Lecture

A treatment of multi-scale hybrid systems involving deterministic and stochastic approaches, coarse graining and hierarchical closures

Alexandros Sopasakis Department of Mathematics and Statistics University of North Carolina at Charlotte

Collaborators:

Markos Katsoulakis, Andy Majda, Dion Vlachos, Peter Plechac, Richard Jordan Work partially supported by NSF-DMS-0606807 and DOE-FG02-05ER25702

Outline • 

Motivation – climate modeling, catalysis, traffic flow

• 

The prototype hybrid system introduction and examples

• 

Part I. Deterministic closures: stochastic averaging, local mean field models

• 

Part II. Stochastic closures: coarse-graining in hybrid systems

• 

The role of rare events and phase transitions

• 

Intermittency and metastability phenomena.

• 

Conclusions

Alexandros Sopasakis

Motivation   Pure Stochastic •  Micromagnetics (P. Plechac) •  Traffic Flow (N. Dundon, T. Alperovich) •  Epidemiology (R. Jordan)

  Coupled Systems •  Catalytic Reactors (M. Katsoulakis, D. Vlachos) •  Climate Modeling (A. Majda, B. Khouider) •  General Biological Systems •  …

Alexandros Sopasakis

Surface processes:

[Vlachos, Schmidt, Aris, J. Chem. Phys 1990] [Lam, Vlachos, Phys. Review B 2001]

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Some challenges and questions: • 

Disparity in scales and models; DNS require ensemble averages for large systems

• 

Model reduction, however no clear scale separation; need hierarchical coarse-graining

• 

Deterministic vs stochastic closures; when is stochasticity important?

• 

Error control, stability of the hybrid algrithm; efficient allocation of computational resources adaptivity, model and mesh refinement

Alexandros Sopasakis

A mathematical prototype hybrid model • 

We introduce the microscopic spin flip stochastic Ising process

• 

Coupled to a PDE/ODE that serves as a caricature of an overlying gas phase dynamics.

Alexandros Sopasakis

The Stochastic Component: • 

We assume a lattice Λ and denote by σ(x) the value of the spin at location x

• 

A spin configuration σ is an element of the configuration space and we write

• 

The stochastic process is a continuous time jump Markov process on with generator L

Alexandros Sopasakis

0 1

0

1

1

The Stochastic Component:

Arrhenius Spin-Flip Dynamics

The Arrhenius spin-flip rate c(x,s) at lattice site x and spin configuration s is given by

with interaction potential and local interaction via Parameters/Constants: •  •  • 

where

is the characteristic time of the stochastic process

L denotes the interaction radius V is usual taken to be some uniform constant

Alexandros Sopasakis

The Stochastic Component: Equilibrium states of the stochastic model are described by the Gibbs measure at the prescribed temperature T

where

and

is the energy Hamiltonian for

with

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The Stochastic Component: The probability of a spin-flip at x during time [t, t+Dt] is

The dynamics as described here leave the Gibbs measure invariant since they satisfy detailed balance,

where

Alexandros Sopasakis

The Stochastic Component In the case of a surface diffusion process we implement spin-exchange Arrhenius dynamics

where

We can mix spin-flip and spin-exchange dynamics based on the requirements/physics of the model. In this case the corresponding generator of the combined mechanism for the dynamics comprised of spin-flip and spin exchange is given via,

Alexandros Sopasakis

The Deterministic Component: We consider the following type of Complex Ginzburg Landau ODE

where The Jacobian of the linearized system has eigenvalues

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The Deterministic Component: Some examples for the ODE CGL: Bistable: Saddle: Linear: where

depends linearly on

Some examples for the PDE

Alexandros Sopasakis

Hybrid System Coupling

The stochastic system and the ODE are coupled via, respectively: • 

The external field,

• 

The area fraction

(or total coverage) defined as the spatial average

of the stochastic process s,

Alexandros Sopasakis

Part I. Deterministic Closure The main requirement is the ergodicity property of the stochastic process

where Due to special structure of g (depends linearly on we have

Where

s)

can be found from the known Gibbs equilibrium measure

Alexandros Sopasakis

Part I. Deterministic Closure The main requirement is the ergodicity property of the stochastic process

Due to special structure of g (depends linearly on we have

Where

)

can be found from the known Gibbs equilibrium measure

Alexandros Sopasakis

Part I. Deterministic Closure Suppose

is the solution of the hybrid system:

And

is the solution of the

(reduced) averaged system:

• 

On an arbitrary bounded time interval [0, T] with fixed N and tc we have:

Alexandros Sopasakis

Part I. Deterministic Closure   Calculate

Numerically

For a given external potential h we calculate by Monte Carlo

In fact in the nearest neighbor case

can be calculated explicitly

Alexandros Sopasakis

  Calculate

Part I. Deterministic Closure analytically

In the long range interaction limit, we obtain the usual mean-field formula where ,-./01../23.145/6737879.1/: ;-
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Fine Lattice Λ



Micro cells: 1 2 3 4

Coarse cells

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Part II. Stochastic Closures The coarse grained generator for the Markovian process h is defined to be,

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