Collision rules

A minima] set of symmetric binary and triple collision rules conserving both momentum and particle number-... 

The multiparticle collision rule was first introduced in the context of a lattice model with a stochastic streaming rule in Ref. 12. 

The unit vector n may be taken to lie on the surface of a sphere and the angles a may be chosen from a set Q of angles. For instance, for a given a, the set of rotations may be taken to be = a, —a. This rule satisfies detailed balance. Also, a may be chosen uniformly from the set Q = a 0 < a < 71. Other rotation rules can be constructed. The rotation operation can also be carried out using quaternions [13]. The collision rule is illustrated in Fig. 1 for two particles. From this figure it is clear that multiparticle collisions change both the directions and magnitudes of the velocities of the particles. 

Figure 1. Application of the multiparticle collision rule for two particles with initial velocities vi and V2 leading to postcollision velocities vj and vj, respectively. Intermediate velocity values in the rule (vi — V) and (v2 — V) are shown as thin solid lines while cb(vi — V) and cb(v2 — V) are shown as dashed lines.

For the collision rule using rotations by a about a randomly chosen axis, these expressions may be evaluated approximately to give [26]... 

Multiparticle collision dynamics as formulated earlier has an ideal gas equation of state. Ihle, Tiizel, and Kroll [118] have generalized the collision rule to... 

Here V = (N Vj + N2S2) / N + N2) is the mean velocity of the pair of cells. By summing over the particles in cell 1 it is easy to verify that velocities normal to [Pg.137]

Although this collision rule conserves momentum and energy, in contrast to the original version of MPC dynamics, phase space volumes are not preserved. This feature arises from the fact that the collision probability depends on AV so that different system states are mapped onto the same state. Consequently, it is important to check the consistency of the results in numerical simulations to ensure that this does not lead to artifacts. 

Collision The particles at all lattice sites undergo a collision that conserves the total number of particles and the total momentum at each site. The collision rules may or may not be deterministic. 

In the linearized LB equation Eq. (7), the ensemble averaged effect of the particle-particle collision is now represented by a relaxation of the distribution function ft to the equilibrium function /fq, where the matrix Ly does not necessarily have to correspond to an existing set of collision rules. The question now arises if L can be simplified even further to the form Ly = aSy, so that the LB equation takes the form (with x — -St/a)... 

Although LB therefore nowadays may be considered as a solver for the NS equations, there is definitely more behind it. The method originally stems from the lattice gas automaton (LGA), which is a cellular automaton. In a LGA, a fluid can be considered as a collection of discrete particles having interaction with each other via a set of simple collision rules, thereby taking into account that the number of particles and momentum is conserved. 

In the LB technique, the fluid to be simulated consists of a large set of fictitious particles. Essentially, the LB technique boils down to tracking a collection of these fictitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the so-called collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

Inputting solid particles at fixed positions, of different sizes simulates a solid phase in the fluid lattice (Fig. 4). The number of fluid particles per node and their interaction law (collisions) affect the physical properties of real fluid such as viscosity. Particle movements are divided into the so called propagation step (spatial shift) and collisions. Not all particles take part in the collisions. It strongly depends on their current positions on the lattice in a certain LGA time step. In order to avoid an additional spurious conservation law [13], a minimum of two- and three-body collisions (FHP1 rule) is necessary to conserve mass and momentum along each lattice line. Collision rules FHP2 (22 collisions) and FHP5 (12 collisions) have been used for most of the previous analyses [1],[2],[14], since the reproduction of moisture flow in capillaries, in comparison to the results from NMR tests [3], is then the most realistic. 

For a single fluid, existing two-dimensional models are all variations of the original FHP lattice-gas (Frisch et al., 1986). The cellular space is built as a hexagonal lattice. At most, six moving particles may reside in a cell at a time. Several variants have been constructed differing in the number of particles at rest and in the collision rules. 

Molecular motors enter the model implicitly by specifying the interaction rules between two rods. Since the diffusivity of molecular motors is about 100 times larger than that of microtubules, as a first approximation we neglect spatial variations of the molecular motor density. While the varying concentration of molecular motors affects certain quantitative aspects [7], our analysis captures salient features of the phenomena and the collision rules are spatially homogeneous. All rods are assumed to be of equal length I and diameter d I,... 

For physically relevant states, the propagation and collision rules for the behavior of such a set of cells as time goes on may mirror what would happen with a physical system. This is why cellular automata are appealing. Another advantage is that due to the locality mentioned above, the relevant computer programs may be effectively parallelized, which usually significantly speeds up computations. The most interesting cellular automata are those for which the rules are of a nonlinear character (cf. Chapter 15). 

Fig. 7.16. Operation of a cellular automaton-a model of gas. The particles occupy the lattice nodes (cellsl. Their displacement from the node symbolizes which direction they are heading on with the velocity equal to 1 length unit per 1 time step. In the left scheme (a), the initial situation is shown. In the right scheme the result of the one step propagation and one step collision is shown. Collision only took place in one case (at 03 2). collision rule has been applied (of the lateral outgoing). The game would...

On the other hand, the LBG method can capture both mesoscopic and macroscopic scales even larger than those that can be modeled by discrete-particle methods. This advantage is due to computational simplicity of the method, which comes from coarse-grained discretization of both the space and time and drastic simplification of collision rules between particles. We can look at the validity of these simplifications by comparing them with more realistic discrete-particles simulation. We regard both DPD and LBG as being complementary computational tools for modeling the slow dynamics in porous media over wide spatio-temporal scales. 

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