Multiple time-scales methods

We have shown, using the method of matched asymptotic expansions, that in the outer domain there is an adjustable variable, and in the inner region there is another such variable. The composite solution is, therefore, a function of these two variables. Exploitation of this function is the essential idea behind the Multiple Time Scale method. Interested readers should refer to Nayfeh (1973) for exposition of this technique. 

It is also interesting to note that although the local, two-body equilibrium solution is indeed a particular solution to Eq. (6.77), substitution of the equilibrium solution into Eq. (6.79) leads to nonsensical results. Therefore, the equilibrium solution is not self-consistent to leading order. There are numerous extensive treatments on the development, modifications, and treatment of the Boltzmann transport equation in gas kinetic theory (see the Further Reading section at the end of Chap. 3). More modern approaches, originally due to Bogolubov, consider the time scales of the expansion procedure more carefully. These so-called multiple time-scales methods are powerful procedures... 

E. A. Frieman, J. Math. Phys., 4, 410,1963 G. H. Su, J. Math. Phys., 5, 1273, 1964. [See this book for more discussion and examples of multiple time-scale methods appUed to reduced forms of the LiouvUle equation.]... 

Methods for Dealing with the Intrinsic Multiple Time Scale Problem in Molecular Dynamics... 

By now it should be clear that this kind of operator algebra can be a useful method for generating integrators. We show, in the following, how it can be applied to generate a wide variety of methods for treating the multiple time scale problem. 

New Sampling Methods for the Extrinsic Multiple Time Scale Problem... 

Watanabe, M., Karplus, M. Dynamics of Molecules with Internal Degrees of Freedom by Multiple Time-Step Methods. J. Chem. Phys. 99 (1995) 8063-8074 Figueirido, F., Levy, R. M., Zhou, R., Berne, B. J. Large Scale Simulation of Macromolecules in Solution Combining the Periodic Fast Multiple Method with Multiple Time Step Integrators. J. Chem. Phys. 106 (1997) 9835-9849 Derreumaux, P., Zhang, G., Schlick, T, Brooks, B.R. A Truncated Newton Minimizer Adapted for CHARMM and Biomolecular Applications. J. Comp. Chem. 15 (1994) 532-555... 

In such cases, the MEHMC method could be employed in combination with an enhanced sampling method that deforms the effective energy surface (but preserves the location of the potential minima), such as that in [29, 97]. Likewise, it may be worthwhile to explore the use of a reversible multiple-time-scale molecular dynamics propagator [103] with MEHMC to accelerate the dynamical propagation. 

Section 2.4 alluded to the possibility of expanding the methods presented in Chapter 2 to account for the presence of multiple singular perturbation parameters in a system of differential equations. This appendix is concerned with this topic, and, to this end, let us consider a multiple-time-scale (multiply perturbed) system in the standard form... 

The analysis of multiple-time-scale systems can, however, be carried out by extending the methods used for analyzing two-time-scale systems presented in Section 2.2. In analogy with two-time-scale systems, in the limiting case as e —> 0, the dimension of the state space of the system in Equations (B.l) collapses... 

Note that this method ultimately leads to a set of state-space realizations for the reduced-order models for each time scale of a multiple-time-scale system, but does not identify the slow and fast variables associated with the individual... 

Schulten238 outlined the development of a multiple-time-scale approximation (distance class algorithm) for the evaluation of nonbonded interactions, as well as the fast multipole expansion (FME). efficiency of the FME was demonstrated when the method outperformed the direct evaluation of Coulomb forces for 5000 atoms by a large margin and showed, for systems of up to 24,000 atoms, a linear dependence on atom number. 

Notwithstanding the algorithmic developments described in preceding sections, and the laudable efforts by the simulation community to achieve full exploitation of available parallel hardware, the problem identified earlier remains, in macromolecular structures at least that is, the current time and spatial scales are in many instances inappropriate for the target physical questions. Because information spanning several orders of magnitude beyond the currently accessible picosecond-nanosceond time frame is needed from MD simulations, the development of more effective multiple-time-scale MD methods is seen as crucial. 

Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotter-factorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Parrinello simulations [20] and ab initio molecular dynamics algorithms [21]. 

It can be concluded that ETD is a beneficial extension of static classification methods when online response to changes is required and the measurement devices exhibit a delayed behaviour. The DWT provides a useful framework for characterizing changes of a time series locally and on multiple time scales and could therefore beneficially be used for ETD. 

A related topic is the issue of time scales. Dynamic simulations of atomic behavior generally require time steps that are short enough to capture the vibrational modes of the system, whereas changes at the macroscopic scale usually occur over vastly longer time scales. Coupling between such widely varying time scales is a very important challenge, but it is not within the scope of this review. However, the problem of multiple-time-scale simulations will be discussed briefly in the discussion of dynamical methods. 

In an effort to lengthen the feasible simulation time scale of molecular simulations, Streett and co-workers introduced the multiple time-step method in 1978. " These authors recognized that the components of the force that vary most rapidly, and hence require small time steps for numerical resolution, are typically associated with atom pair interactions at small separations. This spatial localization is important because each of the N particles in the simulation has such an interaction with only a few, say [Pg.372]

We continue in the ensuing chapters with several tutorials tied together by the theme of how to exploit and/or treat multiple length scales and multiple time scales in simulations. In Chapter 5 Thomas Beck introduces us to real-space and multigrid methods used in computational chemistry. Real-space methods are iterative numerical techniques for solving partial differential equations on grids in coordinate space. They are used because the physical responses from many chemical systems are restricted to localized domains in space. This is a situation that real-space methods can exploit because the iterative updates of the desired functions need information in only a small area near the updated point. 

In addition to the liquid structure of pure ILs, the time scales and molecular processes involved in the solvation of polar solutes in ILs have been examined by Kim and coworkers, who established the existence of multiple time scales in the relaxation of the solute using classic MD methods. Solvation dynamics in ILs and aqueous IL solutions have also been investigated in detail by Margulis and co-workers. In particular, they have been able to identify the molecular origins of the red edge effect as arising from the dynamic heterogeneity of these liquids on time scales relevant to optical spectroscopy. 

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