Multiple time-scale simulations

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. 

Procacci P, March M and Martyna G J 1998 Electrostatic calculations and multiple time scales in molecular dynamics simulation of flexible molecular systems J. Chem. Phys. 108 8799-803... 

Since many systems of interest in chemistry have intrinsic multiple time scales it is important to use integrators that deal efficiently with the multiple time scale problem. Since our multiple time step algorithm, the so-called reversible Reference System Propagator Algorithm (r-RESPA) [17, 24, 18, 26] is time reversible and symplectic, they are very useful in combination with HMC for constant temperature simulations of large protein systems. 

Wisdom, J. The Origin of the Kirkwood Gaps A Mapping for Asteroidal Motion Near the 3/1 Commensurability. Astron. J. 87 (1982) 577-593 Tuckerman, M., Martyna, G. J., Berne, J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys. 97 (1992) 1990-2001 Tuckerman, M., Berne, J. Vibrational Relaxation in Simple Fluids Comparison of Theory and Simulation. J. Chem. Phys. 98 (1993) 7301-7318 Humphreys, D. D., Friesner, R. A., Berne, B. J. A Multiple-Time Step Molecular Dynamics Algorithm for Macromolecules. J. Chem. Phys. 98 (1994) 6885-6892... 

A variety of techniques have been introduced to increase the time step in molecular dynamics simulations in an attempt to surmount the strict time step limits in MD simulations so that long time scale simulations can be routinely undertaken. One such technique is to solve the equations of motion in the internal degree of freedom, so that bond stretching and angle bending can be treated as rigid. This technique is discussed in Chapter 6 of this book. Herein, a brief overview is presented of two approaches, constrained dynamics and multiple time step dynamics. 

Dynamic simulations were aimed at capturing the multiple-time-scale behavior revealed by the theoretical developments presented above. Figures 7.5 and 7.6 show the evolution of the mole fraction of n-butane and of the temperature on selected column stages for a small step change in the reboiler duty. Visual inspection of the plots indicates that the temperatures exhibit a fast transient,... 

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]. 

M. Tuckermann, B.J. Berne and G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992) 1990 P. Procacci and B.J. Berne, Computer simulation of solid C60 using multiple time-step algorithms, J. Chem. Phys., 101 (1994) 2421. 

Tobias D J, K Tu and M L Klein 1997. Atomic-scale Molecular Dynamics Simulations of Lipid Membranes Current Opinion in Colloid and Interface Science 2-15-26 Tuckerman M, B J Berne and G J Martyna 1992. Reversible Multiple Time Scale Molecular Dynamics. Journal of Chemical Physics 97.1990-2001... 

B. J. Berne, in Multiple Time Scales, J. U. Brackbill and B. I. Cohen, Eds., Academic Press, Orlando, 1985, p. 419. Molecular Dynamics and Monte Carlo Simulations of Rare Events. 

Theory, Experiment, and Reaction Rates. A Personal View. J. D. Doll and A. F. Voter, Annu. Rev. Phys. Chem., 38, 413 (1987). Recent Develoinnents in the Theory of Surface Diffusion. B. J. Berne, in Multiple Time Scales, J. U. Brackbill and B. I. Cohen, Eds., Academic Press, Orlando, FL, 1987, pp. 419-436, Molecular Dynamics and Monte Carlo Simulation of Rare Events. P. HSnggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys., 62, 251 (1990). Reaction-Rate Theory Fifty Years After Kramers. 

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. 

Two important challenges exist for multiscale systems. The first is multiple time scales, a problem that is familiar in chemical engineering where it is called stiffness, and we have good solutions to it. In the stochastic world there doesn t seem to be much knowledge of this phenomenon, but I believe that we recently have found a solution to this problem. The second challenge—one that is even more difficult—arises when an exceedingly large number of molecules must be accounted for in stochastic simulation. I think the solution will be multiscale simulation. We will need to treat some reactions at a deterministic scale, maybe even with differential equations, and treat other reactions by a discrete stochastic method. This is not an easy task in a simulation. 

The modeling and simulation of Cl and SoS is challenged by the key characteristics of these systems i) coexistence of multiple time scales, from infrastructure evolution to real-time contingencies it) multiple levels of interdependencies and lack of fixed boundaries, i.e. CIs are made of multiple layers (management, information control, energy, physical infrastructure) Hi) broad spectrum of hazards and threats iv) different types of physical flows, i.e. mass, information, power. 

Usually refers to a method of solving Newton s equations of classical mechanics numerically, in order to propagate the positions and velocities of a system of molecules forward in time and thus to explore the phase space of the system. See Molecular Dynamics and Hybrid Monte Carlo in Systems with Multiple Time Scales and Long-range Forces Reference System Propagator Algorithms Molecular Dynamics DMA Molecular Dynamics Simulations of Nucleic Acids Molecular Dynamics Studies of Lipid Bilayers and Molecular Dynamics Techniques and Applications to Proteins. 

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