Stochastic rotation

For consistency we refer to this model as multiparticle collision (MPC) dynamics, but it has also been called stochastic rotation dynamics. The difference in terminology stems from the placement of emphasis on either the multiparticle nature of the collisions or on the fact that the collisions are effected by rotation operators assigned randomly to the collision cells. It is also referred to as real-coded lattice gas dynamics in reference to its lattice version precursor. 

T. Ihle and D. M. Kroll, Stochastic rotation dynamics a Galilean-invariant mesoscopic model for fluid flow, Phys. Rev. E 63, 020201(R) (2001). 

C. M. Pooley and J. M. Yeomans, Kinetic theory derivation of the transport coefficients of stochastic rotation dynamics, J. Phys. Chem. B 109, 6505 (2005). 

Stochastic rotation, multiparticle collision dynamics and, 93 Stochastic transition ... 

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. 

In all molecular crystals, there are both vibrations and stochastic rotations or translational motions of the molecules. 

In the following sections of this chapter, we first treat intramoleular vibrations then, in much more detail, phonons using typical examples and finally, very briefly, stochastic rotational motions ( reorientations ) and translational diffusion of molecules. Although the experimental methods for the characterisation of dynamics in molecular crystals are in principle no different from those used to investigate inorganic crystals, we shall briefly describe inelastic neutron diffraction, Raman scattering, infrared and far-infrared spectroscopy, as well as NMR spectroscopy to the extent necessary or useful for the specific understanding of molecular and lattice... 

The last item concerns not only space and time discretization but also drastic simplification of the collision operator and particle motion rules as well, e.g., by employing cellular automata, lattice gas models [37,38], or stochastic models such as stochastic rotation dynamics [39]. 

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. 

Dfle, T. and KroU, D.M., Stochastic Rotation Dynamics I Formalism, Gahlean invariance, and Green-Kubo relations, Phys. Rev. E, 67 (6), 066705, 2003. 

Here, two basic models of rotation of a methylene group about a C-C bond are discussed. One simple model involves a stochastic rotational diffusion about a bond this occurs if the potential barrier between several equilibrium sites is smaller than ksT, In other words, rotation about a bond occurs by successive random jumps of small amplitude. This rotation is specified by a correlation time Tj — being the diffusion constant... 

For the numerical simulation of flowing polymers, several mesoscopic models have been proposed in the last few years that describe polymer (hydro-)dynamics on a mesoscopic scale of several micrometers, typically. Among these methods, we like to mention dissipative particle dynamics (DPD) [168], stochastic rotation dynamics (sometimes also called multipartide collision dynamics) [33], and lattice Boltzmann algorithms [30]. Hybrid simulation schemes for polymer solutions have been developed recenfly, combining these methods for solvent dynamics with standard particle simulations of polymer beads (see Refs [32, 169, 170]). Extending the mesoscopic fluid models to nonideal fluids including polymer melts is currently in progress [30, 159,160,171]. 

The first technique is known as the stochastic rotational dynamics (SRD) method or multiparticle collision dynamics, which is a particle-based algorithm suited to account for hydrodynamic interactions on the mesoscale. The coarse-grained solvent is described as ideal-gas particles that propagate via streaming and collision steps, which are constructed such that the dynamics conserves mass, momentum, and energy. 

In the streaming step, the solvent particle trajectories are ballistic. Collisions are introduced by sorting the particles into cubic lattice cells and performing a stochastic rotation of the relative velocities in each cell. In contrast to DPD, the SRD method acts on all particles within the same collision cell. 

All particles in the cell are subject to the same rotation, but the rotations in different cells and at different times are statistically independent. There is a great deal of freedom in how the rotation step is implemented, and any stochastic rotation matrix which satisfies semi-detailed balance can be used. Here, we describe the most commonly used algorithm. In two dimensions, R is a rotation by an angle a, with probability 1 /2. In three dimensions, a rotation by a fixed angle a about a randomly chosen axis is typically used. Note that rotations by an angle -a need not be considered, since this amounts to a rotation by an angle a about an axis with the opposite orientation. If we denote the randomly chosen rotation axis by R, the explicit collision rule in three dimensions is... 

If the attempt is accepted, perform a stochastic rotation with the scaled rotation matrix 5R. Otherwise, use the rotation matrix R. 

The kinetic contribution to the pressure, TSd = NksT/V, is clearly present in all MFC algorithms. For SRD, this is the only contribution. The reason is that the stochastic rotations, which define the collisions, transport (on average) no net momentum across a fixed dividing surface. More general MFC algorithms (such as those discussed in Sect. 6) have an additional contribution to the virial equation of state. Flowever, instead of an explicit force F, as in (9), the contribution from the... 

The use of the rotation rule (2) together with an average over the sign of the stochastic rotation angle yields... 

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