Stochastic rotation dynamics

In SRD, the solvent is modeled by a large number N of point-like particles of mass m which move in continuous space with a continuous distribution of velocities. The algorithm consists of individual streaming and collision steps. In the streaming step, the coordinates, r,(t), of all solvent particles at time t are simultaneously updated according to 

In order to define the collisions, particles are sorted into cells, and they interact only with members of their own cell. Typically, the system is coarse-grained into cells of a regular, typically cubic, grid with lattice constant a. In practice, lengths are often measured in units of a, which corresponds to setting a = 1. The average number of particles per cell, M, is typically chosen to be between three and 20. The actual number of particles in a cell at a given time, which fluctuates, will be denoted by Nc. The collision step consists of a random rotation R of the relative velocities 5v, = V, — u of aU the particles in the 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 

In its original form [18,19], the SRD algorithm was not Galilean invariant. This is most pronounced at low temperatures or small time steps, where the mean free path, A = At k T/m, is smaller than the cell size a. If the particles travel a distance between collisions which is small compared to the cell size, essentially the same 

In addition to restoring Galilean invariance, this grid-shift procedure accelerates momentum transfer between cells and leads to a collisional contribution to the transport coefficients. If the mean free path A is larger than a/2, the violation of Galilean invariance without grid shift is negligible, and it is not necessary to use this procedure. 

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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). 

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]. 

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

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. 

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 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... 

For slower motions in more viscous media where XrAco > 1, the ESR spectrum depends dramatically on the combined influence of molecnlar motion and magnetic interactions. The slow-motional ESR line shapes, in principle, provide a more detailed picture of rotational dynamics compared with fast-motional line shapes, and can be fully analyzed using a theoretical approach based on numerical solution of the stochastic Lionville equation. The theoretical approach resulting in the Schneider-Freed set of programs enables the calculation of theoretical line shapes of ESR spectra of nitroxides subjected to anisotropic rotational... 

Dynamics on longer time scales determines spectral line shapes and requires more coarse-grained models rooted in a stochastic approach. For semirigid systems the relevant set of stochastic coordinates can be restricted to the set of orientational coordinates RS10W = Cl, which can be described, in turn, in terms of a simple formulation for a diffusive rotator, characterized by a diffusion tensor D [16], i.e. 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

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