7OyFEb-lecture8 - Symplectic Integration Introduction...

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Symplectic Integration Introduction Realistic Objectives for Molecular Dynamics Simulations In general, the aim of a MD simulation is to identify qualitative and statistical properties of molecular motions rather than to reproduce a precise trajectory. Large system size and small time steps lead to compounding of roundo ff error. Chaotic dynamics cause nearby trajectories to diverge rapidly. Initial conditions are poorly known, even when experimental structural data is available. Model misspecification: Numerous approximations: Newtonian mechanics, solvent models, periodic BCs. Force field parameters are estimated. Jay Taylor (ASU) APM 530 - Lecture 8 Fall 2010 1 / 104
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Symplectic Integration Introduction Chaotic Dynamics Local instabilities in MD simulations lead to rapid but bounded divergence of trajectories. Figures show the RMSD between simulations of a water tetramer. Divergence is initially exponential. Saturation occurs within 10 ps. The timestep has little e ff ect. From Schlick (2006). Jay Taylor (ASU) APM 530 - Lecture 8 Fall 2010 2 / 104
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Symplectic Integration Introduction Implications for Numerical Analysis The choice of the numerical integration scheme used in a MD simulation is influenced by several considerations: Simulation of large macromolecules on nanosecond or longer timescales imposes severe demands on time and memory. Stability is at least as important as order of accuracy. Geometric invariants of the system dynamics can sometimes be exploited. Flexible implementation for use with constrained or stochastic dynamics, or with multiple-timestep methods. Jay Taylor (ASU) APM 530 - Lecture 8 Fall 2010 3 / 104
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Symplectic Integration Introduction The St¨ ormer-Verlet Algorithm One of the most popular integrators used in MD is the St¨ ormer-Verlet algorithm (position Verlet): X n +1 / 2 = X n + t 2 V n V n +1 = V n + F ( X n +1 / 2 ) X n +1 = X n +1 / 2 + t 2 V n +1 The position Verlet algorithm has the following properties: explicit; second-order accurate; symplectic and time-reversible. Jay Taylor (ASU) APM 530 - Lecture 8 Fall 2010 4 / 104
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Symplectic Integration Hamiltonian Mechanics Hamiltonian Form of Newtonian Mechanics Recall that Newton’s equation of motion can be expressed as a Hamiltonian system: ˙ p = H q , ˙ q = H p where q is the coordinate vector; p = M ˙ q is the momentum vector; the Hamiltonian H gives the total energy of the system: H ( p , q ) = 1 2 p T M 1 p + U ( q ) . Jay Taylor (ASU) APM 530 - Lecture 8 Fall 2010 5 / 104
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Symplectic Integration Hamiltonian Mechanics Hamiltonian Systems More generally, we say that a system of ODE’s on an open domain R d × R d is an autonomous Hamiltonian system if there is a smooth function H : R such that ˙ p = H q , ˙ q = H p . Here d is the degrees of freedom of the system.
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