Mathematical analysis of temperature accelerated dynamics David - PowerPoint PPT Presentation

Mathematical analysis of temperature accelerated dynamics David Aristoff (joint work with T. Leli` evre) School of Mathematics University of Minnesota August 12, 2013 D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 1 / 18

Introduction Molecular dynamics We consider overdamped Langevin dynamics : � 2 β − 1 dW t , dX t = −∇ V ( X t ) dt + (1) used to model the evolution of the position vector X t of N particles in an energy landscape defined by the potential energy V : R 3 N → R . The dynamics (1) are obtained as a limit as m → 0 or γ → ∞ of the Langevin dynamics dX t = m − 1 P t dt (2) dP t = −∇ V ( X t ) dt − γ m − 1 P t dt + � 2 γβ − 1 dW t . This energy landscape typically has many metastable states, corresponding to basins of attraction of the gradient dynamics dy / dt = −∇ V ( y ). In applications it is of interest how X t moves between these basins – this is the so-called metastable dynamics . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 2 / 18

Introduction Definition. We write D for a generic basin of attraction of dy / dt = −∇ V ( y ). 0 1 b Figure : The basin D = [0 , b ] of attraction of 1 w.r.t. dy / dt = −∇ V ( y ). D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 3 / 18

Introduction Let S : R 3 N → N be a function which labels the basins of attraction of dy / dt = −∇ V ( y ). So each basin D has the form D = S − 1 ( i ), i ∈ N . Figure : A trajectory of X t and S ( X t ), with two basins labeled 0 and 1. The metastable dynamics is then S ( X t ) t ≥ 0 . Problem: Efficiently generate approximations ˆ S ( t ) t ≥ 0 of S ( X t ) t ≥ 0 . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 4 / 18

Introduction The quasistationary distribution With metastable dynamics, the time scale to reach “local equilibrium” in a basin is much smaller than the time scale to exit the basin. The notion of local equilibrium can be formalized using the quasistationary distribution (QSD) . Definition. The superscript in X µ t means X t has initial distribution given by µ : X 0 ∼ µ . Definition. The QSD ν in D satisfies the following: t →∞ P ( X µ t ∈ A | X µ ν ( A ) = lim s ∈ D ∀ s ∈ [0 , t ]) for any probability measure µ supported in D and any measurable A ⊂ D . Given that X t remains in D , the distribution of X t converges exponentially fast to the QSD in D , no matter the initial distribution of X 0 . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 5 / 18

Generating exit events from a basin If X 0 is distributed according to the QSD in D , then the first exit time of X t from D is exponentially distributed and independent of the exit position: Theorem. Define τ = inf { t > 0 : X ν t / ∈ D } . Then P ( τ > t ) = e − λ t and τ, X τ are independent . The theorem can be used to tackle the following: Subproblem. Efficiently generate an exit event of X ν t from a given basin D . The idea is to iterate the subproblem solution to generate the metastable dynamics approximation ˆ S ( t ) t ≥ 0 . The assumption X 0 ∼ ν should not be drastic because the time scale for X t to reach the QSD in D is much smaller than the time scale for X t to exit D . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 6 / 18

Generating exit events from a basin Generating exit events Definition. Let { ∂ D i } i =1 , 2 ,..., n be a measurable partition of ∂ D . ∂ D 4 x 4 ∂ D 1 ∂ D 3 x 3 x 1 x 0 x 2 ∂ D 2 Figure : The basin D of attraction of x 0 . Each ∂ D i is a neighborhood of a saddle point, x i ( i ≥ 1), of V in ∂ D . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 7 / 18

Generating exit events from a basin Generating exit events Definition. Let { ∂ D i } i =1 , 2 ,..., n be a measurable partition of ∂ D . Definition. Define r.v.’s τ and I by τ = inf { t > 0 : X ν X ν t / ∈ D } and I = i ⇔ τ ∈ ∂ D i . Define λ and p i by λ − 1 := E [ τ ] and p i := P ( I = i ) . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 7 / 18

Generating exit events from a basin Generating exit events Definition. Let { ∂ D i } i =1 , 2 ,..., n be a measurable partition of ∂ D . Definition. Define r.v.’s τ and I by τ = inf { t > 0 : X ν X ν t / ∈ D } and I = i ⇔ τ ∈ ∂ D i . Define λ and p i by λ − 1 := E [ τ ] and p i := P ( I = i ) . An exit event from D , starting at the QSD, is represented by the pair ( τ, I ), with τ the exit time and I the exit pathway. D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 7 / 18

Generating exit events from a basin Generating exit events Definition. Let { ∂ D i } i =1 , 2 ,..., n be a measurable partition of ∂ D . Definition. Define r.v.’s τ and I by τ = inf { t > 0 : X ν X ν t / ∈ D } and I = i ⇔ τ ∈ ∂ D i . Define λ and p i by λ − 1 := E [ τ ] and p i := P ( I = i ) . An exit event from D , starting at the QSD, is represented by the pair ( τ, I ), with τ the exit time and I the exit pathway. Starting at the QSD, the expected time to exit D is λ − 1 and the probability to exit through ∂ D i is p i . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 7 / 18

Generating exit events from a basin The last theorem leads to the following: Theorem. Let { T i } i =1 , 2 ,..., n be independent r.v.’s with P ( T i > t ) = e − λ p i t . Then � � 1 ≤ i ≤ n T i , arg min min 1 ≤ i ≤ n T i ∼ ( τ, I ) . The preceding applies to any dynamics whenever the QSD in D exists. The theorem can be used to sample exit events from D provided that estimates of the parameters λ p i are available. From now on we consider only overdamped Langevin dynamics and assume: Assumption. V is a Morse function, D is the basin of attraction of x 0 w.r.t. dy / dt = −∇ V ( y ), and each ∂ D i is a neighborhood in ∂ D of a single (index one) saddle point, x i , of V on ∂ D . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 8 / 18

Generating exit events from a basin The Arrhenius law Under the preceding assumption λ p i can be estimated by the so-called Arrhenius law: The Arrhenius law λ p i ≈ η i e − β ( V ( x i ) − V ( x 0 )) for large β (3) Here η i is a known function of the eigenvalues of the Hessian matrix of V at the saddle point x i and minimum x 0 . In particular η i is β -independent. The Arrhenius law is assumed valid when β ( V ( x i ) − V ( x 0 )) >> 1 . If the locations of the saddle points are known a priori, the theorem along with equation (3) can be used to sample exit events from D . D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 9 / 18

exit events in TAD Temperature accelerated dynamics (TAD) Notation. Let β hi and β lo be a high and low temperature. We use superscripts hi and lo to denote objects at β hi and β lo . (E.g. ν lo is the QSD in D at temperature β lo .) We recall again: Subproblem. Generate an exit event of X ν lo from D at temperature β lo . t In TAD 1 , the exit event at β lo is generated by simulating multiple exit times and pathways at β hi , then extrapolating what would have happened at β lo . In TAD, the saddle point locations are not assumed to be known a priori, and it is not necessary that all the saddle points be found. Furthermore in TAD it is not required that any of the η i be calculated. 1 proposed in A.F. Voter and M.R. Sørensen , J. Chem. Phys. 112 (2000). D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 10 / 18

exit events in TAD Exit Algorithm (for generating an exit event of X ν lo from D ). t Let N = 1, T stop = ∞ and iterate the following steps: 1. Let X ( N ) be an sample of ν hi , the QSD in D at temperature β hi . 0 at temperature β hi until the first time, τ ( N ) , at which it exits D . 2. Evolve X ( N ) t 3. Now X ( N ) τ ( N ) ∈ ∂ D i for some i ∈ { 1 , . . . , n } . If X ( k ) τ ( k ) / ∈ ∂ D i ∀ 1 ≤ k < N , let � τ (1) + . . . + τ ( N ) � e − ( β hi − β lo )( V ( x i ) − V ( x 0 )) T lo = extrapolated low temp exit time i T lo min = min { T lo min , T lo I lo min = i ⇔ T lo min = T lo i } , update fastest low temp exit event i 1 ≤ i ≤ n e − ( β hi − β lo )( V ( x i ) − V ( x 0 )) T stop = T lo min / min update stopping time The minimum above can be replaced with any a priori lower bound. 4. If τ (1) + . . . + τ ( N ) < T stop , let N = N + 1 and return to Step 1. Otherwise store the exit event ( T lo min , I lo min ). D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 11 / 18

exit events in TAD Remarks: At low temperatures the QSD in D can be efficiently sampled 2 . By construction, the fastest extrapolated low temperature exit event will be found by time T stop . The Exit Algorithm is expected to be accurate when the Arrhenius law is valid: 1 ≤ i ≤ n β hi ( V ( x i ) − V ( x 0 )) ≫ 1 . min (4) The Exit Algorithm will be efficient when also β hi ≪ β lo . (5) To see that latter, recall the stopping time T stop is updated via 1 ≤ i ≤ n e − ( β hi − β lo )( V ( x i ) − V ( x 0 )) , T stop = T lo min / min (6) and notice from (4) and (5) the denominator in the RHS of (6) is ≫ 1. 2 See for example G. Simpson and M. Luskin , M2AM (to appear) arxiv:1204.0819. D. Aristoff (UMN) CEMRACS 2013 August 12, 2013 12 / 18