A multiparticle fluid model
A.4 Ballistic annihilation
This section presents another application of our multiparticle model: the bal-listic annihilation problem. Balbal-listically controlled reactions provide simple
ex-Figure A.4: snapshot, in a multiparticle simulation of a von Karman street.
amples of non-equilibrium systems with complex kinetics[Elskens et Frisch 1985, BNet al. 1993, Rey et al. 1995, Rey et al. 1998]. They consist of a system of particles moving freely with given velocities until they experience a collision.
When two particles meet, they instantaneously annihilate each other and disap-pear from the system. In this problem, N-body correlations are expected to play a role on N(t), the particle number decay. The behavior of N(t) is assumed to be described by a power lawN(t)∼t−α, in the long time regime.
In one dimension, systems with only two possible velocities +v or −v have been studied by Elskens and Frisch[Elskenset Frisch 1985] and yield α = 1/2.
The case of a general velocity distribution has been treated analytically by Droz et al.[Rey et al. 1995]. It was shown that different dynamical behaviors can occur depending on the initial velocity distribution and that fluctuations play a very important role, invalidating the predictions of a mean-field approach.
Beyond one dimension, the situation is much more complex. For a continuous space time system, a numerical integration of the Boltzmann equation for the time evolution of the velocity distribution with a uniform initial condition leads to the value α= 0.91.[BN et al. 1993]
A recent two-dimensional molecular dynamics study (with up to 105 particles) by [Trizac] gives a decay exponent α whose value is affected by finite size effects and varies between 0.86 and 0.89 depending on the size of the sample. Moreover, it is observed that the kinetic energy distribution function evolves in time towards a Maxwellian, although the results of the simulations are very noisy.
Due to the way our multiparticle is defined, intrinsic fluctuations are present in the dynamics and ballistic annihilation can be used as a test to check whether the particle correlations are dealt with in a physical way. It is easy to add to the previously defined rules a new mechanism implementing the annihilation of each pair of particles arriving simultaneously at the same site with oppo-site velocities. More precisely, the annihilation rule we use is the following:
Fi → max(Fi −Fi′,0), where, i′ is the direction opposite to i. Note that an annihilation probability less than 1 can also be implemented with a more compli-cated rule.[Chopard et al. 1994]. However the results are found identical up to a rescaling of time.
Simulations have been made for various systems sizes and an initial number of particles per site equals 10, on average. The results of the decay process are given in figure A.5. As one sees, all systems with size larger than 64×64 give the same decay exponent α = 0.875±0.005. This value is in very good agreement with the value obtained by standard molecular dynamic simulations. Moreover, our algorithm is very efficient since the CPU time for the system of size 1024×1024 with initially 107 particles is about 58 seconds on a IBM-SP2 parallel computer with 10 processors. This is several order of magnitude faster than the molecular dynamics computation.
It is interesting to compare this results with those obtained with standard CA or LB simulation. A two-dimensional FHP[Chopard etDroz 1998] cellular
0 1 2 3 4
Figure A.5: Decay laws for the ballistic annihilation simulations, using the mul-tiparticle lattice gas model. The various plots correspond to the lattice sizes indicated in the box. The decay exponent α is given by the slopes of the lines which are all within x= 0.875±0.005, except for the smallest lattice.
automata model with annihilaton gives an exponent α= 1/2. This is the 1D ex-ponent, which can be explained by the fact that, in this case, particle annihilation takes place before collisions can couple the two spatial dimensions.
For the standard, real-valued LB approach in 2D, annihilation is modeled by adding the terms −FiFi′ to the collision operation, where, again, i′ denotes the direction opposite to i. Numerical simualtions then yield α = 1, corresponding to the naive rate equation approach ˙ρ∼ −ρ2.
Therefore, the multiparticle model clearly captures an subtle behavior of bal-listic annihilation. It can be noticed in fig. A.6 that, despite the fast particle decay, multiparticle collisions are present all along the dynamics.
A.5 Conclusion
We have proposed a multiparticle algorithm that conciliates the advantage of both the CA and LB approaches: numerical stability, presence of fluctuations, little statistical noise, due to the large number of particles per site, and flex-ibility to tune model parameters. Although significantly slower than its LB counterpart, our dynamics can be implemented in an efficient way on parallel computers and is much simpler and more flexible than the other multiparti-cle models[Chatagny et Chopard 1993, Boghosian et al. 1997] proposed so far to extend the CA dynamics without an exclusion principle. In a slightly differ-ent spirit, we may also mdiffer-ention the recdiffer-ent multiparticle models by Masselot et al[Masselot etChopard 1998a] and Malevanetset al[Malevanets etKapral 1998].
Our model gives a remarkably good prediction for the ballistic annihilation problem and exhibits the expected hydrodynamical behaviors in the Poiseuille
1 10 100 1000 time
100 102 104 106 108
multiparticle annihilation
256x256 (starting with 500 particles per direction)
total particles
configurations changed by the collision
# N_i>1
750 1000 1250 1500 1750 20000 2 4 6 8 10 zoom (since fisrt 0-value)
Figure A.6: Multiparticle collision frequencies during the ballistic annihilation simulation (256×256 cells). One can observe that even if there are a small number of particles in the whole system, rich collisions still occur and the multiparticle aspect (Ni > 1) remains. The scale is logarithmic for the early part of the simulation, and linear for the zoomed inset (as some experimental value may have become 0 for time step >750).
and von Karman flows. A complete analytical derivation of the expression for the viscosity, taking into account the stochastic part of our algorithm is still needed.