Interdisciplinary Applied Mathematics

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The comparison of the efficiency of the FMM algorithm to that of the particle-particle algorithm depends on the number of particles in the simulation, the maximum level of refinement, and the number of multipoles (Sadus, 1999). (Schmidt and Lee, 1991) analyzed the different scenarios. Typically, the FMM algorithm is advantageous for simulations involving tens of thousands of molecules. Simple particle-particle algorithms assisted by time-saving concepts such as a neighbor list are adequate for atoms or simple polyatomic molecules. However, the rigorous evaluation of the properties of real macromolecules by molecular simulation requires thousands of interaction sites per molecule. Alternative computational strategies are required to deal with this increased level of complexity. It is in this context that hierarchical tree algorithms (e.g., FMM) are likely to play an increasingly important role in molecular simulation.

16.1.3 Thermostats

For several reasons (e.g., drift during equilibration, drift as a result of force truncation and integration errors, heating due to external or frictional forces), it is necessary to control the temperature of the system in MD simulations. In a canonical ensemble of finite systems, the instantaneous kinetic temperature fluctuates (Frenkel and Smit, 2002). In fact, if the average kinetic energy per particle is kept constant (as is done in the isokinetic MD scheme (Evans and Morriss, 1990)), then the true constant-temperature ensemble would not be simulated. In practice, the difference between isokinetic and canonical schemes is often negligible (Frenkel and Smit,    2002).    In    this section    we    discuss    some    of the    most    commonly em

ployed temperature coupling schemes.

The Berendsen Thermostat

The Berendsen algorithm simulates weak coupling with first-order kinetics to an external heat bath with a given temperature T0. According to this algorithm the deviation of the system temperature from T0 is slowly corrected according to the following equation (van der Spoel et al., 2004):

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