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

Venue: | Physica D |

Citations: | 18 - 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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Ten Lectures on Wavelets
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Citation Context ...ff s ) (10) and for any smooth (C 1 ) function w we have hwi ff (t) \Gamma w(t) = O(ff s ) (11) where s is a fixed integer with s AE 1. One could, for example, chose for ae the Meyer scaling function =-=[5]-=-. Note that, in the frequency domain, the smoothing operator (9) corresponds to a low pass filter with cut-off frequency ! c = O(1=ff). From now on we will always identify hwi ff with a smooth functio... |

972 |
Tildesley, Computer Simulation of Liquids
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Citation Context ...nsity function ae ens (q; p) := exp (\GammaH (q; p)=ffi) R R exp (\GammaH (q; p)=ffi) dqdp : Smoothed Dynamics 11 This is always true for systems with n very large; n the number of degrees of freedom =-=[1]-=-. Note that identifying microcanonical and macrocanonical ensemble averages is common practice in molecular dynamics [1]. Under the assumption that a given Hamiltonian system is ergodic, equipartition... |

586 | Introduction to Mechanics and Symmetry
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(Show Context)
Citation Context ...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. In the context of molecular dynamics, the potential g(q) T Kg(q)=(2ffl 2 ) stands for covalent bond stretching and bond-ang... |

414 |
An Introduction to Differentiable Manifolds and Riemannian Geometry
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Citation Context ...(q) T = 0, B(q) = b q (q), and the composed matrix [G(q) T B(q) T ] is invertible and well conditioned. The existence of such a coordinate system follows, at least locally, from the Frobenius Theorem =-=[2]. The-=- corresponding conjugate momenta are given by [G(q) T B(q) T ] " p 1 p 2 # = p which results in the Hamiltonian H(q; p) = p T 1 GM \Gamma1 G T p 1 2 + p T 2 BM \Gamma1 B T p 2 2 + V + 1 ffl 2 q T... |

403 | Numerical integration of cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes
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Citation Context ...(Q) = 0; G(Q)M \Gamma1 P = 0 g (14) provided that the matrix (3) is invertible as assumed earlier. This approximation has been used, for example, in MD simulations to remove the bond stretching modes =-=[22]-=-, [26]. The solutions on M 0 are now smooth. However, the approximation (8) introduces, in general, an error of order O(ffi) over bounded time intervals (see Section 6). While, for example, this error... |

216 |
Geometric singular perturbation theory for ordinary differential equations
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Citation Context ...is related to the high frequency modes in the Fourier spectrum of the solutions. Smoothed Dynamics 4 Differential equations of the form (1) fall into the class of singularly perturbed systems of type =-=[6]-=- d dt z = 1 ffl f(z; ffl) (8) Solutions of (8) satisfy, in general, jjz(t)jj = O(1) and jj d dt z(t)jj = O(ffl \Gamma1 ) ; i.e., they are bounded but vary rapidly in t. Thus the step-size of a numeric... |

189 |
Averaging Methods in Nonlinear Dynamical Systems
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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 ... |

118 |
Numerical Hamiltonian Problems
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- 1994
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Citation Context ... = 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 shown in [18], the numerical solutions can asymptotically be con... |

108 |
A theoretical perspective of dynamics, structure and thermodynamics
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Citation Context ...namics 26 the fastest degrees of motion. As well known from other mean force field approaches (for example, Brownian motion), fluctuations play an important role in reproducing 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... |

59 |
Dynamics of Proteins and Nucleic Acids
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- 1987
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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. ... |

51 |
Computer "Experiments" on Classical Fluids
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- 1967
(Show Context)
Citation Context ...gnificant amount of computational work for the numerical integration over time intervals of order O(1). For example, the lengths of a molecular dynamics simulation with an explicit method like Verlet =-=[28]-=- is for that reason restricted to a few tens of picoseconds up to a few nanoseconds, depending on the size of the problem [14]. This means that the time scale of the process that can be simulated is l... |

50 | Symplectic numerical integrators in constrained Hamiltonian systems
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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... |

50 | Symplectic integration of constrained Hamiltonian systems
- Leimkuhler, Reich
- 1994
(Show Context)
Citation Context ...his 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 the Hamiltonian system (32) with ... |

45 |
Statistical Mechanics, Pergamon
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(Show Context)
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 ... |

42 |
Motion under a strong constraining force
- Rubin, Ungar
- 1957
(Show Context)
Citation Context ...(ffi ffl 2 ) and hpi p ffl (t) \Gamma P (t) = O(ffi ffl 2 ) over bounded intervals of time. The approximation of (1) by a constrained Hamiltonian systems has been considered before (see, for example, =-=[21]-=-,[27]). In a naive approach, one would introduce the new variables:= 1 ffl 2 Kg(q) and rewrite (1) as d dt q = M \Gamma1 p d dt p = \GammarV (q) \Gamma G(q) T ffl 2 K \Gamma1s= g(q) (12) In the limit ... |

37 |
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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... |

31 |
Classical statistical mechanics of constraints: A theorem and application to polymers
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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... |

18 |
Effect of constraints on the dynamics of macromolecules
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Citation Context ...0; G(Q)M \Gamma1 P = 0 g (14) provided that the matrix (3) is invertible as assumed earlier. This approximation has been used, for example, in MD simulations to remove the bond stretching modes [22], =-=[26]-=-. The solutions on M 0 are now smooth. However, the approximation (8) introduces, in general, an error of order O(ffi) over bounded time intervals (see Section 6). While, for example, this error turns... |

11 |
Momentum conserving symplectic integrators
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- 1994
(Show Context)
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... |

8 |
Problems with different time scales for ordinary differential equations
- Kreiss
- 1979
(Show Context)
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... |

8 |
Brownian dynamics study of a polymer chain of linked rigid bodies
- Pear, Weiner
- 1979
(Show Context)
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 ... |

6 |
Elastic molecular dynamics with flexible constraints
- Zhou, Reich, et al.
(Show Context)
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... |

6 |
Invariant manifolds and the initialization problem for some atmospheric equations
- Kopell
- 1985
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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... |

5 |
Symplectic integrators and the conservation of angular momentum
- Zhang, Skeel
- 1995
(Show Context)
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 ...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... |

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Conformational free energy from thermodynamic integration simulations
- Kuczera
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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 ... |

3 |
Integration methods for molecular dynamics, to appear
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- 1995
(Show Context)
Citation Context ...e-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 where the st... |

3 |
Invariant manifolds of numerical integration schemes applied to stiff systems of singular perturbation type -- Part I: RK-methods, Research Report 92-14, Seminar fur Angewandte Mathemathik
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- 1995
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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... |