Stochastic collisions method

Another method of controlling the temperature that can be used in CP MD is the stochastic thermostat of Andersen.27 In this approach the velocity of randomly selected nucleus is rescaled this corresponds in a way to the stochastic collisions with other particles in the system. Therefore, this approach is often called a stochastic collision method. The Andersen thermostat has recently been shown28 to perform very well in the Car-Parinello molecular dynamic simulations of bimolecular chemical reactions. 

Given that the two techniques in some ways complement each other in their ability to explore phase space, it is not surprising that there has been some effort to combine the two methods. Some of the techniques that we have considered in this chapter and in Chapter 7 incorporate elements of the Monte Carlo and molecular d)mamics techniques. Two examples are the stochastic collisions method for performing constant temperature molecular dynamics, and the force bias Monte Carlo method. More radical combinations of the two techniques are also possible. 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

To carry out this method, values are chosen for Tq, the desired temperature, and v, the mean frequency with which each particle experiences a stochastic collision. If successive collisions are uncorrected, then the distribution of time intervals between two successive stochastic collisions, P(v, t), is of the Poisson fonn. 

The first maj or extension of the stochastic particle method was made by O Rourke 5501 who developed a new method for calculating droplet collisions and coalescences. Consistent with the stochastic particle method, collisions are calculated by a statistical, rather than a deterministic, approach. The probability distributions governing the number and nature of the collisions between two droplets are sampled stochastically. This method was initially applied to diesel sprays13171... 

The lowest-lying potential energy surfaces for the 0(3P) + CH2=C=CH2 reaction were theoretically characterized using CBS-QB3, RRKM statistical rate theory, and weak-collision master equation analysis using the exact stochastic simulation method. The results predicted that the electrophilic O-addition pathways on the central and terminal carbon atom are dominant up to combustion temperatures. Major predicted end-products are in agreement with experimental evidence. New H-abstraction pathways, resulting in OH and propargyl radicals, have been identified.254... 

Another MD formulation for constant-temperature and -pressure calculations based on the constant temperature formulation of Nos6 (77) has been described recently. The stochastic collisions which permit energetic fluctuations has been replaced by the dynamic method of scaling of velocities of the atoms, in addition to the scaling of velocities by V1/3. The method, which is completely dynamical, requires one to choose appropriate values for Q and M which respectively determine the time scale of the temperature and volume fluctuations. 

In recent years, new discrete-particle methods have been developed for modeling physical and chemical phenomena occurring in the mesoscale. The most popular are grid-type techniques such as cellular automata (CA), LG, LBG, diffusion- and reaction-limited aggregation [37] and stochastic gridless methods, e.g., DSMC used for modeling systems characterized by a large Knudsen number [41], and SRD [39]. Unlike DSMC, in SRD collisions are modeled by simultaneous stochastic rotation of the relative velocities of every particle in each cell. 

Two methods that have been developed that do maintain correct canonical averaging are the stochastic collision and the extended systems approaches. Both are covered in detail elsewherel . We report here only on some of the salient features from the extended systems approach since this approach is used primarily for constant temperature MD simulations for heterogeneous catalytic materials. 

Atomistic MD becomes very ineffident for low-density problems, for example, for long, single chains. For this method to be useful, we need to compute long enough for the chain to evolve on the maaoscopic time and space scales while using a specified miaoscopic particle interaaion law. The time step, however, is limited. If the chain units are too far apart, it could take many millions of time steps to simulate a few collisions. The expense becomes unacceptably high. Thus, MD can be too ineffeaive for dilute systems. Stochastic MC methods, event-driven molecular dynamics, or a hybrid pivot MC/MD generation procedure can be much more successful. 

The analysis of simulation studies has shown that despite the use of a stochastic optimisation method, the presented model of the safe ship control in collision situation at sea, using an ant algorithm, enables obtaining reproducible solutions. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

One aspect of MD simulations is that all molecules, including the solvent, are specified in full detail. As detailed above, much of the CPU time in such a simulation is used up by following all the solvent (water) molecules. An alternative to the MD simulations is Brownian dynamics (BD) simulation. In this method, the solvent molecules are removed from the simulations. The effects of the solvent molecules are then reintroduced into the problem in an approximate way. Firstly, of course, the interaction parameters are adjusted, because the interactions should now include the effect of the solvent molecules. Furthermore, it is necessary to include a fluctuating force acting on the beads (atoms). These fluctuations represent the stochastic forces that result from the collisions of solvent molecules with the atoms. We know of no results using this technique on lipid bilayers. 

There are many systems that can fluctuate randomly in space and time and cannot be described by deterministic equations. For example. Brownian motion of small particles occurs randomly because of random collisions with molecules of the medium in which the particles are suspended. It is useful to model such systems with what are known as stochastic differential equations. Stochastic differential equations feature noise terms representing the behavior of random elements in the system. Other examples of stochastic behavior arise in chemical reaction systems involving a small number of molecules, such as in a living cell or in the formation of particles in emulsion drops, and so on. A useful reference on stochastic methods is Gardiner (2003). 

Another method that introduces a very simplified dynamics is the Multi-Particle Collision Model (or Stochastic Rotation Model) [130]. Like DSMC particle positions and velocities are continuous variables and the system is divided into cells for the purpose of carrying out collisions. Rotation operators, chosen at random from a set of rotation operators, are assigned to each cell. The velocity of each particle in the cell, relative to the center of mass velocity of the cell, is rotated with the cell rotation operator. After rotation the center of mass velocity is added back to yield the post-collision velocity. The dynamics consists of free streaming and multi-particle collisions. This mesoscopic dynamics conserves mass, momentum and energy. The dynamics may be combined with full MD for embedded solutes [131] to study a variety of problems such as polymer, colloid and reaction dynamics. 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

Chandler and co-workers successfully used stochastic dynamics in their studies of -alkanes and cyclohexane isomerization in solution. The method used is based on the BGK theory. The assumption is that the primary form of interaction between the solvent molecules and the isomerizing system is in the form of hard collisions. These collisions, when they occur, randomize the velocity of one of the isomerizing molecule s atoms. The computational implementation of this is quite simple at random times, based on the collision frequency (which is taken to be proportional to the solvent viscosity), instantaneously change one random atom s velocity to one selected from the Boltzmann distribution at the temperature of interest. Then continue running dynamics until the next collision occurs, at which time another... 

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