dimension reduction in “heat bath” models raz kupferman the hebrew university
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Dimension Reduction in “Heat Bath” Models Raz Kupferman The Hebrew University. Part I Convergence Results Andrew Stuart John Terry Paul Tupper R.K. Ergodicity results: Paul Tupper’s talk. - PowerPoint PPT PresentationTRANSCRIPT
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Dimension Reduction in “Heat Bath” Models
Raz KupfermanThe Hebrew University
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Part IConvergence Results
Andrew StuartJohn Terry
Paul TupperR.K
Ergodicity results: Paul Tupper’s talk.
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Set-up: Kac-Zwanzig Models
A (large) mechanical system consisting of a “distinguished” particle interacting through springs with a collection of “heat bath” particles.
The heat bath particles have random initial data (Gibbsian distribution).
Goal: derive “reduced” dynamics for the distinguished particle.
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Motivation
•Represents a class of problems where dimensionreduction is sought. Rigorous analysis.
•Convenient test problem for recent dimension reduction approaches/techniques:
•optimal prediction (Kast)
•stochastic modeling•hidden Markov models (Huisinga-Stuart-Schuette)•coarse grained time stepping (Warren-Stuart, Hald-K)•time-series model identification (Stuart-Wiberg)
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The governing equations
(Pn,Qn): coordinates of distinguished particle
(pj,qj): coordinates of j-th heat bath particle
mj: mass of j-th particle
kj: stiffness of j-th spring
V(Q): external potential
The Hamiltonian:
€
H =1
2Pn
2 + V (Qn ) +p j
2
2m j
+ k j (q j − Qn )2
j=1
n
∑j=1
n
∑
The equations of motion:
€
˙ Q n = Pn
˙ P n = −V '(Qn ) + k j (q j − Qn )j
∑˙ q j = p j m j
˙ p j = −k j (q j − Qn )
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Initial data
Heat bath particles have random initial data: Gibbs distribution with temperature T:
€
f ( p,q) = Z−1 exp −H( p,q,P,Q)
T
⎡ ⎣ ⎢
⎤ ⎦ ⎥
i.i.d N(0,1)
€
p j (0) = m jT η j
q j (0) = Q(0) + T k j ξ j
Initial data are independent Gaussian:
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Generalized Langevin Equation
€
˙ Q n = Pn
˙ P n = −V '(Qn ) + k j (q j − Qn )j
∑˙ q j = p j m j
˙ p j = −k j (q j − Qn )
Solve the (p,q) equations and substitute back into the (P,Q) equation
€
Kn (t) = k j cos(ω j t)j∑ , ω j = k j m j
Memory kernel:
€
Fn (t) = T k j1 2 ξ j cos(ω j t) + η j sin(ω j t)[ ]
j∑
Random forcing:
“Fluctuation-dissipation”:
€
E Fn (t)Fn (s) = T Kn (t − s)
€
˙ ̇ Q n + V '(Qn ) = − Kn (t − s) ˙ Q n (s)ds + Fn (t)0
t
∫
(Ford-Kac 65, Zwanzig 73)
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Choice of parameters
Heat baths are characterized by broad and dense spectra. Random set of frequencies:
€
ω j ~ U[0,na ], a∈ (0,1), Δω = na n → 0
Ansatz for spring constants:
€
k j = f 2(ω j ) Δω
Assumption:f2(ω) is bounded and decays faster than 1/ω.
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€
˙ ̇ Q n + V '(Qn ) = − Kn (t − s) ˙ Q n (s)ds + Fn (t)0
t
∫
Generalized Langevin Eq.
€
Kn (t) = f 2(ω j )cos(ω j t)j∑ Δω
≈ f 2(ω)cos(ωt) dt0
∞
∫
Memory kernel:
(Monte-Carlo approximation of the Fourier transform of f2)
€
Fn (t) = T f (ω j ) ξ j cos(ω j t) + η j sin(ω j t)[ ]j
∑ Δω
≈ T f (ω) cos(ωt) dB1(ω) + sin(ωt) dB2(ω)[ ]0
∞
∫
Random forcing:
(Monte-Carlo approximation of a stochastic integral)
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Lemma
1. For almost every choice of frequencies (ω-a.s.) Kn(t) converges pointwise to K(t), the Fourier cosine transform of f2(ω).
2. KnK in L2(,L2[0,T])
Theorem: (ω-a.s.) the sequence of random functions Fn(t) converges weakly in C[0,T] to a stationary Gaussian process F(t) with mean zero and auto-covariance K(t); (FnF).
Proof: CLT + tightness
can be extended to “long term” behavior: convergence of empirical finite-dimensional distributions (Paul Tupper’s talk)
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ExampleIf we choose
€
f 2(ω) =2α
π
1
α 2 + ω2
then
€
Kn (t) → K(t) = exp(−α t )
and, by the above theorem, Fn(t) converges weakly to the Ornstein-Uhlenbeck (OU) process U(t) defined by the Ito SDE:
€
dU = −αU dt + 2αT dB
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Convergence of Qn(t)
€
˙ ̇ Q n + V '(Qn ) = − Kn (t − s) ˙ Q n (s)ds + Fn (t)0
t
∫
Theorem: (ω-a.s.) Qn(t) converges weakly in C2[0,T] to the solution Q(t) of the limiting stochastic integro-diff, equation:
€
˙ ̇ Q + V '(Q) = − K(t − s) ˙ Q (s)ds + F(t)0
t
∫
Proof: the mapping (Kn,Fn)Qn is continuous
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Back to example
€
f 2(ω) =2α
π
1
α 2 + ω2
if
€
˙ ̇ Q + V '(Q) = − e−α ( t−s) ˙ Q (s)ds + U(t)0
t
∫
then Qn(t) converges to the solution of
which is equivalent to the (memoryless!!) SDE
€
dQ = P dt
dP = −V '(Q) + R[ ] dt
dR = (−αR − P) dt + 2αT dB
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Numerical validation
Empirical distribution of Qn(t) for n=5000 and various choices of V(Q) compared with the invariant measure of the limiting SDE
single well
double well
triple well
extremely long correlation time
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•“Unresolved” component of the solution are modeled by an auxiliary, memoryless, stochastic variable.
•Bottom line: instead of solving a large, stiff system in 2(n+1) variables, solve a Markovian system of 3 SDEs!
•Similar results can be obtained for nonlinear interactions. (Stuart-K ‘03)
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Part IIFractional Diffusion
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Found in a variety of systems and models: (e.g., Brownian particles in polymeric fluids, continuous-time random walk)In all known cases, fractional diffusion reflects the divergence of relaxation times; extreme non-Markovian behaviour.
€
E ΔQ(t)2
~ tγ γ ≠1
Fractional (or anomalous) diffusion:
Question: can we construct a heat bath models that generated anomalous diffusion?
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Reminder
€
˙ ̇ Q n + V '(Qn ) = − Kn (t − s) ˙ Q n (s)ds + Fn (t)0
t
∫
€
Kn (t) = k j cos(ω j t)j∑ , ω j = k j m j
Memory kernel:
€
Fn (t) = T k j1 2 ξ j cos(ω j t) + η j sin(ω j t)[ ]
j∑
Random forcing:
€
ω j ~ U[0,na ]
€
k j = f 2(ω j ) Δω
Parameters:
€
f 2(ω) =C
ω1−γIf we take
€
Kn (t) → K(t) =C
tγthe
npower law decay of memory kernel
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Theorem: (ω-a.s.) Qn(t) converges weakly in C1[0,T] to the solution Q(t) of the limiting stochastic integro-diff, equation:
€
˙ ̇ Q + V '(Q) = − K(t − s) ˙ Q (s)ds + F(t)0
t
∫
The limiting GLE
€
K(t) =C
tγ
F(t) is a Gaussian process with covariance K(t); derivative of fractional Brownian motion (1/f-noise)
(Interpreted in distributional sense)
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Solving the limiting GLE
€
˙ ̇ Q + V '(Q) = −˙ Q (s)
(t − s)γds + F(t)
0
t
∫
For a free particle, V’(Q)=0, and a particle in a quadratic potential well, V’(Q)=Q, the SIDE can be solved using the Laplace transform.
Free particle: Gaussian profile, variance given by sub-diffusive (Mittag-Leffler) function of time, var(Q)~t.
Quadratic potential: sub-exponential approach to the Boltzmann distribution.
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Numerical results
Variance of an ensemble of 3000 systems, V(Q)=0(compared to exact solution of the GLE)
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Quadratic well: evolving distribution of 10,000 systems (dashed line: Boltzmann distribution)
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What about dimensional reduction?
Even a system with power-law memory can be well approximated by a Markovian system with a few (less than 10) auxiliary variables.
How? Consider the following Markovian SDE:
€
˙ ̇ Q = −V '(Q) + GT u
˙ u = −G ˙ Q − Au + C ˙ W
u(t) : vector of size mA: mxm constant matrixC: mxm constant matrixG: constant m-vector
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€
˙ ̇ Q = −V '(Q) + GT u
˙ u = −G ˙ Q − Au + C ˙ W
Solve for u(t) and substitute into Q(t) equation:
€
K~
(t) = GT exp(−At) G€
˙ ̇ Q + V '(Q) = − K~
(t − s) ˙ Q (s)ds + F~
(t)0
t
∫
where
€
F~
(t) = GT exp(−At) u(0) + GT exp(−A(t − s))C0
t
∫ dWs
Goal: find G,A,C so that fluc.-diss. is satisfied and the kernel approximates power-law decay.
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The RHS is a rational function of degree (m-1) over m. Pade approximation of the Laplace transform of the memory kernel (classical methods in linear sys. theory).
Even nicer if kernel has continued-fraction representation
It is easier to approximate in Laplace domain:
€
ˆ K (s) = GT (A + sI)−1G
(and the Laplace transform of a power is a power).
€
A =
1 2 2 2 L 2
2 5 6 6 L 6
2 6 9 10 L 10
M M M O M
2 6 10 14 L 4m − 3
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
G =
1
1
1
M
1
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
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Laplace transform of memory kernel (solid line) compared with continued-fraction approximation for 2,4,8,16 modes (dashed lines).
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Variance of GLE (solid line) compared with Markovian approximations with 2,4,8 modes.
Fractional diffusion scaling observed over long time.
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Comment:
Approximation by Markovian system is not only a computational tools. Also an analytical approach to study the statistics of the solution (e.g. calculate stopping times).
Controlled approximation (unlike the use of a “Fractional Fokker-Planck equation”).
Bottom line:
Even with long range memory system can be reduced (with high accuracy) into a Markovian system of less than 10 variables (it is “intermediate asymptotics but that what we care about in real life).