random dynamical systems with microstructurekmelnick/esijuly2011/feresslides.pdfplan of talk...
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Random dynamical systems with microstructure
Renato FeresWashington University, St. Louis
ESI, July 2011
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Acknowledgements
Different aspects of this work are joint with:
Tim Chumley (Central limit theorems)
Scott Cook (Random billiards with Maxwellian limits)
Jasmine Ng (Spectral properties of Markov operators)
Gregory Yablonsky (Earlier engineering work)
Hong-Kun Zhang (Current work on most of these topics)
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Plan of talk
Billiards with random microstructure
Billiards: Hamiltonian flows on manifolds with boundaryMicrostructure: geometric structure on the boundary
The Markov operator
Derived from the random microstructureDefines generalized billiard reflection
Properties of the Markov operator
Stationary distributionsSpectral gap and moments of scatteringThe billiard LaplacianConditioningCase studies
CLT and Diffusion
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Random billiards with microstructure
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Random billiards with microstructure
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Wall and molecule subsystems
Configuration spaces: Riemannian manifolds with corners
Mwall, Mmol ∶=Mmol ×R ×Tk
and potential functions:
Uwall ∶Mwall →RUMol ∶Mmol →R
The total system has configuration space M and potential
U ∶M →R.
When subsystems sufficiently far away, M ≅Mwall ×Mmol and
U = Uwall +Umol.
Outside of product region = interaction zone. Motion:
∇c′(t)dt
= −gradc(t)U
with specular collisions at boundary.
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Interaction region (microscopic): definitions
S ∶=Mmol × 0 ×Tk ×Mwall boundary of inter. zone;
E(q, v) ∶= 1
2∥v∥2q +U(q) energy function on N ∶= TM ;
NS ∶= TSM , N(E) ∶= E−1(E), NS(E) ∶= NS ∩N(E);θv(ξ) ∶= ⟨v, dτvξ⟩ contact form on N ;
dθ symplectic form on N ;
XE Hamiltonian vector field: XE dθ = −dE;
η ∶= (grad E)/∥grad E∥2 (in Sasaki metric on N);
Ω ∶= (dθ)m Liouville volume form on N ;
ΩE ∶= η Ω flow invariant volume on energy surfaces;
T ∶NS →NS the return (billiard) map to S.
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Geodesic flow example (Mwall
trivial)
Mwall = single point
Mmol = 0,1Mmol ∶= 0,1 ×R ×T1
S = 0,1 × 0 ×T1
Potentials are constant
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Example: a dumbbell molecule
Mwall = single point
Mmol = SO(2)Mmol ∶= SO(2) ×R ×T1
S = SO(2) × 0 ×T1
Potentials are constant
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Example with potential (Mmol
trivial)
Coordinates: x =√m1/m (x1 − l/2), y =√m2/mx2.
E(x, y, x, y) = m
2(x2 + y2 + k
m1
x2) .
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Random microstructures a la Gromov
B ∶= billiard table: Riemannian manifold with boundary;
F (∂B) orthonormal frame bundle with group O(k);Nwall ∶= TMwall state space of wall system;
V ∶= P(O(k) ×Nwall) space of probability measures;
V is naturally an O(k)-space.
Random microstructure on ∂B: O(k)-equivariant map
G ∶ F (∂B) → P(O(k) ×Nwall).Example: G = (ξ, ζ) constant, where ξ ∈ P(O(k)) is rotation invariantand ζ is Gibbs canonical distribution.
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Gibbs canonical distribution at temperature T
An invariant volume form on Nwall of physical significance:
ζ ∶= e−βE
Z(β)ΩEwall∧ dE
where β = 1/κT . (κ = Boltzmann constant.) Density ρ is obtained bymaximizing Boltzmann entropy:
H(ρ) ∶= −∫Nwall
ρ log ρΩwall
under constraint ∫NwallEρΩwall = E0. (β = Lagrange multiplier.)
Maximal uncertainty about state given mean value of E.
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The Markov operator P of a microstructureH ∶= half-space in dimension k + 1;
N+mol∶= TMmol ×H;
π ∶ N+S ∶= N+mol×Tk ×Mwall → N+
molprojection to first factor;
λ ∈ P(Tk) Lebesgue;
ζ ∈ P(Nwall) a fixed probability (say, the Gibbs measure);
T ∶N+S → N+S the return map.
Define the map P ∶ P(N+mol) → P(N+
mol), by
µ ↦ µP ∶= (π T )∗(µ⊗ λ⊗ η).
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Markov chains (dynamics under partial state info)
Standard finite state Markov chains with detailed balance: M is a
groupoid, V is the set of units, T is the inverse operation, v ↦ ηv are
the transition probabilities, and µ η is T -invariant.
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Example of P (constant speed)
Transition probabilities operator:
(Pf)(θ) = ∫ 1
0
f(Ψθ(r)) dr.
Surface microstructure defined by a billiard table contour. The
coordinate r is random (uniform between 0 and 1).
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Stationary distributions
Definition: µ ∈ P(N+mol) is stationary if µP = µ.
Theorem
Let P ∶ P(N+mol) → P(N+
mol) be the Markov operator associated to the
Gibbs canonical distribution on Nwall with temperature parameter β.Then the Gibbs canonical distribution on N+
molwith the same
parameter β is stationary.
Proof.
Use e−β(Emol+Ewall) = e−βEmole−βEwall and invariance of the symplecticvolume form on NS(E) under the return map T .
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Example: Let Ce−β
2m1w
2
dw ds be fixed state of wall
Equilibrium state of molecule: dµ(v) = C cosθ∣v∣2e−β
2m2∣v∣
2
dθ d∣v∣If no moving parts (fixed speed ∣v∣ = 1): dµ(θ) = 1
2cos θ dθ.
No dependence on shapes.
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The operator P on functions
Define action of P on functions by ν(Pf) = (νP )(f).Definition
The molecule-wall system is symmetric if there are volume preservingautomorphisms J and K of N+S that:
respect the product N+S = N+mol×Nwall;
induce the same map J on N+mol
;
J T = T −1 J (time reversibility)
K T = T K (symmetry).
Let µ be the stationary measure and H ∶= L2(N+mol
, µ).Theorem
If system is symmetric, P is a self-adjoint operator on H of norm 1.
The symmetry condition essentially always holds. Typically, wefind in the examples that P is a Hilbert-Schmidt operator.
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Problem: relate structural features and spectrum of P
For example, in purely geometric settings (no moving parts on thewall, no potentials) want to relate shape and spectrum.
Of special interest: spectral gap.
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Case studies (analytical and numerical)
A simple two masses system;
Adding a quadratic potential;
Billiard systems with no energy exchange;
The method of conditioning;
Moments of scattering and spectral gap;
Systems with weak scattering and the billiard Laplacian.
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A simple two-masses system - I
Main system parameter: γ ∶= √m2
m1
= tanα.
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A simple two-masses system - II
Define: x = √m1
mx1, w ∶= √m1
mv1, dζ(w) ∶= C exp (− 1
2w2/σ2)dw dx;
Theorem
P has a unique stationary distribution µ on (0,∞), given by
dµ(v) = σ−2v exp(− v2
2σ2)dv.
P is a Hilbert-Schmidt operator on L2((0,∞), µ) of norm 1;
ηPn → µ exponentially inTV-norm for all initial η.
If φ is C3 on (0,∞), then
(Lφ)(z) ∶= limγ→0
(Pγϕ) (z) − ϕ(z)2γ2
= (1z− z)ϕ′(z) + ϕ′′(z)
holds for all z > 0.
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A simple two-masses system - III
1 2 3 4 50
0.5
1
speed
prob
abili
ty d
ensi
ty
Evolution of an initial probability measure, µ0, having a step function
density. The graph in dashed line is the limit density v exp(−v2/2)and the other graphs, from right to left, are the densities of µ0P
n at
steps n = 1,10,50,100. Here m1/m2 = 100.
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A simple two-masses system - IV
0 1 2 3 4 5
0
1
speed
Comparison of the second eigendensity of P (numerical) and the
second eigendensity of the billiard Laplacian L: (1 − z2/2)ρ(z). Used
γ = 0.1; the numerical value for the second eigenvalue of P was found
to be 0.9606, to be compared with 1 + 2γ2(−2) = 0.9600 derived from
eigenvalue −2 of L.
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A simple two-masses system - V
0 0.04 0.08 0.12 0.16 0.20
0.04
0.08
0.12
0.16
mass ratio parameter γ
spec
tral
gap
Asymptotics of the spectral gap of P for small values of the
mass-ratio parameter γ. The discrete points are the values of the gap
obtained numerically. The solid curve is the graph of f(γ) = 4γ2,
suggested by comparison with L.
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The bumps family - I (parameter K = l/R > 0)
dµ(θ) = 1
2sin θ dθ and H = L2 ([0, π] , µ);
PK = the Markov operator for bumps with curvature K.
Theorem
PK is a self-adjoint, compact operator on H of norm 1;
For small K, the spectral gap of PK is g(K) = 1
3K2 +O(K3);
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The billiard Laplacian
The (reduced) billiard map is an area-preserving map
T ∶ S2 → S2.
Regard PK as defined on L2(S2,A). Let ∆ be the spherical Laplacian.
Theorem
Let Φ be a compactly supported smooth function on S2 ∖ N,Sinvariant under rotations about the z-axis in R
3. Then
PKΦ −Φ = K2
6∆Φ +O(K3).
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Moments of scattering for bumps family
Define the jth moment of scattering
Ej(θ) = Eθ [(Θ − θ)j]where Θ is the random post-collision angle given θ.
and PΦ −Φ = ∑nj=1
Φ(n)
n!Ej +O(En+1) if Φ smooth.
Proposition
If sin θ > 3K/2 (middle range of angles), the moments satisfy:
If n is odd, En(θ) = Kn+1
2(n+2)cot θ +O(Kn+3);
If n is even, En(θ) = Kn
n+1+O(Kn+2).
It follows that
PKΦ −Φ1
3K2
= 1
2 sin θ
d
dθ(sin θ
dΦ
dθ) +O(K)
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E2(θ)/γK for K = 2/3,1/3,1/6
E(ΘK − θ)2/(spectral gap) → 1 (constant function)
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 30.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
This gives an asymptotic interpretation of the spectral gap as the
mean square deviation from specularity.
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General bumps family, big K
Theorem
For the general bumps family P is quasi-compact.
4 10 16 22
0.1
0.2
0.3
0.4
Normalized curvature K
spec
tral
gap
The solid line is the graph of 2/K and the values marked with an
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The technique of conditioning
Wall does not affect J-even eigenvalues, but brings J-odd eigenvalues
closer to 0.
0.1 0.2 0.3 0.4 0.5
0.2
0.25
0.3
0.35
0.4
0.45
0.5
height of vertical wall
spectr
al
gap
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More complicated shapes
−0.5 0 0.50.1
0.2
0.3
distance between bumps, h
spec
tral
gap
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P does not specify shape (up to homothety)
Two billiard cells with the same Markov operator:
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Gas transport in channels - 3 levels of description
Microscopic model (deterministic motion)
Random flight in channel (Markov process on set of directions)
Diffusion limit: gas concentration u(x, t) along R should satisfy
∂u
∂t=D
∂2u
∂x2
Relate: (1) microstructure, (2) spectrum of P , (3) diffusion constant.
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Transition to diffusion—the Central Limit Theorem
Consider the following experiment. Let
r = radius of channel;
v = constant particle speed;
L = half channel length;
Release the particle from middle point with distribution ν. Measurethe expected exit time, τ(aL, r, v) as a→∞.
Proposition
Suppose P on L2([0, π], µ) has positive spectral gap and µ is ergodicfor P . Then
τ(aL, r, v) ∼ 1
D
a2
lna
where D = 4rvπ
ξ(P ).We wish to understand how D depends on P .
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D and the spectrum of P
Let Π be the (projection-valued) spectral measure of P on [−1,1]:P = ∫ 1
−1λ dΠ(λ).
Fix β > 1 and let Z ∣a = Zχ∣Z∣≤a/ lnβ a.
Spectral measure of Z on [−1,1]: ΠZ(⋅) = lima→∞1
lna⟨Z ∣a,Π(⋅)Z ∣a⟩.
Theorem
D0 ∶= diffusion const. for i.i.d. process with angle distribution µ. Then
D =D0∫ 1
−1
1 + λ
1 − λdΠZ(λ).
If P has discrete spectrum, ΠZ(λi) ∶= lima→∞1
lna∣⟨Z ∣a, φi⟩∣2.
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An elementary example
θ = initial angle, define integer k, s ∈ [0,1), and probability p:
2h
b tan θ= k + s, p = ⎧⎪⎪⎨⎪⎪⎩
s if k is odd
1 − s if k is even
h
b
θ π−θ pp
1-p
1-p
θ
Special case: θ = π/4, b > 2h. Then k = 0, s = 2h/b. For a longchannel of diameter 2r and particle speed v, the random flight tendsto Brownian motion with
σ2 =√2rv ( b
2h− 1) .
An application of the central limit theorem for Markov chains.
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