SDEs in large dimension and numerical methods Part 1: Sampling the - PowerPoint PPT Presentation

Introduction Free energy Thermodynamic integration Adaptive biasing techniques SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T. Lelièvre CERMICS - Ecole des Ponts ParisTech & Matherials project-team - INRIA RICAM Winterschool, December 2016

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Motivation The aim of molecular dynamics simulations is to understand the relationships between the macroscopic properties of a molecular system and its atomistic features. In particular, one would like to evaluate numerically macroscopic quantities from models at the microscopic scale. Many applications in various fields: biology, physics, chemistry, materials science. Various models: discrete state space (kinetic Monte Carlo, Markov State Model) or continuous state space (Langevin). The basic ingredient: a potential V which associates to a configuration ( x 1 , ..., x N ) = x ∈ R 3 N atom an energy V ( x 1 , ..., x N atom ) . The dimension d = 3 N atom is large (a few hundred thousand to millions).

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Empirical force field Typically, V is a sum of potentials modelling interaction between two particles, three particles and four particles: � � � V = V 1 ( x i , x j ) + V 2 ( x i , x j , x k ) + V 3 ( x i , x j , x k , x l ) . i < j i < j < k i < j < k < l 5 ǫ = 1 , σ = 1 For example, 4 V 1 ( x i , x j ) = V LJ ( | x i − x j | ) 3 V LJ ( r ) where 2 � 12 − � σ �� σ � 6 � V LJ ( r ) = 4 ǫ is 1 r r the Lennard-Jones potential. 0 -1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 r

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion + thermostat: Langevin dynamics: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt − γ M − 1 P t dt + � 2 γβ − 1 d W t where γ > 0. Langevin dynamics is ergodic wrt � � − β p t M − 1 p µ ( d x ) ⊗ Z − 1 exp d p with p 2 d µ = Z − 1 exp ( − β V ( x )) d x � where Z = exp ( − β V ( x )) d x is the partition function and β = ( k B T ) − 1 is proportional to the inverse of the temperature.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion + thermostat: Langevin dynamics: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt − γ M − 1 P t dt + � 2 γβ − 1 d W t where γ > 0. Langevin dynamics is ergodic wrt � � − β p t M − 1 p µ ( d x ) ⊗ Z − 1 exp d p with p 2 d µ = Z − 1 exp ( − β V ( x )) d x � where Z = exp ( − β V ( x )) d x is the partition function and β = ( k B T ) − 1 is proportional to the inverse of the temperature. In the following, we focus on the overdamped Langevin (or gradient) dynamics � 2 β − 1 d W t , d X t = −∇ V ( X t ) dt + which is also ergodic wrt µ .

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Macroscopic quantities of interest These dynamics are used to compute macroscopic quantities: (i) Thermodynamic quantities (averages wrt µ of some observables): stress, heat capacity, free energy,... � T � R d ϕ ( x ) µ ( d x ) ≃ 1 E µ ( ϕ ( X )) = ϕ ( X t ) dt . T 0 (ii) Dynamical quantities (averages over trajectories): diffusion coefficients, viscosity, transition rates,... M E ( F (( X t ) t ≥ 0 )) ≃ 1 � F (( X m t ) t ≥ 0 ) . M m = 1 Difficulties: (i) high-dimensional problem ( N ≫ 1); (ii) X t is a metastable process and µ is a multimodal measure.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Metastability: energetic and entropic barriers A two-dimensional schematic picture 2 2.0 1.5 1.5 1 1.0 0.5 0.5 0 0.0 x y -0.5 -0.5 -1 -1.0 -1.5 -1.5 -2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0 2000 4000 6000 8000 10000 x Iterations 3 3 2 2 1 1 0 0 x y -1 -1 -2 -2 -3 -3 -6 -4 -2 0 2 4 6 0 2000 4000 6000 8000 10000 x Iterations − → • Slow convergence of trajectorial averages • Transitions between metastable states are rare events

Introduction Free energy Thermodynamic integration Adaptive biasing techniques A toy model for solvation Influence of the solvation on a dimer conformation [Dellago, Geissler] . �� Compact state. Stretched state. The particles interact through a pair potential: truncated LJ for all particles except the two monomers (black particles) which interact through a double-well potential. A slow variable is the distance between the two monomers.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques A toy example in material sciences The 7 atoms Lennard Jones cluster in 2D. (a) C 0 , V = − 12 . 53 (b) C 1 , V = − 11 . 50 (c) C 2 , V = − 11 . 48 (d) C 3 , V = − 11 . 40 Figure: Low energy conformations of the Lennard-Jones cluster. − → simulation

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Simulations of biological systems Unbinding of a ligand from a protein (Diaminopyridine-HSP90, Courtesy of SANOFI) Elementary time-step for the molecular dynamics = 10 − 15 s Dissociation time = 0 . 5 s Challenge: bridge the gap between timescales

SDEs in large dimension and numerical methods Part 1: Sampling the - PowerPoint PPT Presentation

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