Stochastic collisions

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 probability that any particular particle suffers a stochastic collision in a time interval of At is v At. 

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. 

Cons tant-temperat u re-constant-pressure calculations. The trajectory for the atoms and volume are generated according (u the solution of the equations of motion for the lagrangian (1) discussed earlier in connection with constant-pressure calculations. In addition, stochastic collisions are introduced to allow for fluctuations in en-... 

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 The stochastic collisions which permit energetic fluctuations has been replaced by the dynamic method of scaling of velocities of the atoms, in addition to the scaling of velocities by V1/3.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. 

Fig. 10. The densities obtained at different cooling rates at T = 0 and p = 1 for the Lennard-Jones glass. The cooling rate is the value of the stochastic collision frequency per particle, v. The number of runs employed to obtain the reported average is shown inside each point. (From Fox and Andersen (67).)...

The above stochastic collision model then leads to a generalization, Eq. (253), of the Fokker-Planck equation for the evolution of the phase distribution function for mechanical particles, where the velocities acquire a fractional character [30], rather than both the displacements and the velocities as in Eq. (235). In the present context, all these comments apply, of course, to rotational Brownian motion. 

Reactions on metals, including many oxidation processes, are known to proceed in a way very different from stochastic collision types, which can be described by mass action (or acting surfaces ). The number of systems in which collective effects or topochemical type processes (via nucleation and growth of nuclei) are proved to determine the kinetic behavior is increasing. Despite the extensive literature on reactions in oscillatory regimes and spatially-structured reactions on surfaces (Gorodetskii et al., 2005 Latkin et al., 2003 Peskov et al., 2003), such facts have not yet found an adequate reflection in the area under consideration. 

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. 

Fig. II. Radial distribution functions for systems obtained by continuous cooling at p = 1. The stochastic collision frequencies were 3.2 for the -I- and 0.04 for the O points. (From Fox and Andersen (67).)...

Brownian dynamics (BD) models diffusional systems in which the particles undergo Brownian motion. In such systems the particles, whose mass and size are larger than those of the solvent molecules, are subjected to stochastic collisions and to the viscous drag exerted by these molecules. This leads to the apparently random motion of the particles, which is diffusion, first recorded in the 19th century by Brown. 

The mean-field effect of the environment can be included in biomolecular simulations simply by adding an expression for the solvation free energy of an instantaneous solute conformation to a given molecular mechanics force field [1]. Such an implicit solvent potential addresses the thermodynamic component of solute-solvent interactions. Kinetic and hydrodynamic properties may be reintroduced through the use of Langevin dynamics where coupling with a temperature bath is implemented through stochastic collisions and solvent friction [2,3,18]. 

More realistic kinetic behavior in implicit solvent simulations can be obtained with the Langevin thermostat [18] where stochastic collisions and friction forces provide kinetic energy transfer to and from the solute in an analogous fashion to explicit solute-solvent interactions. As a result, kinetic transition rates similar to rates from explicit solvent simulations can be recovered with an appropriate choice of the friction constant [2]. 

G. Brenn, St. Kalcmdcrsld, I. Ivanov Investigation of the stochastic collisions of drops produced by Rayleigh breakup of two laminar Uquid jets, Phys. Fluids 9, 349—364 (1997). 

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. 

---