Title: Dimension Reduction in Heat Bath Models Raz Kupferman The Hebrew University
1Dimension Reduction in Heat Bath ModelsRaz
KupfermanThe Hebrew University
2Part IConvergence ResultsAndrew StuartJohn
TerryPaul TupperR.K
Ergodicity results Paul Tuppers talk.
3Set-up Kac-Zwanzig Models
The heat bath particles have random initial data
(Gibbsian distribution).
Goal derive reduced dynamics for the
distinguished particle.
4Motivation
- 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)
5The governing equations
6Initial data
7Generalized Langevin Equation
8Choice of parameters
Assumptionf2(?) is bounded and decays faster
than 1/?.
9(No Transcript)
10- Lemma
- For almost every choice of frequencies (?-a.s.)
Kn(t) converges pointwise to K(t), the Fourier
cosine transform of f2(?). - Kn?K in L2(?,L20,T)
11Example
12Convergence of Qn(t)
13Back to example
14Numerical validation
Empirical distribution of Qn(t) for n5000 and
various choices of V(Q) compared with the
invariant measure of the limiting SDE
15- Unresolved component of the solution are
modeled by an auxiliary, memoryless, stochastic
variable. - Bottom line instead of solving a large, stiff
system in 2(n1) variables, solve a Markovian
system of 3 SDEs! - Similar results can be obtained for nonlinear
interactions. (Stuart-K 03)
16Part IIFractional Diffusion
17Fractional (or anomalous) diffusion
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.
Question can we construct a heat bath models
that generated anomalous diffusion?
18Reminder
19The limiting GLE
F(t) is a Gaussian process with covariance K(t)
derivative of fractional Brownian motion
(1/f-noise)
(Interpreted in distributional sense)
20Solving the limiting GLE
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.
21Numerical results
Variance of an ensemble of 3000 systems,
V(Q)0(compared to exact solution of the GLE)
22Quadratic well evolving distribution of 10,000
systems (dashed line Boltzmann distribution)
23What 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.
24Solve for u(t) and substitute into Q(t) equation
where
Goal find G,A,C so that fluc.-diss. is satisfied
and the kernel approximates power-law decay.
25The 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
26Laplace transform of memory kernel (solid line)
compared with continued-fraction approximation
for 2,4,8,16 modes (dashed lines).
27Variance of GLE (solid line) compared with
Markovian approximations with 2,4,8
modes. Fractional diffusion scaling observed over
long time.
28Comment 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).