Stochastic dynamics simulations algorithms

Here, we will explain the most effective of them, the so-called shooting algorithm. For simplicity we will focus on how to do it for deterministic trajectories such as those obtained from a molecular dynamics simulation. We note, however, that very similar algorithms can be applied to stochastic trajectories [5, 7, 8]. 

When one implements an MC stochastic dynamics algorithm in this model (consisting of random-hopping moves of the monomers by one lattice constant in a randomly chosen lattice direction), the chosen set of bond vectors induces the preservation of chain connectivity as a consequence of excluded volume alone, which thus allows for efficient simulations. This class of moves... 

For most macroscopic dynamic systems, the neglect of correlations and fluctuations is a legitimate approximation [383]. For these cases the deterministic and stochastic approaches are essentially equivalent, and one is free to use whichever approach turns out to be more convenient or efficient. If an analytical solution is required, then the deterministic approach will always be much easier than the stochastic approach. For systems that are driven to conditions of instability, correlations and fluctuations will give rise to transitions between nonequihbrium steady states and the usual deterministic approach is incapable of accurately describing the time behavior. On the other hand, the stochastic simulation algorithm is directly applicable to these studies. 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

EDMD and thermodynamic perturbation theory. Donev et developed a novd stochastic event-driven molecular dynamics (SEDMD) algorithm for simulating polymer chains in a solvent. This hybrid algorithm combines EDMD with the direa simulation Monte Carlo (DSMC) method. The chain beads are hard spheres tethered by square-wells and interact with the surrounding solvent with hard-core potentials. EDMD is used for the simulation of the polymer and solvent, but the solvent-solvent interaction is determined stochastically using DSMC. 

A last increasingly popular family of MC simulations, not to be confused with the Metropolis method, exploits sampling from non-Boltzmann distributions to simulate kinetic events which are extremely rare compared to the typical molecular timescales, e.g., as is the case of charge transfer (dynamic or Kinetic MC [105], KMC) or reactive events (Gillespie s stochastic simulation algorithm [106]). 

Another dynamical method for simulating diffusional motion is commonly referred to as stochastic dynamics. A stochastic dynamics equivalent to the leap-frog position algorithm was derived by van Gunsteren and Berendsen, and is given by... 

Whereas in mixed MD/MC simulations, some of the atoms are moved by pure MD, and other particles are moved by pure MC, it is also possible to constmct algorithms in which the displacement itself is determined in part by a deterministic factor and in part by a stochastic factor. In this class, we can further distinguish essentially three techniques Langevin or stochastic dynamics, hybrid Monte Carlo, and force bias Monte Carlo and related techniques. 

Kanazawa, K., Koller, D., Russell, S. (1995). Stochastic simulation algorithms for dynamic probabilistic networks. In Proceedings of UAl 1995. UAL... 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

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