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Atomistic simulations of rare events Atomistic simulations of rare events using the using the gentlest ascent gentlest ascent dynamics dynamics Amit Samanta Rare events Amit Samanta GAD Ad-atom Applied and Computational Mathematics, diffusion Princeton University, Princeton, USA. Quasi- Newtonian Joint work with Conclusions Prof. Weinan E (Princeton), Xiang Zhou (Brown) 28 March 2012 Max Planck Institute for the Physics of Complex Systems Dresden, Germany

A simple fact: Nature works on disparate time scales A direct manifestation: In nature, dynamics often proceed in the Atomistic simulations of form of rare events . rare events using the gentlest Focus : exploring a smooth energy surface for local minima, 1 ascent dynamics saddles starting from one initial point Amit Samanta Goal : set of dynamical equations that converge to saddle points 2 Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions 1 J. P. Doye and D. J. Wales, J. Chem. Phys. (2002)

A simple fact: Nature works on disparate time scales A direct manifestation: In nature, dynamics often proceed in the Atomistic simulations of form of rare events . rare events using the gentlest Focus : exploring a smooth energy surface for local minima, 1 ascent dynamics saddles starting from one initial point Amit Samanta Goal : set of dynamical equations that converge to saddle points 2 Rare events Challenge : 3 GAD problem nonlocal in nature but only local information available Ad-atom (1-form Fokker Planck, Witten Laplacian, etc not useful) diffusion follow minimum eigenmode close to saddle point - but can easily Quasi- become unstable (degenerate eigenvalues) Newtonian how to move out of basin of attraction - need better sampling Conclusions techniques no global convergence 1 J. P. Doye and D. J. Wales, J. Chem. Phys. (2002)

A simple fact: Nature works on disparate time scales A direct manifestation: In nature, dynamics often proceed in the Atomistic simulations of form of rare events . rare events using the gentlest Focus : exploring a smooth energy surface for local minima, 1 ascent dynamics saddles starting from one initial point Amit Samanta Goal : set of dynamical equations that converge to saddle points 2 Rare events Challenge : 3 GAD problem nonlocal in nature but only local information available Ad-atom (1-form Fokker Planck, Witten Laplacian, etc not useful) diffusion follow minimum eigenmode close to saddle point - but can easily Quasi- become unstable (degenerate eigenvalues) Newtonian how to move out of basin of attraction - need better sampling Conclusions techniques no global convergence System dimensions : Lennard-Jones cluster ( LJ n ) 4 n = 4 atoms, 6 saddle points 1 1 J. P. Doye and D. J. Wales, J. Chem. Phys. (2002)

A simple fact: Nature works on disparate time scales A direct manifestation: In nature, dynamics often proceed in the Atomistic simulations of form of rare events . rare events using the gentlest Focus : exploring a smooth energy surface for local minima, 1 ascent dynamics saddles starting from one initial point Amit Samanta Goal : set of dynamical equations that converge to saddle points 2 Rare events Challenge : 3 GAD problem nonlocal in nature but only local information available Ad-atom (1-form Fokker Planck, Witten Laplacian, etc not useful) diffusion follow minimum eigenmode close to saddle point - but can easily Quasi- become unstable (degenerate eigenvalues) Newtonian how to move out of basin of attraction - need better sampling Conclusions techniques no global convergence System dimensions : Lennard-Jones cluster ( LJ n ) 4 n = 4 atoms, 6 saddle points 1 n = 10 atoms, > 160 , 000 saddle points 1 1 J. P. Doye and D. J. Wales, J. Chem. Phys. (2002)

Many transition events : Nanoindentation Atomistic simulations of rare events Made of very hard indenter material – Tungsten, using the Diamond, Boron nitride, R Aluminium nitride indenter gentlest ascent h Material whose dynamics hardness is to be substrate substrate measured Amit Samanta Measured quantities:- indentation load (P), indentation depth (h) Important Parameters:- indentation rate, indenter tip radius (R), elastic modulus, temperature Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions W. Gerberich and W. Mook, Nat. Mat. (2005)

Many transition events : Nanoindentation Atomistic simulations of rare events Made of very hard indenter material – Tungsten, using the Diamond, Boron nitride, R Aluminium nitride indenter gentlest ascent h Material whose dynamics hardness is to be substrate substrate measured Amit Samanta Measured quantities:- indentation load (P), indentation depth (h) Important Parameters:- indentation rate, indenter tip radius (R), elastic modulus, temperature Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions W. Gerberich and W. Mook, Nat. Mat. (2005)

Gentlest Ascent Dynamics H = ∇ 2 V ( x ) Idea: F = −∇ V ( x ) , Atomistic move along direction n , minimize along other dimensions simulations of rare events ˜ F = F ⊥ − F � , F � = ( F , n ) n , F ⊥ = F − F � using the gentlest ascent dynamics Amit Samanta Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions W. E and X. Zhou, Nonlinearity (2011) A. Samanta and W. E, J Chem Phys (2012)

Gentlest Ascent Dynamics H = ∇ 2 V ( x ) Idea: F = −∇ V ( x ) , Atomistic move along direction n , minimize along other dimensions simulations of rare events ˜ F = F ⊥ − F � , F � = ( F , n ) n , F ⊥ = F − F � using the gentlest Equations of motion: ascent dynamics x = F ( x ) − 2 ( F , n ) n ˙ Amit Samanta γ ˙ n = − H n + ( n , H n ) n Rare events Lemma :The stable fixed points of this dynamics are the index-1 saddle GAD points of V . (Local minima of V are saddle points of GAD) Ad-atom diffusion Quasi- Newtonian Conclusions W. E and X. Zhou, Nonlinearity (2011) A. Samanta and W. E, J Chem Phys (2012)

Gentlest Ascent Dynamics H = ∇ 2 V ( x ) Idea: F = −∇ V ( x ) , Atomistic move along direction n , minimize along other dimensions simulations of rare events ˜ F = F ⊥ − F � , F � = ( F , n ) n , F ⊥ = F − F � using the gentlest Equations of motion: ascent dynamics x = F ( x ) − 2 ( F , n ) n ˙ Amit Samanta γ ˙ n = − H n + ( n , H n ) n Rare events Lemma :The stable fixed points of this dynamics are the index-1 saddle GAD points of V . (Local minima of V are saddle points of GAD) Ad-atom diffusion Quasi- Newtonian Conclusions shallow wells change in stability simple, amendable to mathematical analysis, can be extended to higher index saddles, non-gradient systems, efficient numerical schemes W. E and X. Zhou, Nonlinearity (2011) A. Samanta and W. E, J Chem Phys (2012)

Convergence to One Saddle Point Atomistic simulations of x = F ( x ) − 2 ( F , n ) n ˙ rare events using the n = − H n + ( n , H n ) n gentlest ˙ ascent dynamics 2-dimensional example : V ( x , y ) = sin ( π x ) sin ( π y ) Amit Samanta Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions

Convergence to One Saddle Point Atomistic simulations of x = F ( x ) − 2 ( F , n ) n ˙ rare events using the n = − H n + ( n , H n ) n gentlest ˙ ascent dynamics 2-dimensional example : V ( x , y ) = sin ( π x ) sin ( π y ) Amit Samanta Rare events 2 GAD 0.8 1.5 Ad-atom randomly initialized 0.6 diffusion 1 0.4 direction vector 0.5 0.2 Quasi- Newtonian 0 0 time step important −0.2 Conclusions −0.5 −0.4 guess direction determines −1 −0.6 convergence −1.5 −0.8 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Configuration Space Density Distribution Atomistic x = F ( x ) − 2 ( F , n ) n + σ ˙ ˙ w simulations of rare events using the γ ˙ n = − H n + ( n , H n ) n gentlest ascent dynamics 2-dimensional example : V ( x , y ) = sin ( π x ) sin ( π y ) Amit Samanta Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions

Configuration Space Density Distribution Atomistic x = F ( x ) − 2 ( F , n ) n + σ ˙ ˙ w simulations of rare events using the γ ˙ n = − H n + ( n , H n ) n gentlest ascent dynamics 2-dimensional example : V ( x , y ) = sin ( π x ) sin ( π y ) Amit Samanta 4 4 0.8 0.8 Rare events 3 3 0.6 0.6 GAD 2 2 0.4 0.4 Ad-atom 1 0.2 1 0.2 diffusion 0 0 0 0 Quasi- −1 −0.2 −1 −0.2 Newtonian −0.4 −2 −0.4 −2 Conclusions −0.6 −0.6 −3 −3 −0.8 −0.8 −4 −4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 randomly initialized direction vector System spends considerable amount of time near saddle points.

Variants: MD-GAD Atomistic x = v ˙ simulations of rare events v = F − 2( F , n ) n ˙ using the gentlest ascent γ ˙ n = − H n + ( n , H n ) n dynamics incorporate thermostat, barostat Amit Samanta Rare events GAD Ad-atom diffusion Quasi- Newtonian Conclusions A. Samanta and W. E, J Chem Phys (2012)