Preservation of Thermodynamic Structure in Model Reduction and - - PowerPoint PPT Presentation

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Hans Christian Öttinger Department of Materials, ETH Zürich, Switzerland Preservation of Thermodynamic Structure in Model Reduction and Coarse Graining

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F kT Z ln – =

Don’t derive! Evaluate!

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Outline

Thermodynamic structure Coarse graining Model reduction

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GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling

metriplectic structure (P. J. Morrison, 1986)

dx dt

• L x

( ) δE x ( ) δx

• ⋅

M x ( ) δS x ( ) δx

• ⋅

+ =

L antisymmetric, M Onsager/Casimir symmetric, positive-semidefinite Jacobi identity L x ( ) δS x ( ) δx

• ⋅

= M x ( ) δE x ( ) δx

• ⋅

=

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Physics of the Poisson Bracket

d dt

• rj

pj 0 1 1 – ∂H ∂rj

• ∂H

∂pj

• ⋅

= cosymplectic matrix A B , { } ∂A ∂rj

• ∂B

∂pj

• ⋅

∂A ∂pj

• ∂B

∂rj

• ⋅

–    

j 1 = N



= drj dt

• ∂H

∂pj

• pj

m

• vj

= = = dpj dt

• ∂H

∂rj

• –

Fj = = dA dt

• ∂A

∂rj

• ∂A

∂pj

• 0 1

1 – ∂H ∂rj

• ∂H

∂pj

• ⋅

⋅ =

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Outline

Thermodynamic structure Coarse graining Model reduction

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Generalized Canonical GENERIC – Ensemble

px z ( ) exp λk x ( )Πk z ( )

k



–       ∝

list of relevant observables (slow state variables)

Πk z ( ):

list of Lagrange multipliers (adjusted to give proper averages)

λk x ( ):

Example:

xk ρ r ( ) = Πk z ( ) miδ r ri – ( )

i



= xk Πk  x =

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Coarse Graining: Static Building Blocks relative entropy

E E0  x = Ljk Πj Πk { , }  x =

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Intermediate Exam No.1 Has this messy room a large or small entropy?

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Coarse Graining: Dissipative Bracket

D 1 2τ

• ∆τx

( )2   =

Einstein

A B , [ ] 1 2kBτ

• ∆τAf ∆τBf

  =

Frictional properties are related to time-dependent fluctuations:

dS dt

• δS

δx

• M δS

δx

• ⋅

⋅ S S , [ ] = =

emergence of irreversibility

Green-Kubo

τ1 τ2

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A B , [ ] 1 2kBτ

• ∆τAf ∆τBf

  = Don’t derive! Evaluate!

Systematic Coarse

Graining: “Four Lessons and A Caveat” from Nonequilibrium Statistical Mechanics

Hans Christian Öttinger

Abstract

With the guidance offered by nonequilibrium statistical thermodynamics, simulation techniques are elevated from brute-force computer experiments to systematic tools for extracting complete, redundancy-free, and consistent coarse-grained information for dynamic systems. We sketch the role and potential of Monte Carlo, molecular dynamics, and Brownian dynamics simulations in the thermodynamic approach to coarse graining. A melt of entangled linear polyethylene molecules serves us as an illustrative example. MRS BULLETIN • VOLUME 32 • NOVEMBER 2007 • www/mrs.org/bulletin

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Outline

Thermodynamic structure Coarse graining Model reduction

emergence no emergence

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Dirac’s Bracket Construction

δAp δx

• δA

δx

• L δAp

δx

• ⋅

L δA δx

• ⋅

invariant manifold new bracket:

δAp δx

• L δBp

δx

• ⋅

⋅ Ap Bp , { } =

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Kramers’ Escape Problem

h ξ

1

• 1

1

∂p ∂t

• D ∂

∂ξ

• p

∂hε ∂ξ

• ∂p

∂ξ

• +

    = hε h ε ⁄ = peq e

hε –

Z ⁄ = u p peq ⁄ = ds dξ ⁄ e

hε

= ∂u ∂t

• De

hε ∂

∂ξ

• e

h – ε∂u

∂ξ

• 

   = ∂u ∂t

• D e

2hε ∂2u

∂s2

• =

Invariant manifold (1d): uy

1 ys + = M ZD kB

• e

–

2hε ∂

∂s

• u ∂

∂s

• e

2hε

= δS δu

• kB

Z

• e

–

2 – hε

u ln =

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Solution to Kramers’ Escape Problem

e

2hε ∂

∂s

• u ∂

∂s

• e

2hε a s

( ) s = Mred ZD kB

• sa s

( ) s d

s ˆ – s ˆ



1 –

– = Z 2πε h'' 1 ( )

• ≈

s ˆ πε 2 h'' 0 ( )

• e1 ε

⁄

≈ Mred 2D kBZs ˆ3

• uR

uL – u ln

R

u ln L –

• =

δSred δu

• kBs

ˆ 2

• e

–

2 – hε

u ln

R

u ln L – ( ) = u ·R u ·L – D Zs ˆ

• –

uR uL – ( ) = =

reaction kinetics

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Summary

Thermodynamic structure Coarse graining Model reduction

emergence, Green-Kubo formula, no emergence, invariant manifolds, Dirac’s don’t derive – evaluate!, efficient simulations GENERIC, metriplectic construction, Kramers’ escape problem, fortuitous cancellation mechanisms

preservation