## Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems (1995)

Venue: | Physica D |

Citations: | 17 - 8 self |

### BibTeX

@ARTICLE{Reich95smootheddynamics,

author = {Sebastian Reich},

title = {Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems},

journal = {Physica D},

year = {1995},

volume = {89},

pages = {28--42}

}

### OpenURL

### Abstract

We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step-size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion...

### Citations

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Citation Context ..., since P T 1 G(Q)M \Gamma1 G(Q) T P 1 = O(ffl 4 ) ; the equations (33) are equivalent to (31) up to terms of order O(ffi ffl 2 ). Standard perturbation results for differential equations (see, e.g., =-=[23]-=-) imply that the same is true for the solutions over bounded intervals of time. 2 Corollary 1. An order O(ffl 2 ) approximation of the smoothed dynamics of (1) over bounded intervals of time is given ... |

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1994], Numerical Hamiltonian Problems
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Citation Context ...1=2 = P k\Gamma1=2 \Gamma \Deltat [rV (Q k ) + G(Q k ) Tsk ] 0 = g(Q k+1 ) (38) of the Verlet scheme [28] which requires now the solution of an implicit equation in the variablesk . It has been shown =-=[12]-=- that this scheme preserves the symplectic structure [13],[24] of Hamiltonian flows, is time-reversible, and conserves first integrals related to symmetries of the system [29],[19]. Furthermore, as sh... |

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Citation Context ...nn constant. (6) is typically satisfied for Hamiltonian systems of type (1) with n large; i.e. n AE 1. Hamiltonian systems of type (1) arise typically in the context of molecular dynamics simulations =-=[14]-=- (which provides the main motivation of this paper) and in the spatial discretization of Hamiltonian (hyperbolic) PDEs [13] like, for example, the Sine-Gordon equation by spectral or related methods. ... |

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Citation Context ... O(ff l ). Thus Z \Deltaw(t) dt = ff 2 Z \Deltaw(ff 2 ) d = ff 2 O(ff l ) : 2 Let us finally review a few results from statistical mechanics. A Hamiltonian system with Hamiltonian H is called ergodic =-=[16]-=- if the time average hAi := lim T!1 1 2T Z T \GammaT A(q(t); p(t))dt (19) of an observable A(q; p) is equal to the microcanonical (constant energy E = H(q; p)) ensemble average [16] hAi ens := Z Z ae ... |

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Citation Context ... demonstrate its properties by means of a simple numerical example. Another approach to the long-time integration of highly oscillatory Hamiltonian system has been taken by Simo and his collaborators =-=[25]-=-. They advocate the direct discretization of (1) by an implicit energy-momentum method and the usage of a large step-size. However, there do not exist rigorous stability and order of convergence resul... |

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Citation Context ... the system d dt Q 2 = B(Q)M \Gamma1 B(Q) T P 2 d dt P 2 = \Gammar Q 2 V (Q) + rQ 2 V F (Q) +rQ 2 P T 2 B(Q)M \Gamma1 B(Q) T P 2 2 (31) Remark. The potential (30) has been introduced before by Fixman =-=[7]-=- in the context of statistical mechanics. He showed that (30) has to be included into the constrained formulation (13) to make sure that, in the limit ffl ! 0, the unconstrained system (1) and the con... |

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Citation Context .... It has been shown [12] that this scheme preserves the symplectic structure [13],[24] of Hamiltonian flows, is time-reversible, and conserves first integrals related to symmetries of the system [29],=-=[19]-=-. Furthermore, as shown in [18], the numerical solutions can asymptotically be considered as the exact solution of a perturbed constrained Hamiltonian system. The same scheme can also be applied to th... |

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Citation Context ...e sure that, in the limit ffl ! 0, the unconstrained system (1) and the constrained system (13) possess the same reduced density function ae ens (q 2 ; p 2 ). Similar results can be found in [27] and =-=[17]-=-. Smoothed Dynamics 17 6 Constraint Formulations The coordinates (q 1 ; p 1 ; q 2 ; p 2 ) were only introduced for theoretical purposes. This leaves us with the task of reformulating (31) in terms of ... |

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Citation Context ...n of order O(ffl 2 ) to hq 1 i p ffl . 2 Remarks. (i) Corollary 1 implies that the smoothed dynamics of (1) cannot, in general, be approximated by the slow solutions of (1) as introduced by Kreiss in =-=[9]-=-. One can show that, up to terms of order (ffl 2 ), the slow solutions of (1) are given by the constrained equations (13) which differ from (34) by the Fixman potential (30) and thus by a term of orde... |

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Citation Context ...tic, time-reversible, and momentum conserving. The method (39) is computational expensive. An effective implementation of (39) and the discretization of (32) by less expensive methods can be found in =-=[3]-=- and [11]. Note that one could also discretize (32) by a proper modification of the energy-momentum methods proposed in [25]. Example 2. In this example we consider a three-bead-two-bond structure whe... |

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Citation Context ... In particular, solutions of (1) oscillate highly about the manifold M 0 . Thus, as we will show in Section 2, the manifold M 0 does not even satisfy the weaker assumption of normal-hyperbolicity [6],=-=[8]-=-. This leaves us with the task of finding a different approach to the long-time integration of (1). In this paper, we attempt to do so by introducing the notion of the smoothed dynamics of highly osci... |

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Citation Context ...lesk . It has been shown [12] that this scheme preserves the symplectic structure [13],[24] of Hamiltonian flows, is time-reversible, and conserves first integrals related to symmetries of the system =-=[29]-=-,[19]. Furthermore, as shown in [18], the numerical solutions can asymptotically be considered as the exact solution of a perturbed constrained Hamiltonian system. The same scheme can also be applied ... |

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Citation Context ...energy H(q 2 ; p 2 ) of the remaining variables (q 2 ; p 2 ) is a function that satisfies r q 2 H(q 2 ; p 2 ) = hr q 2 Hi (1) ens (q 2 ; p 2 ) and r p 2 H(q 2 ; p 2 ) = hr p 2 Hi (1) ens (q 2 ; p 2 ) =-=[10]-=-. Furthermore, let ae (2) ens (q 2 ; p 2 ) denote the density function corresponding to the (Hamiltonian) free energy H(q 2 ; p 2 ), then the total ensemble average hAi ens := Z Z Z Z A(q 1 ; p 1 ; q ... |

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Citation Context ...lutions on M ffl satisfy now dz=dt = O(1), time-steps of order O(1) can be used in a Smoothed Dynamics 5 numerical integrator provided that the equations are discretized by a proper (implicit) method =-=[15]-=-. However, Assumption 2 is not satisfied for singularly perturbed Hamiltonian systems. In particular, solutions of (1) oscillate highly about the manifold M 0 . Thus, as we will show in Section 2, the... |

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Citation Context ...ducing correct rate constants [4]. This requires an embedding of the mean force field into stochastic dynamics [4]. For our particular system (32), this will be discussed in a forthcoming publication =-=[20]-=-. Acknowledgements. This work was started while the author was visiting the Beckman Institute in Urbana-Champaign. We like to thank Klaus Schulten for providing a very stimulating environment and Robe... |