Post on 31-May-2020
Kinetic Theories in Multiscale Modeling of Polycrystals
Maria Emelianenko
Dept. of Mathematical SciencesGeorge Mason University
FRG, U. Maryland, March 4, 2009
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 1 / 33
Acknowledgements:
David KinderlehrerDept. of Math. Sciences, Carnegie Mellon University
Shlomo Ta’asanDept. of Math. Sciences, Carnegie Mellon University
Dmitry GolovatyDept. of Theoretical and Applied Math, University of Akron
Tony RollettDept. of Materials Science and Engineering, Carnegie Mellon University
Greg RohrerDept. of Materials Science and Engineering, Carnegie Mellon University
Yariv EphraimDept. of ECE, George Mason University
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 2 / 33
Introduction Grand view
Grand challenges in understanding polycrystallinemicrostructure
A central problem in materials science is the understanding and control ofmicrostructure: ensemble of grains that comprise polycrystalline materials.Performance is influenced by the types of grain boundaries in the material and the waythat they are connected.
Structure sensitive properties:
Superconducting Critical Current Density
Electromigration Damage Resistance
Stress Corrosion Cracking
Electrical activity
Creep Behavior
”Insensitive”:
Average elastic energy densityFracture follows grain boundaries.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 3 / 33
Introduction Motivation
Questions: are microstructures all like soap froth? C.S. Smith (1951)
MgO looks very much unlike, but appearances can be deceiving.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 4 / 33
Introduction Motivation
Motivation
Experiment: evolution of grain boundary character distribution (GBCD)GBCD = relative areas of grain boundaries sorted by misorientation angles and normal
Gorzkowski et al. Zeitschrift fur Metallkunde, 96 (2005) 207.
Recent discovery
Grain boundary character (GBCD) is a scale invariant steady state characteristic of amaterial.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 5 / 33
Introduction Microscopic dynamics
Foundations
Curvature driven growth of a network of grain boundaries
Mullins equation:
vn = µ(∂2γ
∂θ2+ γ)κ on Γ(i)
Herring condition for triple junctions:∑TJ
(∂γ
∂θn + γt) = 0, at TJ’s
Interfacial energy depends on the misorientationangle and normal: γ(θ, α)
Mullins-von Neumann law: dAndt
= c(n − 6)extended to Rn MacPherson & Srolovitz, Nature 2007
Other simulation techniques: Potts model, MonteCarlo, mean field theory etc
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 6 / 33
Introduction Mesoscopic view
Bridging micro and macro scales
People have known for a long time that grain boundary populations vary with interfacialenergy, as well as other features. We want to suggest that this relationship isquantitative and predictable.
misorientation angle-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
ener
gy
1.000
1.001
1.002
1.003
1.004
1.005
1.006
population
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
Shallow Well: simulation
Individualgraindynamics
Cumulativeprobabilitydistributions
Evolutionequations fordistributionfunctions
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 7 / 33
Introduction Mesoscopic view
Coarsening: Numerical simulation and experiment
Movie
Movie
Some grains grow, some shrink. Is this the whole story?
Movies courtesy of D. Kinderlehrer, I. Livshits, S. Taasan, K. Barmak.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 8 / 33
Introduction Mesoscopic view
Motivation
Processes that alter GB distributions:
Continuous partWill not be discussed in this presentation.Incremental changes in the areas of faces:as grain boundaries migrate, areas of GBfaces increase or decrease.
Doesn’t affect misorientations, can bemodeled using calculus on manifolds.
Discontinuous (jump) partThe focus of present work.Critical events:- collapse and creation of grain faces- collapse of small grains
Affects the misorientations andinterfacial energies. Stochastic tools aremore effective.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 9 / 33
Introduction Mesoscopic view
Motivation
Processes that alter GB distributions:
Continuous partWill not be discussed in this presentation.Incremental changes in the areas of faces:as grain boundaries migrate, areas of GBfaces increase or decrease.
Doesn’t affect misorientations, can bemodeled using calculus on manifolds.
Discontinuous (jump) partThe focus of present work.Critical events:- collapse and creation of grain faces- collapse of small grains
Affects the misorientations andinterfacial energies. Stochastic tools aremore effective.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 9 / 33
Introduction Goals
Goals
Stable distribution has been identified (GBCD). Now need to identify ways to model it.
Grand challenge: Develop predictive theory for the evolution of materials texturethat will
identify stationary statisticsprovide a dynamical model based on critical eventsquantify rates at which critical events take placehelp explain relationship between energy and GB distributions
What we talk about today: Theoretical approaches to modeling evolution ofcritical events.
Modeling correlated populations via Boltzmann approachGeneralized continuous time random walk frameworkMarkov modulated Poisson processesNumerical features, results, limitations
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 10 / 33
Introduction Goals
Goals
Stable distribution has been identified (GBCD). Now need to identify ways to model it.
Grand challenge: Develop predictive theory for the evolution of materials texturethat will
identify stationary statisticsprovide a dynamical model based on critical eventsquantify rates at which critical events take placehelp explain relationship between energy and GB distributions
What we talk about today: Theoretical approaches to modeling evolution ofcritical events.
Modeling correlated populations via Boltzmann approachGeneralized continuous time random walk frameworkMarkov modulated Poisson processesNumerical features, results, limitations
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 10 / 33
1-dimensional model Focusing on critical events
Simplified model
3D2D
αi - misorientationsγ(α) - interfacial energies
1D
xj − ”triple junctions”(xj , xj+1) − ”grain boundaries”lj = xj+1 − xj − ”GB areas”{αi}i=1,...,n − ”misorientations”
Given initial orientations αi and total system energy in the form
En(t) =∑
f (αi )(xi+1(t)− xi (t))
define equations of motion through gradient flow dynamics
xi = f (αi )− f (αi−1), i = 0, . . . , n.
vi = li = xi+1 − xi = f (αi+1)− 2f (αi ) + f (αi−1)
where f (α) plays the role of an interfacial energy.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 11 / 33
1-dimensional model Validation
Stable statistics
We consider the following statistics:
ρor (α, t) =∑
k δ(α− αk) pdf of orientationsρlen(l , t) =
∑i δ(l − li ) pdf of lengths
ρw (α, t) =∑
k lkδ(α− αk) weighted pdf of orientations (GBCD)ρvel(v , t) =
∑j δ(v − vj) pdf of velocities
Distributions harvested from simulation stabilize early in the process. Gives evidence fora well-defined dynamic process.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 12 / 33
1-dimensional model Conformity with experiment
Influence of interfacial energies
Conforming to experimental observations, distribution of orientation parameters α isinversely correlated with interfacial energies f (α).
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
orientations
gra
in b
ou
nd
ary
char
acte
r d
istr
ibu
tio
n
2d data [11]Boltzmann fitenergy f(x)1d sim.data
Figure: GBCD distribution for 1d model, 2d large-scale simulation and its
Boltzmann fit.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 13 / 33
Boltzmann approach Simple model
Modeling critical events via inelastic Boltzmann equation
We can think of a critical event as a collision of inelastic particles: particle ≡ GB.
(α2, v2) + (α1, v1) ⇒ (α1, v∗), where v∗ = v1 + v2 + f (α2)− f (α1),
~~
~~
(a)
(b)
(l∗(+1), v∗
(+1), α∗
(+1))(l∗, v∗, α∗)(l∗(−1), v∗
(−1), α∗
(−1))
(l(+1), v(+1), α(+1))(l, v, α)(l(−1), v(−1), α(−1))
(l∗(−2), v∗
(−2), α∗
(−2))
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 14 / 33
Boltzmann approach Simple model
Continuity equation:
∂ρ(l , v , α, t)
∂t+ v
∂ρ(l , v , α, t)
∂l= {gain} − {loss}.
where the right hand side term can be written as an integral over all possible collisions:
−2
N(t)
∫A+
v ′ρ(0, v ′, α′, t)ρ(l , v − v ′ + f (α)− f (α′), α, t) dα′dv ′
+2
N(t)
∫A−
v ′ρ(0, v ′, α′, t)ρ(l , v , α, t) dα′dv ′.
subject to restricting conditions:A− := {f (α) ≤ v ′ + 2f (α′)} ∩ {f (α′) ≤ v + 2f (α)} ∩ {v ′ < 0},A+ := {f (α) ≤ v ′ + 2f (α′) ≤ v + 3f (α)} ∩ {v ′ < 0} .
∆
~~
t+ t∆
~~
t
~~
t− t
(l + v∆t, v, α)
(0, v′, α′)
(l, v∗, α)
(v′∆t, v′, α′) (l − v∗∆t, v∗, α)
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 15 / 33
Boltzmann approach Simple model
Simple model performance for quadratic potential
0 1 20
0.02
0.04
0.06
T=5
0
orientations
0 0.05 0.10
0.5
1lengths
−5 0 50
0.1
0.2velocities
0 1 20
0.05
0.1
T=3
00
0 0.05 0.10
0.5
1
−5 0 50
0.2
0.4
0 1 20
0.05
0.1
T=6
00
0 0.05 0.10
0.5
1
−5 0 50
0.5
1
Lengths and orientations fitexperimental results well, butvelocities start to deviate at laterstages. Possible correlations?
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 16 / 33
Boltzmann approach Extended model
Extended space model
Is there a remedy for complexity and correlation issues?Then let us expand the state space to include incidence relations: ρ(l , α(−1), α, α(+1), t).
Grain boundaries ≡ colliding particles.
~~
~~
(a)
(b)
α α(+1)
β = α(−1)β(−1) β(+1) = α α(+1)
β(−1)
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 17 / 33
Boltzmann approach Evolution equation
By following these simple collision rules, we can write a Boltzmann-type collisionequation for the density function:
∂ρ(l , α(−1), α, α(+1)
)∂t
+(f
(α(−1)
)+ f
(α(+1)
)− 2f (α)
) ∂ρ(l , α(−1), α, α(+1)
)∂l
= W ,
where W = W+ −W− with
W+ := 1N(t)
∫B
(2f (s)− f (α)− f (α(−1))
)ρ
(0, α(−1), s, α
)ρ
(l , s, α, α(+1)
)ds
+ 1N(t)
∫B
(2f (s)− f (α)− f (α(+1))
)ρ
(0, α, s, α(+1)
)ρ
(l , α(−1), α, s
)ds.
Similarly
W− := 1N(t)
∫B
(f (α) + f (s)− 2f (α(−1))
)ρ
(0, s, α(−1), α
)ρ
(l , α(−1), α, α(+1)
)ds
+ 1N(t)
∫B
(f (α) + f (s)− 2f (α(+1))
)ρ
(0, α, α(+1), s
)ρ
(l , α(−1), α, α(+1)
)ds.
where B := R+ × R3.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 18 / 33
Boltzmann approach Evolution equation
Extended model performance for quadratic potential
0 1 20
200
400orientations
T=5
0
0 2 4
x 10−3
0
5000
10000lengths
−2 0 20
500
1000velocities
0 1 20
200
400
T=3
00
0 2 4
x 10−3
0
1000
2000
−2 0 20
500
1000
0.5 1 1.50
20
40
60
T=6
00
0 2 4
x 10−3
0
50
100
150
−0.1 0 0.10
100
200
Lengths, orientations and velocitiesare in agreement with experimentalresults.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 19 / 33
Recap
Quick summary
Recap: We have discussed two models capable of describing the evolution of ρ(α, l , t).
1 Advantages:
both models have clear physical interpretation;extended Boltzmann model completely describes system evolution
2 Drawbacks:
simple model does not take into account correlationsextended approach is informationally complex.
Continuous time random walk theory can possibly eliminate some of these difficulties,while also giving more insight into characteristics of underlying stochastic process.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 20 / 33
Random walk intro Simple walk
Simple random walk model and its master equations
The walker takes a step in a random direction with fixed probabilities pi , s.t.∑
pi = 1.
Symmetric walk in 1d: pi = 1/2
Symmetric walk in 2d: pi = 1/4
Pi (t + ∆t) =1
2Pi−1(t) +
1
2Pi+1(t)
Pi,j(t + ∆t) =1
4(Pi,j−1(t) + Pi,j+1(t) + Pi−1,j(t) + Pi+1,j(t))
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 21 / 33
Random walk intro Simple walk
Simple random walk model and its master equations
The walker takes a step in a random direction with fixed probabilities pi , s.t.∑
pi = 1.
Symmetric walk in 1d: pi = 1/2
Symmetric walk in 2d: pi = 1/4
Pi (t + ∆t) =1
2Pi−1(t) +
1
2Pi+1(t)
Pi,j(t + ∆t) =1
4(Pi,j−1(t) + Pi,j+1(t) + Pi−1,j(t) + Pi+1,j(t))
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 21 / 33
Random walk intro Simple walk
Master equation and PDE
In the symmetric 1d random walk case, the master equation is given by:
Pi (t + ∆t) =1
2Pi−1(t) +
1
2Pi+1(t)
In the continuum limit, ∆t → 0, ∆x → 0:
Pi (t + ∆t) = Pi (t) + ∆t∂Pi
∂t+ O(∆t2)
Pi±1(t) = Pi (t)±∆x∂Pi
∂x+ ∆x2 ∂2Pi
∂x2+ O(∆x2)
leads to the diffusion equation∂Pi
∂t= K
∂2Pi
∂x2
Many generalizations are possible.Continuous time random walk: lattice is replaced by continuous state space.Jump sizes and waiting times are drawn from probability distributions.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 22 / 33
Random walk intro Simple walk
Master equation and PDE
In the symmetric 1d random walk case, the master equation is given by:
Pi (t + ∆t) =1
2Pi−1(t) +
1
2Pi+1(t)
In the continuum limit, ∆t → 0, ∆x → 0:
Pi (t + ∆t) = Pi (t) + ∆t∂Pi
∂t+ O(∆t2)
Pi±1(t) = Pi (t)±∆x∂Pi
∂x+ ∆x2 ∂2Pi
∂x2+ O(∆x2)
leads to the diffusion equation∂Pi
∂t= K
∂2Pi
∂x2
Many generalizations are possible.Continuous time random walk: lattice is replaced by continuous state space.Jump sizes and waiting times are drawn from probability distributions.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 22 / 33
CTRW approach Motivation
Continuous time random walk of the lengths and velocities
Grain areas Grain velocities
Simulation results: steady-state statistics of li and vi (exists for all types of f )relative areas relative velocities
0 10 20 30 40 50 600
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 23 / 33
CTRW approach Motivation
Continuous time random walk of the lengths and velocities
Grain areas Grain velocities
Simulation results: steady-state statistics of li and vi (exists for all types of f )relative areas relative velocities
0 10 20 30 40 50 600
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 23 / 33
CTRW approach Theoretical approach
Continuous Time Random Walk framework
The walker moves according to two parameters:(1) Waiting times with pdf w(t) (2) Jump sizes with pdf µ(x).
Chapman-Kolmogorov type master equation:
p(s, t) = p0(s)ψ(t) +
∫ ∫p(s − s′, t − t′)µ(s′)w(t′)dt′ds′
We derive an equivalent form in terms of the memory kernel
Φ(u) =1− w(u)
uw(u):
∫Φ(t − t′)
∂
∂tp(x , t′)dt′ =
∫[p(x − x ′, t)− p(x , t)]µ(x ′)dx ′
Φ(t) = cδ(t) (equivalent to w(t) ∼ exp(λt): Markov process, no memory
Φ(t) 6= cδ(t): non-Markov process with memory kernel Φ(t).For the choice of Φ(u) = 1
λuβ−1
∂β
∂tβp(x , t) = λ
∫[p(x − x ′, t)− p(x , t)]µ(x ′)dx ′
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 24 / 33
CTRW approach Fractional kinetics
Non-Markov nature of dynamics
Do we have a Markov process in our model problem? No!
Waiting times have fat tails (left), plus jump sizes are non-Gaussian (right)
0.1 0.2 0.3 0.4 0.5
0
5
10
15
20
25
30
35
40
45
50 exp fita*xb fithistogram
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2Jump size distributions for 50000 grains
f(x)=(x−0.5)2
f(x)=|x−0.5|
MSD 〈v2〉 (left) and arrival rate (right) both show nonlinear trend in the long term limit,independent of the form of the potential. Log-log slope changes from β = 1 to β ≈ 0.7.
10−8
10−6
10−4
10−2
100
102
10−6
10−5
10−4
10−3
10−2
10−1
−16 −14 −12 −10 −8 −6 −4 −2 00
1
2
3
4
5
6
7
8
9
log(t)
log(
N(t
))
f(α)=alpha2 or (α−0.5)2 or |alpha−0.5|1/2 or (alpha−0.5)6 or (α−0.5)2(α−1)2
simulation is repeated 10 times for each potential.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 25 / 33
CTRW approach Asymptotics
Closed form solution
For stable jump sizes distribution µ(s) = αs−α−1, 0 < α < 1, s > 0 and waiting timesw(t) ∼ t−1−β , the master equation coincides with the fractional kinetic equation:
∂βp(x , t)
∂tβ= Kγ
∂α
∂xαp(x , t),
It admits a closed form solution in the form
p(x , t) =1
tβ/αWα,β(
x
tβ/α),
Wα,β is the inverse Laplace transform of the
Mittag-Leffler function Eα,β(z) =∞∑j=0
z j
Γ(αj + β)
Fractional derivative in Caputo form
dβ
dtβf (t) =
1
Γ(1− β)
∫ t
0
f ′(τ)
(t − τ)βdτ,
for which
L[ dβ
dtβf (t)
]= uβ f (u)− uβ−1f (0).
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 26 / 33
CTRW approach Numerical results
Numerical results
We look at our simplified model in the coupled 2-dimensional case space of velocitiesand lengths:
∂β
∂tβρ(l , v , t) = −v
∂ρ(l , v , t)
∂l+ λ
∫ ∞
−∞[ρ(l , v − s, t)− ρ(l , v , t)]µ(s)ds.
Using Grunwald-Letnikov definition of fractional derivative:
∂βf (t)
∂tβ= lim
h→0
1
hβ
[t/h]∑k=0
ωβk f (t − kh), where ω
(β)k = (−1)k
(βk
).
We determine µ(s) from the empirical jump probability density and run the model withparameter λ = 1.
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.10
5
10
15
Per
cen
tag
e
Empirical jump sizes distribution
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k
Coefficients ω(1−β)k
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 27 / 33
CTRW approach Numerical results
Comparison of simulation with fractional PDE.
Taking β = 0.7, numerical solution of the fractional equation can be obtained byusing the explicit FTCS difference scheme
p(m+1)i,j = p
(m)i,j + Sβ
m∑k=0
ω(1−β)k I
(m−k)i,j ,
where Sβ = ∆β
(∆x)2, ∆t ≤ 1
43/2−β (∆x)2β and
I(m−k)i,j = vj(p
(m−k)i−1,j − p
(m−k)i+1,j )∆x +
∑sl :j+sl∈[1,Nj]
µl [p(m−k)i,j+sl
− p(m−k)i,j ].
which has been shown to be numerically stable.
FPDE (right) are in agreement with simulation (left).
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 28 / 33
Generalizations Slowing dynamics
Slowing down effect
Departure from fractional random walk theory: non-identical waiting times distributions
Recall: for a random walk theory, one has to assume that all waiting times are i.i.d.variables. In this case, variables are independent, but not identically distributed. Analogyto non-homogeneous Poisson process with decaying frequency.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 29 / 33
Generalizations Three stages of the process
Connections to renewal processes theory
N(t) - cumulative arrivals, λ(t) - arrival rate, w(t) - waiting times pdf
λ(t) = w(t) +
∫ t
0
λ(s)w(t − s)ds, N(t) =
∫ t
0
λ(s)ds
Fractional regime only describes intermediate steps of the process.
Is there a formalism capable of describing the complete evolution process?Can we find an evolution equation for pdfs related to velocities and GBCD in this
context?
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 30 / 33
Generalizations MMPP
Markov-modulated Poisson processes
Relation to hidden Markov chains:
wi ∼ exp(ri ), with mean parameter ri that increases with i due to the slowing effectof coarsening dynamics
Markov modulated Poisson process(MMPP) - Poisson process varying its arrivalrate according to an m-state irreducible continuous time Markov chain.
The observable process can be written as Xt = X0 +∫ t
0AXudu + Mt with a
martingale part Mt and with rate of arrivals depending upon the state of anindirectly observed Markov chain with rate matrix A having entries ri .
MMPP is an attractive model for short- and middle-range autocorrelations because itcan be easily parameterized to produce dependence in the series and remains analyticallytractable.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 31 / 33
Generalizations MMPP
MMPP parameter estimation
X (t) - underlying continuous-time homogeneous Markov chain with state space 1, . . . , r .r - order of the MMPP.Q = qij - generator of the Markov chain.N(t) observed process - variable-rate Poisson process with rate λi when X (t) = i .Ti - time of the i-th Poisson event. Inter-event times Yk = Tk − Tk−1.X (Tk) - embedded discrete-time Markov chain obtained from sampling thecontinuous-time chain at the Poisson event times.(X (Tk), Yk) - Markov renewal process with transition probability matrix F (y) = {Fij(y)},where Fij(y) = P(Yk ≤ y , X (tk−1 + y) = j |X (tk−1 = i). The transition density matrix,obtained from differentiation of Fij w.r.t. y , is given by
f (y) = exp((Q − Λ)y)Λ, Λ = diagλ1, . . . , λr .
Parameters of the stationary MMPP: (Q, Λ) can be estimated by EM algorithm.Preliminary results: r = 10 fits well with 1-d simulation interarrival times for N = 103
grain boundaries.
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 32 / 33
Summary Current work
Work in progress
Further progress can only be achieved through close collaboration with materialsscientists.
Building MMPP formalism for GBCD evolution
Understanding the slowing down effect and its relationship to the fractionaldynamics
Providing physical validation
Studying numerical and analytical issues
Developing the complete predictive theory for the evolution of materials statistical(texture, GBCD etc) and structural properties from microstructure
In particular, it will
identify stationary statisticsprovide a dynamical model based on critical eventsquantify rates at which critical events take placeprovide the set of simple rules to be used for grain boundary engineering
THANKS!
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 33 / 33
Summary Current work
Work in progress
Further progress can only be achieved through close collaboration with materialsscientists.
Building MMPP formalism for GBCD evolution
Understanding the slowing down effect and its relationship to the fractionaldynamics
Providing physical validation
Studying numerical and analytical issues
Developing the complete predictive theory for the evolution of materials statistical(texture, GBCD etc) and structural properties from microstructure
In particular, it will
identify stationary statisticsprovide a dynamical model based on critical eventsquantify rates at which critical events take placeprovide the set of simple rules to be used for grain boundary engineering
THANKS!
Maria Emelianenko (GMU) Kinetic Theories in Multiscale Modeling of Polycrystals March 4, 2009 33 / 33