Multi-particle collision model

A. Malevanets and R. Kapral, Mesoscopic multi-particle collision model for fluid flow and molecular dynamics, in Novel Methods in Soft Matter Simulations, M. Karttunen, I. Vattulainen, and A. Lukkarinen (eds.), Springer-Verlag, Berlin, 2003, p. 113. 

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, we use the term MFC to describe the generic class of particle-based algorithms for fluid flow which consist of successive free-streaming and multi-particle collision steps. The name SRD is reserved for the most widely used algorithm which was introduced by Malevanets and Kapral [18]. The name refers to the fact that the collisions consist of a random rotation of the relative velocities Svi = V, - u of the particles in a collision cell, where u is the mean velocity of all particles in a cell. There are a number of other MFC algorithms with different collision rules [31-33]. For example, one class of algorithms uses modified collision rules which provide a nontrivial coUisional contribution to the equation of state [33,34]. As a result, these models can be used to model non-ideal fluids or multi-component mixtures with a consolute point... 

In a binary mixture of A and B particles, phase separation can occur when there is an effective repulsion between A-B pairs. In the current model, this is achieved by introducing velocity-dependent multi-particle collisions between A and B particles. There are Nx and Mb particles of type A and B, respectively. In two dimensions, the system is coarse-grained into Lj of cells of a square lattice of Unear dimension L and lattice constant a. The generalization to three dimensions is straightforward. 

Particle methods (Molecular Dynamics, Dissipative Particle Dynamics, Multi-Particle Collision Dynamics) simulate a system of interacting mass points, and therefore thermal fluctuations are always present. The particles may have size and structure or they may be just point particles. In the former case, the finite solvent size results in an additional potential of mean force between the beads. The solvent structure extends over unphysically large length scales, because the proper separation of scale between solute and solvent is not computationally realizable. In dynamic simulations of systems in thermal equilibrium [43], solvent structure requires that the system be equilibrated with the solvent in place, whereas for a structureless solvent the solute system can be equilibrated by itself, with substantial computational savings [43]. Finally, lattice models have a (rigorously) known solvent viscosity, whereas for particle methods the existing analytical expressions are only approximations (which however usually work quite well). 

The Zimm model apphes to dilute solutions, and, therefore, to the dynamics of a single solvated chain. It has become a benchmark system, used to test the validity of mesoscopic simulation methods. A single chain, modeled by bead-spring interactions, coupled to a surrounding solvent to account for hydrodynamic interactions, has been successfully simulated via (1) Molecular Dynamics [180-182], (2) Dissipative Particle Dynamics [183,184], Multi-Particle Collision Dynamics [185,186],... 

The main aim of this paper is to review the CDW-EIS model used commonly in the decription of heavy particle collisions. A theoretical description of the CDW-EIS model is presented in section 2. In section 3 we discuss the suitablity of the CDW-EIS model to study the characteristics of ultra-low and low energy electrons ejected from fast heavy-ion helium, neon and argon atom collisions. There are some distinct characteristics based on two-centre electron emission that may be identified in this spectrum. This study also allows us to examine the dependence of the cross sections on the initial state wave function of multi-electron targets and as such is important in aiding our understanding of the ionization process. 

The total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

The lattice Boltzmann method is a mesoscopic simulation method for complex fluid systems. The fluid is modeled as flctitious particles as they propagate and collide over a discrete lattice domain at discrete time steps. Macroscopic continuum equations can be obtained from this propagation-collision dynamics through a mathematical analysis. The nature of particulates and local dynamics also provide advantages for complex boundaries, multi-phase/multicomponent flows, and parallel computation. 

The Smoluchowski theory has also been employed to estimate rate constants, which showed agreement with the multi-step chain growth via OA. In this model, the reaction may occur between any of two NPs in the system. As the size of the NP increases, the collision cross section of the particle enlarges, while the motion rate of the particle rapidly decreases. Combination of these two variables leads to a quick reduction in growth rate by OA, so that it finally stops. The resulting size distribution often exhibits a most probable value, which can be simulated using the Smoluchowski theory. The kinetic model can be presented as follows ... 

RMD Simulation of Chemical Nucleation (22). A series of microscopic computer experiments was performed using the cooperative isomerization model (Eq. 2). This system was selected for the trial simulations for several reasons First, only two chemical species are involved, so that a minimal number of particles is needed. Second, the absence of buffered chemicals (e.g., A and B in the Trimolecular reaction of the next section) eliminates the need for creation or destruction of particles in order to maintain constant populations (19., 22j. Third, the dynamical model of the cooperative mean-field interaction can be examined as a convenient means of introducing cubic or higher nonlinearity into molecular models based on binary collisions. Finally, the need for a microscopic simulation is most apparent for transitions between multi -pie macroscopic states. Indeed, the characterization of spatially localized fluctuations is of obvious importance to the understanding of nucleation phenomena. As for the equilibrium vapor-liquid and liquid-solid transitions, detailed simulations at the molecular level should provide deep physical insight into chemical nucleation processes whkh is unattainable from theory, higher-level simulation, or experiment. 

---