Mathematical strategies and error quantification in the - PowerPoint PPT Presentation

Mathematical strategies and error quantification in the coarse-graining of many-body stochastic systems Markos Katsoulakis University of Massachusetts, Amherst, USA and University of Crete, Greece

1. Coarse-graining of polymers; DPD methods 2. Stochastic lattice dynamics/ KMC Time (s) Microscopic lattice Coarse lattice Microscopics �→ CG system �→ Reconstructed Microscopics

Tsch¨ op, Kremer, Batoulis, B¨ urger and Hahn Acta Polymer. ’98.

Microscopics: United Atom (UA) Model • Continuum model: X ∈ ( R 3 ) N – positions of n atoms on one macromolecule; m macromolecules; N = nm . • Hamiltonian: H N ( X ) = H b ( X )+ H nb ( X )+ H Coul ( X )+ H wall + H kin Bonded Interactions: Gaussian, FENE, etc. short-range � H b ( X ) = U b ( θ i , φ i , r i ) short-range i Non-Bonded Interactions: 12-6 Lennard Jones long-range � U LJ H nb ( X ) = nb ( | x i − x j | ) i,j

1 Equilibrium Gibbs measure at β = kT . µ ( dX ) = 1 Z e − β H ( X ) Π dx i Molecular Dynamics ( via Langevin thermostat) • UA is a typical set-up for CG in polymer science literature: Briels, et. al. J.Chem.Phys. ’01; Doi et. al. J.Chem.Phys. ’02; Kremer et. al. Macromolecules ’06, etc. Also the parametric statistics approach: M¨ uller-Plathe Chem.Phys.Chem ’00.

CG procedure, ”blobs”: for instance Doi, et al. ’02 T X = Q = ( q 1 , . . . , q m ) ∈ Q , where q i ∈ R 3 . Exact CG Hamiltonian ¯ H ( Q ) via Renormalization map : � H ( Q ) = − 1 e − β H ( X ) dX ¯ β log { X | T X = Q }

Break-up of computational task: Simplifying assumptions (i) ¯ H decouples: � H ( Q ) = ¯ ¯ H b + ¯ U b + ¯ ¯ H nb = U nb CG var. U φ (ii) ¯ U b = ¯ b + ¯ b + ¯ U θ U r b where each term depends only on torsion angle φ , rotation angle θ and distance r respectively between successive CG particles. (iii) ¯ U nb depends only on two-body interactions between CG par- ticles; no multi-body interactions included.

How to calculate the CG non-bonded interactions ¯ U nb : McCoy-Curro scheme, Macromolecules ’98. For two isolated small molecules with centers of mass at q 1 , q 2 : � U nb ( | q 1 − q 2 | ) = − 1 e − β H ( X ) dX β log { X | T X =( q 1 ,q 2 ) } • The calculation is computationally feasible but disregards multi- body interactions. • Extension to long chains: Doi et al. J.Chem.Phys. ’02.

Challenges in coarse-graining methods Often: wrong predictions in dynamics, phase transitions, melt structure, crystallization, etc. See for instance: • CG in polymers: sensitive dependence to temperature low vs. high Doi et al. J.Chem.Phys. ’02

• DPD: Pivkin, Karniadakis J. Chem. Phys. (2006): artificial crystallization • ”classical” example: 1-D nearest neigbor Ising vs. Curie-Weiss (or Mean Field)

Mathematics and Numerics of CG 1. Error Quantification and numerical accuracy of CG methods. 2. The role of randomness: need to approximate the measure rather than just H = H ( X ): T ∗ µ micro ( dX ) ≈ µ cg ( dQ ) ∼ e − β ¯ e − β H ( X ) dX ∼ µ micro ( dX ) H ( Q ) dQ �→ 3. The role of multi-body CG interaction terms. 4. ”Reverse map”-reconstruct microscopic info from CG: Mathematical formulation in terms of relative entropy; loss of information during CG–information re-insertion in reverse map. joint work with: P. Plech´ aˇ c (U of TN, ORNL), V. Harmandaris (Max Planck Inst. Polymers, Mainz)

2. Stochastic lattice dynamics–Ising Systems σ ( x ) = 0 or 1: site x is resp. empty or occupied. Hamiltonian: H N ( σ ) = − 1 � x � = y J ( x, y ) σ ( x ) σ ( y ) + h � x σ ( x ) 2 - J : potential with interaction range L , J ( x − y ) = 1 LV � i − j � , x = i/N, y = j/N L possibly short-/long- range interactions. 1 Canonical Gibbs measure: at the inverse temperature β = kT , 1 exp � − β H N ( σ 0 ) � µ Λ , β ( σ = σ 0 ) = P N ( σ = σ 0 ) Z Λ , β

Arrhenius adsoprtion/desorption dynamics: σ ( x ) = 0 or 1: site x is resp. empty or occupied. Generator: L X f ( σ ) = � x c ( x, σ , X )[ f ( σ x ) − f ( σ )] Transition rate: c ( x, σ , X ) = c 0 exp � − β U ( x ) � U ( x ): Energy barrier a particle has to overcome in jumping from a lattice site to the gas phase. - Detailed Balance - U ( x ) = U ( x, σ , X ) = � z � = x J ( x − z ) σ ( z ) − h ( X ) . - strong interactions/low temperature → clustering/phase transitions

Why study this system? 0. Many-particle system, related to realistic models, KMC, etc. 1. Strong interactions/low temperature → clustering/phase tran- sitions. ”Complex” landscape: metastability of islands. How CG performs in predicting phase transitions and various rare events? 2. Equilibrium/ Detailed Balance. How CG performs in transient and long time regimes? 3. Numerous analytic benchmark solutions; a variety of mathe- matical physics tools.

Hierarchical coarse-graining of stochastic lattice dynamics K., Majda, Vlachos, Proc. Nat. Acad. Sci. ’03, JCompPhys ’03; K., Vlachos J.Chem.Phys. ’03 Construct a stochastic process for a hierarchy of “mesoscopic” length or time scales. Coarse-grained Monte Carlo algorithm (CGMC). Coarse observable at resolution q : � η t ( k ) = T σ t ( k ) := σ t ( y ) y ∈ D k In general it is non-markovian

Stochastic closures : can we write a new approximating Markov process for η t ? • ”projective dynamics”: Koleshik, Novotny, Rikvold, PRL ’98; coarse rates for total coverage calculated by sampling; Ergodicity : Are the long-time dynamics reproduced? • Errors can contaminate the simulation at long times; wrong switching times in bistable systems: Hanggi et al PRA ’84 (well- mixed systems). • Connections to lumpable Markov processes

Microscopic Coarse-Grained Process Process Lumping CG + error state state ... state 1 1 2 Reconstruction +error Lumping CG + error state 2 Reconstruction +error Lumping Lumping Lumping Lumping + error + error + error + error CG state state ... state m N-1 N Reconstruction Reconstruction Reconstruction Reconstruction +error +error +error +error Microscopic CG equilibria Error equilibria Estimates

1. CG Schemes at Equilibrium K., Plechac, Rey-Bellet, Tsagkarogiannis, [ M 2 AN, ’07, J. Non. Newt. Fluid Mech. ’08, preprint ] • CG Hamiltonian–Renormalization Group Map : N = mq � H m ( η ) = e − β ¯ e − β H N ( σ ) P N ( d σ | η ) ≡ E [ e − β H N | η ] H (0) • Correction terms around a first ”good guess” ¯ m : m ( η ) − 1 H (0) m ) | η ] , β log E [ e − β ( H N − ¯ H (0) H m ( η ) = ¯ ¯ m = N, N − 1 , ... • Heuristics: Expansion of e ∆ H and log: − E [ ∆ H | η ] 2 + O(( ∆ H ) 3 ) ( ∆ H ) 2 | η � = E [ ∆ H | η ] + E � formal calculations inadequate since: ∆ H ≡ H N − ¯ H (0) = N · O( ǫ ) m H (0) • Rigorous analysis – Cluster expansion : around ¯ m

Systems with short+long-range interactions H N ( σ ) = H l N ( σ ) + H s N ( σ ) ; J : long range potential ∼ H ( l ) N radius L . K : short range potential ∼ H ( s ) with radius S << L . N Examples: Surface processes, epitaxial growth, polymers, etc. CG: approximation of the free energy-landscape. CG prior: ¯ P m ( η ) = P N ( { σ : T σ = η } ) • Splitting strategy: − β � m ( η ) � N ( σ ) − ¯ H l H l m ( η ) ¯ P N ( d σ | η ) e − β ¯ e − β H s H l e − β H N ( σ ) P N ( d σ ) N ( σ ) e = P m ( η )

Case 1: Long- and intermediate-range interactions Approximate CG Hamiltonian: H (0) ( η ) = − 1 J ( k, l ) η ( k ) η ( l ) − 1 � � � � ¯ ¯ ¯ ¯ J (0 , 0) η ( l )( η ( l ) − 1) + h ( l ) η ( l ) 2 2 l ∈ Λ c l ∈ Λ c l ∈ Λ c k � = l M M M H (0) | η � • E � H N − ¯ = 0 Involves two-body CG interaction only: � � ¯ J ( k, l ) η ( k ) η ( l ) = J ( x − y ) σ ( x ) σ ( y ) P N ( d σ | η k , η l ) x ∈ C k ,y ∈ C l Where J ( k, l ) = 1 � � ¯ J ( x − y ) q 2 x ∈ C k y ∈ C l ,y � = x • Analytical version of McCoy-Curro scheme in polymers: � U mcc ( η k , η l ; k − l ) = − 1 e − β H N ( σ ) P N ( d σ | η k , η l ) ¯ β log

Corrections to the Hamiltonian ¯ H (0) �→ Multi-body terms H (0) H (1) H m ( η ) = ¯ ¯ m ( η ) + ¯ m ( η ) + ... � � � H (1) ( η ) = β ¯ [ j 2 k 1 k 2 k 3 ( − E 1 ( k 1 ) E 2 ( k 2 ) E 1 ( k 3 ) + ... k 1 k 2 >k 1 k 3 >k 2 • E r ( k ) ≡ E r ( η ( k )) = (2 η ( k ) /q − 1) r + o q (1) • “Moments” of interaction potential J : � � � j 2 ( J ( x − y ) − ¯ J ( k 1 , k 2 ))( J ( y − z ) − ¯ k 1 k 2 k 3 = J ( k 2 , k 3 )) x ∈ C k 1 y ∈ C k 2 z ∈ C k 3 Computational complexity-Compression of ¯ H (1) • Evaluation of the Hamiltonian: Count Speed-up O( NL d ) Microscopic: H N ( σ ) 1 CG0: ¯ H (0) O( ML d /q d ) O( q 2 d ) H (0) + ¯ CG1: ¯ H (1) O( ML 2 d /q 2 d ) O( q 3 d /L d ) • Decay of J (e.g. Coulomb) �→ J − ¯ J decays faster.