Stochastic particle generation

Different from the molecular dynamics (MD) simulation method (Sect. 4.5), the Brownian dynamics approach does not directly simulate the inter-particle collision. Instead, in the Brownian dynamics, the pseudorandom motion characteristic of the effect of particle-particle interactions is mimicked by a stochastic force generated from random numbers. This makes the Brownian dynamics more efficient than the... 

In trajectory models, the particle turbulent diffusion can be considered by calculating the instantaneous motion of particles in the turbulent flow field. In order to simulate the stochastic characteristics of the instantaneous gas velocity in a turbulent flow, it is required to generate random numbers in the calculation process. 

By carrying out the above procedure from time 0 to time /,Mm, we evidently obtain only one possible realization of the stochastic process. In order to get a statistically complete picture of the temporal evolution of the system, we must actually carry out several independent realizations or runs. These runs must use the same initial conditions of the problem but different starting numbers for the uniform random number generator in order for the algorithm to result in different but statistically equivalent chains. If we make K runs in all, and record the population sizes (k, t) in run k at time t (i = 1,..., m and k = 1,...,K), then we may assert that the average number of particles at time t is... 

An alternative to stochastic reconstruction of multiphase media is the reconstruction based on the direct simulation of processes by which the medium is physically formed, e.g., phase separation or agglomeration and sintering of particles to form a porous matrix. An advantage of this approach is that apart from generating a medium for the purpose of further computational experiments, the reconstruction procedure also yields information about the sequence of transformation steps and the processing conditions required in order to form the medium physically. It is thereby ensured that only physically realizable structures are generated, which is not necessarily the case when a stochastic reconstruction method such as simulated annealing is employed. 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval... 

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