Post on 22-Jul-2020
IntroductionSpace Discretization of SPDE
Numerical Simulations
Controlled-Error Semi-discretization ofMesoscopic Stochastic Equations for Surface
Diffusion
Yannis Pantazisjoin work with M. Katsoulakis
University of Massachusetts, Amherst, USAPartially supported by NSF: DMS and CMMI CDI-Type II
June, 2011
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Outline
Introduction
Space Discretization of SPDE
Numerical Simulations
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Surface Diffusion - Motivation
I Plass et al., Nature, 01’. Self-assembly of Pb on Cu(111)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Surface Diffusion - Motivation
I Patterns in such systems have rich morphologies at mesoscalesthat change dramatically as control parameters vary.
I Typically they form as a result of microscopic particledynamics in a complex energy landscape, in the presence ofstochastic fluctuations.
I Applications:I Formation of nanopatterns in heteroepitaxy: templating,
optical magnetic, and electronic devices.
I Less than 5% difference in energy between stripes, diskpatterns.
I Hence: sensitivity to entropic effects at finite temperatures anddynamics.
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Surface Diffusion – Model Hierarchy
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Surface Diffusion – Model Hierarchy
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Microscopic Models - Visualization
I Microscopic and Coarse-grained lattices
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Microscopic Models - Definition
I Lattice: LNI Configuration: σ = {σ(x) ∈ {0, 1} : x ∈ LN}I Hamiltonian:
H(σ) := −12
∑x , y ∈ LNy 6= x
J(x − y)σ(x)σ(y) +∑x∈LN
h(x)σ(x)
I Surface diffusion is modeled as exchanges of the spins betweentwo adjacent sites every time an exponential clock hits ⇒ Acontinuous-time jump Markov random process, σt , is defined
I Generator:
Lmf (σ) =∑
x , y ∈ LNx 6= y
c(x , y , σ)(f (σ(x,y))− f (σ)
)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Microscopic Models - Definition
I Diffusion rate for Metropolis dynamics:
c(x , y , σ) := d0 exp(βmin{0, (σ(x)− σ(y))(U(x , σ)− U(y , σ)) + J(1)})
I Diffusion rate for Arrhenius dynamics:
c(x , y , σ) = d0(1−σ(x))σ(y)e−β(U0+U(x,σ))+d0σ(x)(1−σ(y))e−β(U0+U(y,σ))
I Potential:
U(x , σ) :=∑
y ∈ LNx 6= y
J(x − y)σ(y)− h(x)
I Potential equals minus the discrete derivative of Hamiltonianat site x
I Mid to long ranged interaction potential: J(x)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Microscopic Models - Coarse Graining
I Coarsening factor: q
I Group sites into cells, Ck , of size qd
I Coarse lattice: Lm, m = N/qI Coarse-Grained (CG) variables:
η̄t(k) :=1
qd
∑x∈Ck
σt(x), k ∈ Lm
I CG Hamiltonian:
H̄(η̄) := −qd
2
∑k,l∈Lm
J̄(k − l)η̄k η̄l +∑k∈Lm
(h̄(k) +J̄(0)
2)η̄k
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Microscopic Models - Coarse Graining
I CG Generator:
Lcg f (η̄) =∑
k,l∈Lm
c̄k,l(η̄)[f (η̄ +1
qd(δl(k)− δk(l)))− f (η̄)]
I CG rate for Metropolis dynamics:
c̄k,l(η̄) = d0qd η̄k(1− η̄l)×
exp
(βmin{0, Ū(l , η̄)− Ū(k, η̄) + qd J̄(0)(η̄l − η̄k +
1
qd)− J̄(1)}
)I CG rate for Arrhenius dynamics:
c̄k,l(η̄) = d0qd η̄k(1− η̄l)e−β(U0+Ū(k,η̄))
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Microscopic Models - Coarse Graining
I Invariant measure:
µq,m,β(d η̄) =1
Zq,m,βe−q
dβH̄(η̄)Pq,m(d η̄)
I Alternative formulation:
µq,m,β(d η̄) =1
Zq,m,βe−qd (Ē(η̄)+O( 1
qd))
I Discrete free energy functional:
Ē (η̄) = βH̄(η̄) + R̄(η̄)
I Entropy:
R̄(η̄) :=∑k∈Lm
{η̄k log(η̄k) + (1− η̄k) log(1− η̄k)}
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Mesoscopic Models - Definition
I Formal SPDE (Katsoulakis et al. [1], [2]):
∂tρ = ∇ ·{L[ρ]∇δE
δρ
}+
1√N∇ ·{√
2L[ρ]Ẇ}
I ρ(x , t): particle densityI N: ”Size” of the system (number of particles)I E [ρ]: Free energy functionalI L[ρ]: MobilityI Ẇ (x , t): Space-time white noise
I Formal Invariant measure (Sponh ’91, [3]):
µN(dρ) =1
ZNe−NE(ρ)dρ
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Mesoscopic Models - Definition
I Free energy functional:
E [ρ] := βH(ρ) + R(ρ)
I Hamiltonian:
H(ρ) := −12
∫ ∫J(x − x ′)ρ(x)ρ(x ′)dxdx ′ +
∫h(x)ρ(x)dx
I Entropy:
R(ρ) :=
∫ρ(x) log(ρ(x)) + (1− ρ(x)) log(1− ρ(x))dx
I Mobility for Metropolis dynamics:
L[ρ] := d0ρ(1− ρ)I Mobility for Arrhenius dynamics:
L[ρ] := d0ρ(1− ρ) exp(−β(U0 + J ∗ ρ))
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Mesoscopic Models - Properties
I Connection with Cahn-Hilliard equation:I Infinite number of particles (mean field) and constant mobility.
I Connection with Cahn-Hilliard-Cook equation (stochasticversion of Cahn-Hilliard):
I Constant mobility (L[ρ] = L).
I Connection with Ginzburg-Landau equation:I Nearest neighborhood potential and constant mobility.
I In order to simulate an SPDE, it is necessary to:I Approximate white noise, discretize space (i.e.
semi-discretization), discretize timeI Numerical methods: finite difference, finite elements, spectral
methods
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Mesoscopic Models - Properties
I Connection with Cahn-Hilliard equation:I Infinite number of particles (mean field) and constant mobility.
I Connection with Cahn-Hilliard-Cook equation (stochasticversion of Cahn-Hilliard):
I Constant mobility (L[ρ] = L).
I Connection with Ginzburg-Landau equation:I Nearest neighborhood potential and constant mobility.
I In order to simulate an SPDE, it is necessary to:I Approximate white noise, discretize space (i.e.
semi-discretization), discretize timeI Numerical methods: finite difference, finite elements, spectral
methods
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Outline
Introduction
Space Discretization of SPDE
Numerical Simulations
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Manifest
I Derive SDE systems of Langevin-type where both the finitetime and infinite time errors are controllable.
1. Derive models where Detailed Balance Condition is satisfied.I Control equilibrium states and the knowledge of invariant
measure is important for sensitivity analysis
2. Weak error at finite time from GC model is controlled (by q).
3. Large Deviations: Action functionals between microscopicprocess and the derived models be the same.
I Transient and long time behavior is controlledI SPDE is embedded into the action functional
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Manifest
I Derive SDE systems of Langevin-type where both the finitetime and infinite time errors are controllable.
1. Derive models where Detailed Balance Condition is satisfied.I Control equilibrium states and the knowledge of invariant
measure is important for sensitivity analysis
2. Weak error at finite time from GC model is controlled (by q).
3. Large Deviations: Action functionals between microscopicprocess and the derived models be the same.
I Transient and long time behavior is controlledI SPDE is embedded into the action functional
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Direct Langevin Model - Definition
I Recall:
∂tρ = ∇ ·{L[ρ]∇δE
δρ
}+
1√N∇ ·{√
2L[ρ]Ẇ}
I Finite difference SPDE semi-discretization ⇒ system of SDEs:
dρ(xk) = uk(ρ)dt +m∑l=1
vk,l(ρ)dWl , k ∈ Lm
I Now, ρ is a vector with kth element ρ(xk)I Connection with GC variables: η̄k ≈ ρ(xk)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Direct Langevin Model - Definition
I Drift term:
uk(ρ) =1
2(Lk+1(ρ) + Lk(ρ))
[∂Ē(ρ)
∂ρ(xk+1)− ∂Ē(ρ)∂ρ(xk)
]− 1
2(Lk(ρ) + Lk−1(ρ))
[∂Ē(ρ)
∂ρ(xk)− ∂Ē(ρ)∂ρ(xk−1)
]I Diffusion matrix elements:
vk,k(ρ) =
√1
q(Lk+1(ρ) + Lk(ρ))
vk+1,k(ρ) = −vk,k(ρ)v,k,l(ρ) = 0 otherwise
I Ē (ρ) = βH̄(ρ) + R̄(ρ): discrete free energy functional
I Lk(ρ): discrete version of the mobility
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Direct Langevin Model - Properties - Weak Error
I Theorem: (by Katoulakis and Szepessy, 06’ [4])∀T finite, ρt solution of Langevin SDE, η̄t CG jump processand g a “mesoscopic” observable quantity. Then,
max0≤t≤T
|Eg(ρt)− Eg(η̄t)| ≤ CToq(1)
I Proof: (A sketch)I Define u(ξ, t) = E[g(ρt)|ρt = ξ]I Then, u satisfies backward Kolmogorov equation
∂tu + Leu = 0, t < T
u(·,T ) = g
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Direct Langevin Model - Properties - Weak Error
I Then
Eg(η̄T )− Eg(ρT ) = Eu(η̄T ,T )− Eu(η̄0, 0)
= E∫ T
0du(η̄t , t)
=
∫ T0
E[Lcgu − ∂tu]dt
=
∫ T0
E[E[Lcgu − Leu|η̄t ]]dt
I Taylor expand Lcgu (as in Langevin approximation) and derivebounds for the derivatives u′, u′′, u′′′ using Bernsteinestimates for the parabolic PDE.
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Direct Langevin Model - Properties - Large Deviations
I Rate function of the SDE system when m→∞:
S(Ψ) =
∫ T0
∫ 10
L[Ψ](∂xH)2dxdt
I where H(x , t) satisfies
Ψt = ∂x
{L[Ψ](
∂xΨ
Ψ(1−Ψ)− β∂x(J ∗Ψ))
}+ 2∂x{L[Ψ]∂xH}
I This rate function is the same as the microscopic ratefunction for surface diffusion with long range interactionsderived in (Asselah et al. 98’, p. 1077, [5]).
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Direct Langevin Model - Properties - DBC
I We’d like to know the invariant measure of {ρ(xk)}k∈Lm .I Try a guess:
µe(dρ) =1
Zee−qĒ(ρ)
∏k∈Lm
dρ(xk)
I This is not true because the generator of the SDE process isnot adjoint (Le 6= L∗e). Indeed,
< Le f , g >L2(µe ) =< f , Leg >L2(µe )
− 12q
∫ ∑k∈Lm
Ck(ρ)
[∂g
∂ρ(xk)f − ∂f
∂ρ(xk)g
]µe(dρ)
I SDE generator:
Le f =∑k
uk∂f
∂ρ(xk)+
1
2
∑k,l∈Lm
(vvT )kl∂2f
∂ρ(xk)∂ρ(xl)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Direct Langevin Model - Properties - DBC
I Interference term:
Ck(ρ) =
[∂Lk+1∂ρ(xk)
+∂Lk−1∂ρ(xk)
+ 2∂Lk∂ρ(xk)
− ∂Lk+1∂ρ(xk+1)
− ∂Lk∂ρ(xk+1)
− ∂Lk∂ρ(xk−1)
− ∂Lk−1∂ρ(xk−1)
]
I Depends only on the mobility functionI It is zero for additive noise (i.e. constant mobility) ⇒ Detailed
balance is satisfied ⇒ Invariant measure is known
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Perturbed Langevin Model 1 - Definition
I Add the interference term as “correction” to the SDE system
dρ(xk) =
(uk(ρ) +
1
2qCk(ρ)
)dt+
m∑l=1
vk,l(ρ)dWl , k ∈ Lm
I DBC is satisfied with invariant measure µe(dρ)!I Cost to be paid
I Drift is perturbed by a term of order O( 1q ) ⇒ finite-time weakerror becomes worse
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Perturbed Langevin Model 1 - Properties
I Asymptotics of the “correction” termI For Metropolis dynamics:
Ck(ρ) = 2[ρ(xk+1) + ρ(xk−1)− 2ρ(xk)] =2
m2∂xxρ(xk) + O(
1
m4)
I Of diffusion type ⇒ It can be absorbed into the free energyfunctional
I For Arrhenius dynamics:
Ck (ρ) = 2
(1− 2ρ(xk )
ρ(xk )(1− ρ(xk ))− β(J̄(0) + J̄(1))
)Lk (ρ)
−(
1− 2ρ(xk+1)ρ(xk+1)(1− ρ(xk+1))
− β(J̄(0) + J̄(1)))Lk+1(ρ)
−(
1− 2ρ(xk−1)ρ(xk−1)(1− ρ(xk−1))
− β(J̄(0) + J̄(1)))Lk−1(ρ)
= −1
m2∂xx
{(1− 2ρ(xk )
ρ(xk )(1− ρ(xk ))− β(J̄(0) + J̄(1))
)Lk (ρ)
}+ O(
1
m4)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Perturbed Langevin Model 1 - Properties
I Asymptotics of the “correction” termI For Metropolis dynamics:
Ck(ρ) = 2[ρ(xk+1) + ρ(xk−1)− 2ρ(xk)] =2
m2∂xxρ(xk) + O(
1
m4)
I Of diffusion type ⇒ It can be absorbed into the free energyfunctional
I For Arrhenius dynamics:
Ck (ρ) = 2
(1− 2ρ(xk )
ρ(xk )(1− ρ(xk ))− β(J̄(0) + J̄(1))
)Lk (ρ)
−(
1− 2ρ(xk+1)ρ(xk+1)(1− ρ(xk+1))
− β(J̄(0) + J̄(1)))Lk+1(ρ)
−(
1− 2ρ(xk−1)ρ(xk−1)(1− ρ(xk−1))
− β(J̄(0) + J̄(1)))Lk−1(ρ)
= −1
m2∂xx
{(1− 2ρ(xk )
ρ(xk )(1− ρ(xk ))− β(J̄(0) + J̄(1))
)Lk (ρ)
}+ O(
1
m4)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Perturbed Langevin Model 2 - Preparation
I Perturb the invariant measure:
µp(dρ) =1
Zpe−q(Ē(ρ)+
1qP̄(ρ))
∏k∈Lm
dρ(xk)
I Compute if DBC is satisfied. Answer is again no.
< Le f , g >L2(µp)=< f , Leg >L2(µp)
+1
2q
∫ ∑k∈Lm
(Pk(ρ)− Ck(ρ))[
∂g
∂ρ(xk)f − ∂f
∂ρ(xk)g
]µp(dρ)
I wherePk(ρ) = (Lk+1(ρ) + Lk(ρ))
[∂P̄
∂ρ(xk+1)− ∂P̄∂ρ(xk)
]− (Lk(ρ) + Lk−1(ρ))
[∂P̄
∂ρ(xk)− ∂P̄∂ρ(xk−1)
]=
2
m2∂x
{∂x
{∂P̄
∂ρ(xk)
}Lk(ρ)
}+ O(
1
m4)
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Perturbed Langevin Model 2 - Definition
I New SDE system:
dρ(xk) =
(uk(ρ) +
1
2qC̄k(ρ)
)dt+
m∑l=1
vk,l(ρ)dWl , k ∈ Lm
I where C̄k(ρ) = Ck(ρ)− Pk(ρ)I Remember for Metropolis dynamics, “correction” term, Ck(ρ),
is the Laplacian of the density thus if we choose
P̄(ρ) =∑k∈Lm
[ρ(xk) log(ρ(xk)) + (1− ρ(xk)) log(1− ρ(xk))]
I thenC̄k(ρ) = O(
1
m4)
I Interpretation: Entropy is perturbed by a factor 1 + 1q ⇒temperature is perturbed by the same amount.
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Perturbed Langevin Model 2 - Definition
I Similar result but less accurate are obtained for Arrheniusdynamics. If we choose
P̄(ρ) =∑k
[ρ(xk) log(ρ(xk)) + (1− ρ(xk)) log(1− ρ(xk))]
− β2(J̄(0) + J̄(1))
2
∑k
[ρ(xk) log(ρ(xk))− (1− ρ(xk)) log(1− ρ(xk))]
− β2
4(J̄(0) + J̄(1))H̄(ρ)
I then
C̄k(ρ) = O(L2
q2m4)
I Now, both entropy and Hamiltonian are perturbed.
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Outline
Introduction
Space Discretization of SPDE
Numerical Simulations
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
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Implementation Details
I Use the Predictor-Corrector Euler method for timediscretization
I Fast, semi-implicit, 1st weak order method
I Computational acceleration by exploiting the FFT-basedcomputation of convolution
I Make the algorithm independent of interaction potential length
I d-dimensional lattices are straightforward to be simulated
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IntroductionSpace Discretization of SPDE
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Benchmarking
I Use attractive potential and Metropolis dynamics.I The expected behavior is small droplets to be merged into a
large drop as time passes.I Minimize the free energy ⇔ minimize the interface of the
droplets.
VIDEO
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
Gaussian_perturbDB_q_4_Metro_stoch_longTime_1.aviMedia File (video/avi)
IntroductionSpace Discretization of SPDE
Numerical Simulations
Pattern formation
I Stable pattern formation requires the interaction potential tohave both attractive and repulsive parts
I Morse potential:
J(x−y ;χ, ra, rr , J0) :=J0
2πr 2aexp
(−||x − y ||
2
2r 2a
)− J0χ
2πr 2rexp
(−||x − y ||
2
2r 2r
)I Arrhenius dynamics were used
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Simulation over mean coverage c0 and χ
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c0 = .2, χ = .4 c0 = .5, χ = .4 c0 = .8, χ = .4
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c0 = .2, χ = .8 c0 = .5, χ = .8 c0 = .8, χ = .8
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
Simulation over inverse temperature β
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β = 8.0 β = 9.5
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β = 11.0 β = 12.5
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDE
Numerical Simulations
References
M. A. Katsoulakis and D. G. Vlachos.
Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interactingpatricles.Journal of Chemical Physics, 119(18):9412–9427, 2003.
S. Are, M.A. Katsoulakis, and A. Szepessy.
Coarse-grained Langevin approximations and spatiotemporal acceleration for kinetic Monte Carlosimulations of diffusion of interacting particles.Chinese annals of mathematics. Series B, 30:653–682, 2009.
H. Spohn.
Large scale dynamics of interacting particles.Texts and Monographs in Physics. Springer-Verlag, Heidelberg, 1991.
M. A. Katsoulakis and A. Szepessy.
Stochastic hydrodynamical limits of particle systems.Commun. Math. Sci., 4(3):513–549, 2006.
A. Asselah and G. Giacomin.
Metastability for the exclusion process with mean-field interaction.J. Stat. Phys., 93:1051–1110, 1998.
Yannis Pantazis join work with M. Katsoulakis University of Massachusetts, Amherst, USA Partially supported by NSF: DMS and CMMI CDI-Type IIControlled-Error Semi-discretization of Mesoscopic Stochastic Equations for Surface Diffusion
IntroductionSpace Discretization of SPDENumerical Simulations