Controlled-Error Semi-discretization of Mesoscopic Stochastic … · 2011-06-17 · Introduction Space Discretization of SPDE Numerical Simulations Manifest I Derive SDE systems ofLangevin-typewhere

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



Outline

Introduction

Space Discretization of SPDE





Surface Diffusion - Motivation

I Plass et al., Nature, 01’. Self-assembly of Pb on Cu(111)




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.




Surface Diffusion – Model Hierarchy




Microscopic Models - Visualization

I Microscopic and Coarse-grained lattices




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)

)




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)




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





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))





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)}




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




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))




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




Outline

Introduction






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




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)




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




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




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.




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]).




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)




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




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


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




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)




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)





I New SDE system:

dρ(xk) =

(uk(ρ) +

1

2qC̄k(ρ)

)dt+

m∑l=1


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.





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.




Outline

Introduction






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




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


Gaussian_perturbDB_q_4_Metro_stoch_longTime_1.aviMedia File (video/avi)



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




Simulation over mean coverage c0 and χ

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

c0 = .2, χ = .4 c0 = .5, χ = .4 c0 = .8, χ = .4

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

c0 = .2, χ = .8 c0 = .5, χ = .8 c0 = .8, χ = .8




Simulation over inverse temperature β

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

β = 8.0 β = 9.5

10 20 30 40 50 60

10

20

30

40

50

60

10 20 30 40 50 60

10

20

30

40

50

60

β = 11.0 β = 12.5




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.


IntroductionSpace Discretization of SPDENumerical Simulations

Documents

Controlled-Error Semi-discretization of Mesoscopic Stochastic … · 2011-06-17 · Introduction Space Discretization of SPDE Numerical Simulations Manifest I Derive SDE systems ofLangevin-typewhere