Preservation of Thermodynamic Structure in Model Reduction and Coarse Graining Hans Christian Öttinger Department of Materials, ETH Zürich, Switzerland Polymer Physics
F = – kT ln Z Don’t derive! Evaluate! Polymer Physics
Outline Thermodynamic structure Coarse graining Model reduction Polymer Physics
GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling metriplectic structure (P. J. Morrison, 1986) dx ( ) δ E x ( ) δ S x ( ) ( ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = L x + M x ⋅ ⋅ dt δ x δ x ( ) δ E x ( ) δ S x ( ) ( ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - M x = 0 L x = 0 ⋅ ⋅ δ x δ x M Onsager/Casimir symmetric, L antisymmetric, positive-semidefinite Jacobi identity Polymer Physics
Physics of the Poisson Bracket ∂ H d r j p j ∂ H - - - - - - - - - - - - - - - - - - - - - - - - - = = = v j - r j ∂ r j d 0 1 dt ∂ p j m - - - - = ⋅ dt p j – 1 0 ∂ H d p j - - - - - - - ∂ H - - - - - - - - - - - - - - ∂ p j = – = F j dt ∂ r j cosymplectic matrix ∂ A ∂ H - - - - - - - - - - - - - - N ∂ r j ∂ r j dA 0 1 ∂ A ∂ B ∂ A ∂ B - - - - - - - = ⋅ ⋅ - - - - - - - - - - - - - - - - - - - - - - - - - - - - A B = – { , } ⋅ ⋅ dt ∂ A – 1 0 ∂ H ∂ r j ∂ p j ∂ p j ∂ r j - - - - - - - - - - - - - - ∂ p j ∂ p j j = 1 Polymer Physics
Outline Thermodynamic structure Coarse graining Model reduction Polymer Physics
Generalized Canonical GENERIC – Ensemble Π k z ( ) : list of relevant observables (slow state variables) λ k x ( ) : list of Lagrange multipliers (adjusted to give proper averages) x x k = Π k p x z exp – λ k x ( )Π k z ( ) ∝ ( ) k Π k z = m i δ r – r i Example: ( ) ( ) i x k = ρ r ( ) Polymer Physics
Coarse Graining: Static Building Blocks x E = E 0 x L jk = { , } Π j Π k relative entropy Polymer Physics
Intermediate Exam No.1 Has this messy room a large or small entropy? Polymer Physics
Coarse Graining: Dissipative Bracket dS δ S - M δ S - - - - - - - - - - - - - - - - - = = S S ⋅ ⋅ [ , ] dt δ x δ x Frictional properties are related to time-dependent fluctuations: 1 1 - ∆ τ A f ∆ τ B f ) 2 - - - - - - - - - - - A B = - - - - - [ , ] D = ∆ τ x ( 2 k B τ 2 τ Green-Kubo Einstein emergence of irreversibility τ 1 τ 2 Polymer Physics
- ∆τ A f ∆τ B f 1 - - - - - - - - - - - A B = [ , ] 2 kB τ Don’t derive! Evaluate! Polymer Physics
S ystematic 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
Outline Thermodynamic structure Coarse graining emergence Model reduction no emergence Polymer Physics
Dirac’s Bracket Construction new bracket: δ A p A p B p = - - - - - - - - - { , } δ x δ A p - L δ B p - - - - - - - - - - - - - - - - - ⋅ ⋅ δ A δ x δ x - - - - - - δ x L δ A p - - - - - - - - - ⋅ δ x L δ A - - - - - - ⋅ δ x invariant manifold Polymer Physics
Kramers’ Escape Problem ∂ h ε ∂ p ∂ p D ∂ h - - - - - - - - - - - - p - - - - - - - - - - - - - - = + ∂ t 1 ∂ξ ∂ξ ∂ξ h ε ∂ – ε ∂ u h ∂ u - - - - - - - - - - - - e - - - - - - = De ∂ t ∂ξ ∂ξ ξ 2 h ε ∂ 2 u -1 0 1 ∂ u - - - - - - - - - - - - - - = D e ∂ s 2 ∂ t h ε = h ε ⁄ – h ε – h ε 2 k B p eq = e Z 2 h ε ∂ ⁄ δ S 2 h ε ZD - u ∂ - - - - - - - - - - - - e = – ln u - - - - - - - - e - - - - - - - - - e M = – δ u Z k B ∂ s ∂ s u = p p eq ⁄ h ε Invariant manifold (1d): u y = 1 + ys ds d ξ = e ⁄ Polymer Physics
Solution to Kramers’ Escape Problem 2 h ε ∂ 2 h ε a s - u ∂ 2 πε - - - - - - - - - e e = s ( ) Z - - - - - - - - - - - - - ≈ ∂ s ∂ s h '' 1 ( ) ˆ – 1 s ZD πε - e 1 ε M red ⁄ ˆ - - - - - - - - s - - - - - - - - - - - - - - - - - - = – sa s ( ) s d ≈ k B 2 h '' 0 ( ) ˆ – s u R u L ˆ δ S red – h ε 2 2 D – k B s M red - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - R ln L = - - - - - - - - - - - - - - - - - - - - e = – ln u – u ( ) ˆ 3 R ln L δ u 2 k B Zs ln u – u D · R · L u R u L reaction kinetics - - - - - - u = – u = – – ( ) ˆ Zs Polymer Physics
Summary Thermodynamic structure preservation GENERIC, metriplectic Coarse graining emergence, Green-Kubo formula, don’t derive – evaluate!, efficient simulations Model reduction no emergence, invariant manifolds, Dirac’s construction, Kramers’ escape problem, fortuitous cancellation mechanisms Polymer Physics
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