W0 (η, k) + R(k) < R, ŵ >
δ λ + R(k) + iω ′ (k)η
Multiplying by R(k) and integrating in k, after rearrangement:
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
R(k)W0 (η, k)
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
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Scalings
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
̂0 (η, k)
R(k)W
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
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Scalings
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
̂0 (η, k)
R(k)W
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
∫
R(k)W0 (η, k)
dk
δ
λ + R(k) + iω ′ (k)η
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̂0 (η, k)dk = W̄0 (η)
Ð→ ∫ W
Scalings
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
̂0 (η, k)
R(k)W
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
∫
R(k)W0 (η, k)
dk
δ
λ + R(k) + iω ′ (k)η
−δ (1 − ∫
̂0 (η, k)dk = W̄0 (η)
Ð→ ∫ W
R(k)2
dk) Ð→ λ + ĉη 3/2
δ λ + R(k) + iω ′ (k)η
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Scalings
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
̂0 (η, k)
R(k)W
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
∫
R(k)W0 (η, k)
dk
δ
λ + R(k) + iω ′ (k)η
−δ (1 − ∫
̂0 (η, k)dk = W̄0 (η)
Ð→ ∫ W
R(k)2
dk) Ð→ λ + ĉη 3/2
δ λ + R(k) + iω ′ (k)η
< R, ŵ >= ∫ R(k)ŵ (η, k, λ)dk Ð→ w (η, λ).
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Scalings
R(k)2
dk)
δ λ + R(k) + iω ′ (k)η
̂0 (η, k)
R(k)W
=∫ δ
dk
λ + R(k) + iω ′ (k)η
< R, ŵ > −δ (1 − ∫
∫
R(k)W0 (η, k)
dk
δ
λ + R(k) + iω ′ (k)η
−δ (1 − ∫
̂0 (η, k)dk = W̄0 (η)
Ð→ ∫ W
R(k)2
dk) Ð→ λ + ĉη 3/2
δ λ + R(k) + iω ′ (k)η
< R, ŵ >= ∫ R(k)ŵ (η, k, λ)dk Ð→ w (η, λ).
Ô⇒
(λ + ĉη 3/2 ) w (η, λ) = W̄0 (η)
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Scalings