Example on 2D potential with 4 wells Simulations by Masha Cameron - PowerPoint PPT Presentation

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization Eric Vanden-Eijnden Courant Institute • Spectral approach to metastability - sets, currents, pathways; • Transition path theory or how to focus on a specific `reactive’ event; • Application to Lennard-Jones clusters reorganization (with Masha Cameron). Monday, October 8, 12

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization Eric Vanden-Eijnden Courant Institute • Spectral approach to metastability - sets, currents, pathways; • Transition path theory or how to focus on a specific `reactive’ event; • Application to Lennard-Jones clusters reorganization (with Masha Cameron). Monday, October 8, 12

Spectral approach to metastability • Metastability associated with presence of one or more groups of small eigenvalues of the generator, corresponding to slow relaxation processes in the system. • Eigenvectors/eigenfunctions associated with these small eigenvalues indicate what the metastable sets are and what the mechanism of transition (reaction) between them is. 1.5 1.2 1 1 0.8 0.5 0.6 position 0 0.4 y 0.2 − 0.5 0 − 1 − 0.2 − 1.5 − 0.4 0 2 4 6 8 10 − 1 − 0.5 0 0.5 1 time 4 x x 10 Monday, October 8, 12

Spectral approach to metastability p • Overdamped Langevin equation: dX = �r V ( X ) dt + 2 ε dW ( X 2 Ω ✓ R n ) • Generator: L = r V · r � ε ∆ • Eigenvalue/eigenfunction: L φ = λφ 0 = λ 0 < λ 1 ≤ λ 2 ≤ · · · • Spectral representation of transition probability distribution: ∞ X p x e − λ k t φ k ( x ) φ k ( y ) C − 1 e − V ( y ) / ε dy, t ( dy ) = k =0 Z e − V ( y ) / ε dy, C = Ω Monday, October 8, 12

Spectral approach to metastability • Suppose there exists a group of K small eigenvalues: 0 = λ 0 < λ 1  λ 2  · · ·  λ K − 1 ⌧ λ K  · · · • Indicative of metastability. More precisely, on timescales such that λ K − 1 t ⇠ 1 , λ K t � 1 the processes described by the eigenvectors of index K or higher will have decayed and the remaining part of the dynamics will be described by the slow first K eigenvectors Note that the transition probability relaxes slowly if there is metastability, but the process itself is fast - rare events are infrequent but when they occur they typically do so fast! 1.5 1.2 1 1 0.8 ∞ 0.5 p x X e − λ k t φ k ( x ) φ k ( y ) C − 1 e − V ( y ) / ε dy, 0.6 t ( dy ) = position 0 0.4 y k =0 0.2 − 0.5 0 − 1 − 0.2 − 1.5 − 0.4 0 2 4 6 8 10 − 1 − 0.5 0 0.5 1 x time 4 x 10 Monday, October 8, 12

Example on 2D potential with 4 wells Simulations by Masha Cameron Monday, October 8, 12

Spectral approach to metastability • Forward Kolmogorov equation as a conservation law: ∂ t ρ x t = � div j x j x t = �r V ρ x t � ε r ρ x p x t ( dy ) = ρ x t , t ( y ) dy t • Spectral decomposition of currents: ∞ X j x e − λ k t φ k ( x ) j k ( y ) , j k ( y ) = C − 1 e − V ( y ) / ε r φ k ( y ) t ( y ) = k =0 Analysis of 1-form in Witten complex Flowlines of current indicative of mechanisms by which slow relaxation occurs dx d τ = j k ( x ) Monday, October 8, 12

Slowest modes in 2D potential with 4 wells Simulations by Masha Cameron Monday, October 8, 12

Transition Path Theory • How to make spectral approach practical as a computational tool ? In applications, one is typically in high dimensional systems whose spectrum is enormously complicated and cannot be calculated explicitly (even numerically). • Global viewpoint of metastability also problematic - the longest timescales may not be the relevant ones (i.e. they could be associated with presence of deadends or dynamical traps), there may be many of them (subgroups into groups), etc.. • Can we focus on a single `reaction’ rather than having to analyze them all thru calculation of the spectrum? (Indeed in a given system there may be specific reactions of interest and we don’t know a priori to which part of the spectrum they are associated) • Can this all be done even if there is no metastability (i.e. no small parameter)? Monday, October 8, 12

Transition Path Theory • Main idea: focus on `reactive’ trajectories associated with a given transition (reaction) A B • Probability distribution of reactive trajectories µ R ( dy ) = C − 1 e − V ( y ) / ε q ( y )(1 − q ( y )) dy • Probability current of reactive trajectories j R ( y ) = C − 1 e − V ( y ) / ε r q ( y ) • Committor function (capacitor): q ( y ) = E y ( τ B < τ A ) Related to Bovier’s potential theoretic approach, but exact (no small parameter)! Can be generalized to non-reversible processes. Monday, October 8, 12

Some low D examples 4 2 1.5 2 1 0 0.5 2 0 − 2 − 0.5 1.5 − 4 − 1 0 1 1 0.5 0 − 0.5 − 1 0 1 2.5 2 1.5 1 0.5 12 0 2.5 ! 0.5 10 ! 1 2 ! 1.5 8 1.5 ! 1.5 ! 1 ! 0.5 0 0.5 1 1.5 6 1 2.5 0.5 2 4 1.5 0 2 1 − 0.5 0.5 0 0 − 1 ! 0.5 − 2 − 1.5 ! 1 ! 1.5 − 1.5 − 1 − 0.5 0 0.5 1 1.5 ! 1.5 ! 1 ! 0.5 0 0.5 1 1.5 Monday, October 8, 12

Transition Path Theory for MJP • Generator = transition rate matrix i, j ∈ S = { 1 , 2 , . . . , N } L i,j • Microscopic reversibility (detailed balance) µ i L i,j = µ j L j,i • Probability distribution of reactive trajectories µ R i = µ i q i (1 − q i ) • Probability current of reactive trajectories Generator of loop erased reactive paths f R L R i,j = µ i L i,j ( q j − q i ) + i,j = L i,j ( q j − q i ) + • Committor function: q i = E i ( τ B < τ A ) Can again be generalized to non-reversible processes. Monday, October 8, 12

Transition Path Theory for MJP Committor Effective current 1 A 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 B Example of a maze (not metastable!) Monday, October 8, 12

Reorganization of LJ cluster after self-assembly with Masha Cameron • Double funnel landscape - ground state is not accessible directly by self-assembly and requires dynamical reorganization; • Can be described by a MJP using the network calculated by Wales; icosahedron • ~1e4 local minima of potential; • ~1e4 saddle points between these minima; • Network of orbits (minimum energy paths); • Transition matrix ; L i,j = ν e − ∆ V i,j / ε octahedron • Spectral analysis difficult (large network, many small eigenvalues). • Can be analyzed by TPT. • LDT only applicable at extremely low temperatures. At small temperature, systems remains strongly metastable, but pathway and rate of transition are different than those predicted by LDT (entropic effects). Monday, October 8, 12

Reorganization of LJ cluster after self-assembly with Masha Cameron T= 0.05 342 - 354 Percentage of pathways 958 - 1 3223 - 354 958 - 607 396 - 5162 The highest potential barrier Percentage of pathways 3223 - 354 T= 0.15 3886 - 354 351 - 354 3184 - 2831 396 - 1502 1208 - 3299 342 -354 248 - 3299 958 - 1 355 - 354 4950 - 2933 958 - 607 The highest potential barrier Monday, October 8, 12

Reorganization of LJ cluster after self-assembly with Masha Cameron T= 0.05 Number of transition pathways increase with T They become shorter and go thru higher barriers T= 0.15 Monday, October 8, 12

Reorganization of LJ cluster after self-assembly with Masha Cameron T= 0.05 Number of transition pathways increase with T They become shorter and go thru higher barriers T= 0.15 Monday, October 8, 12

Reorganization of LJ cluster after self-assembly with Masha Cameron T= 0.05 T= 0.15 Monday, October 8, 12

Conclusions • TPT can be used to focus and analyze a specific `reaction’. • Gives rate (mean frequency of transition) and mechanism via analysis of current. • No small parameter needed - reduces to LDT or Bovier’s approach in right limits, but can be used outside the range of applicability of these asymptotic theories. • Can be and has been used in many other examples. Monday, October 8, 12

Some other applications ▷ ¡ Thermally induced magnetization reversal in submicron ferromagnetic elements with Weinan E and Weiqing Ren a) 1 S 2 S 3 b) C 3,4 V 1 V 1 5,6 3,4 S 1 S 4 V 2 V 2 3 Practical side of LDT - Dynamics 2 V 1 V 1 7,8 1,2 V 2 V 2 1 4 can be reduced to a Markov jump m 2 C 5,6 0 C 1,2 process on energy map, whose V 2 V 2 8 5 V 1 V 1 9,10 15,16 nodes are the energy minima and V 2 V 2 6 7 S 8 S 5 whose edges are the minimum V 1 V 1 11,12 13,14 C 7,8 energy paths. S 6 S 7 � 1 � 1 0 1 m 1 ▷ ¡ Hydrophobic collapse of a polymeric chain by dewetting transition with Tommy Miller and David Chandler Rate limiting step is entropic - creation of a water bubble Monday, October 8, 12