Steady Distributions in Aggregation Process of Sticky Particles

1 Prog. Theor. Phys. Vol. 78, No.1, July 1987, Progress Letters

Steady Distributions in Aggregation Process of Sticky Particles Hisao HAY AKAWA, Minoru Y AMAMOTO* and Hideki TAKAYASU**

Department of Physics, Kobe University, Kobe 657 *Department of Fundamental Science of Materials Graduate School of Science and Technology, Kobe University, Kobe 657 ** Department of Earth Sciences, Kobe University, Kobe 657 (Received January 21, 1987)

Recently, there has been a rapidly growing interest in the statistical treatments of dissipative systems. In contrast to the statistical mechanics in thermal equilibrium, that for systems far from equilibrium has not yet been established. For systems far from equilibrium the irreversible growth processes have been investigated intensively, since they represent one of the most characteristic phenomena in dissipative systems. A number of models have been proposed to study the irreversible growth processes, such as the Eden model/} the diffusion limited aggregation (DLA)2) and the clustercluster aggregation (CCA).3) These models often have dendritic structures characterized by fractals,4} having fractal dimension D less than the spatial dimension in which the growth processes take place. In this paper, we investigate an aggregation system with the balance between injection and dissipation. Let us consider an ensemble of particles moving in one dimensional space. A particle collides with an adjacent particle to form a cluster, and a new particle is injected at a random position to maintain the number of clusters, each of which may be a particle before or a cluster after collision. The number density of clusters, net), is thus conserved on the average, expressed as (n(t»=l.

(1)

If there are sources and confluences in a physical system, the system should reach an

equilibrium after all with a number of clusters between injected particles and confluenced clusters. When we observe the time evolution of systems, Eq. (1) holds at any time after nCt) is saturated. Each particle or cluster is characterized by its . mass and momentum. The mass and momentum are conserved in each collision, so that the whole processes are described in a deterministic way. Our model gives one of simplified description of dynamic behavior of aerosol. In aerosol systems, there are. a plenty of species of natural sources of particles, for instance, nucleations from supersaturated gas and chemical reactions. In our simulations, we restrict ourselves to the case that all clusters move in a

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Aggregation processes of sticky particles in one dimension are simulated. We consider the situation that particles are injected into a system to keep the number of clusters constant and find a power law for the mass distribution expressed as n(m)cx.m-1.3 under a conservation law of momenta.

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r-----------------------,

Table 1. The velocity and mass distributions for initial and injected particles. Uniform random distribution expresses the case that we have an equal probability when we extract one of samples between 0 and 1.

Case A B C D

o

~

____

~

2

__

~

____

3

~

____

4

velocity

mass

uniform random uniform random Maxwellian Maxwellian

uniform random constant uniform random constant

same direction. We do not consider geometrical structures of clusters because collision cross sections are independent of linear size of clusters in one dimension. Let us note that our model represents collisions of free molecule region where the mean free path is extremely larger than linear sizes of clusters, thus we do not have to take into account steric hinderance by clusters. We find that the mass distribution in this system asymptotically approaches a power law (see Figs. 1 and 2) .

~~

5

P(;;;;'m)~m-a,

collision times (x 105 )

a=0.33±0.04,

(2)

where PC;;;;. m) is the cumulative probabil· ity for mass larger than m. To simulate 1. this system we introduce 1500, 2000, 2500, 3000 and 3500 clusters and let them collide 360000,420000,480000,560000 and 800000 times respectively. In our simulation, we examine the system in the following four cases of the velocity and mass distribution for initial and injected particles (see Table I). These four cases are found to give asymptotic mass distributions with same power indices each other within statistical error. In all cases the mass distributions are represented by power laws in the intermediate mass range, while a cutoff at large m and a deviation from the power law at small m result because of finite steps of numerical simulation and too small a number of collisions, respectively. We can therefore conclude that the asymptotic mass distribution is insensitive to the initial distributions of mass and velocity but is dictated by the balance between injection and dissipation. Let us compare the present result with the result of a model of rivers 5 ) or the voter mode1 6 ) equivalent to the model of rivers. In the model of rivers, rain is falling steadily and uniformly on a slope. As the rain drops drip down at random along the Fig. 2.

The convergency of the power index.

The

result is obtained by the same condition of Fig.

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Fig. 1. Log-log plots of the cumulative distribution of mass. This figure expresses the result of Case C with 3000 clusters. We can see the asymptotic power law which is represented by P(;;;; m) ~ m-O•33 , where a, b, c, d, e, f and g express the mass distribution after 4000, 40000, 80000, 160000, 240000, 400000, 560000 collisions respectively.

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slope, they collide with each other to make a stream. Here Eq. (1) is satisfied with a balance between the injection by rain and the dissipation by the confluence of streams, where n(t) denotes the number density of rivers. Equation (1) holds both for our model and the model of rivers, which lead to power law size distributions. In the model of rivers, the power law index a=1/3, which can be obtained theoretically, is close to 0.33, which is obtained by our calculations. It is interesting that power indices agree with each other in spite of the following two differences between two models: (1) The model of rivers contains randomness, but our model is deterministic. (2) The velocity of a cluster depends on the mass thereof in our model, whereas the flow velocity is independent of the flow flux in the model of rivers. In other words, if we express the relative velocity between colliding two clusters as v~mP where m denotes the reduced mass, p=O in the model of river and p= -1/2 in our model. Vicsek et aF) investigated a system in which particles are fed into the system at a constant rate and larger clusters are removed according to some rules. Although the power law distribution of the cluster size was suggested in their calculation, they did not mention the significance of the conservation law, such as Eq. (1), because they were mainly concerned with the scaling form between the feeding rate and the cluster size distribution. Our model is applicable not only to the aggregation process but also to other. physical phenomena. The first example is that the system of random trains of shocks described by the Burgers equation at a limit of large Reynolds number can be identified with a system of sticky particles.8 ) We can regard each shock front as a particle having mass mi and velocity Ci, which correspond respectively to the distance between the adjacent intersections of the velocity slopes with the spatial axis and ----+ to the speed of the shock front (see Fig. o o o o 3), and momentum miCi. The interval between the intersection points, mi, and u the velocity of a shock front, Ci, are invariant in time. After the Burgers ullJ system reduces to the trains of shocks, there are no shock formations, since negative slopes of the solution vanish in x this system. In this case, we have known that Eq. (2) is replaced by an exponential form. 8 ) Our model describes the case where shocks are steadily generated by an external force. Kida Fig. 3. Turbulent velocity field as a train of ranand Sugihara9 ) investigated a forced dom triangular shocks for a solution of the Burgers equation (below), and a schematic Burgers turbulence where a random picture of an equivalent system of sticky partiexternal force is imposed to the Burgers cles in one dimensional space where each partiequation. As they restricted themselves cle has a velocity c,=(u/')+u/r»/2 and mass to the case that the self-correlation time m,(above). Let us note that both u and c of the random force is much shorter than denote the values in a moving system with the sound velocity. the characteristic time of velocity field,

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The authors wish to thank Professor O. Nagai, Professor T. Tatsumi, Professor S. Hayakawa and Mr. J. J. Kimfor valuable discussions.

M. Eden, Proc. 4th Berkeley Symp. on Mathematical Statics and Probability, ed. F. Neyman (Univ. Calf. Press, Berkeley, 1961), vol. IV, p. 223. 2) T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47 (1981), 1400. 3) P. Meakin, Phys. Rev. Lett. 51 (1983), 1119. M. Kolb, R. Botet and R. Jullien, Ibid. 1123. 4) B. B. Manderbrot, The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982). 5) H. Takayasu and I. Nishikawa, Proc. of 1st. Int. Symp. for "Science on Form'; ed. S. Ishizaka (KTK Sci. Pub., 1986), p. 15. 6) D. Griffeath, "Additive and Cancellative Interacting Particle Systems", Lecture Notes in Math. 724 (Springer-Verlag, 1979). 7) T. V. Vicsek, P. Meakin and F. Family, Phys_ Rev. A32 (1985), 1122. 8) ]. M. Burgers, Proc. Acad, Sci. Arnst. 53 (1950), 247, 393, 718, 732. J. M. Burgers, The Nonlinear Diffusion Equation (D. Reidal Publishing Co., 1974). T. Tatsumi and S. Kida, ]. Fluid. Mech. 55 (1972), 659. 9) S. Kida and M. Sugihara, ]. Phys. Soc. Jpn. 50 (1981), 1785. 10) M. Karder, G. Parisi and Y. C. Zheng, Phys. Rev. Lett. 56 (1986), 889. 11) J. D. Klett, ]. Atmos. Sci. 32 (1975), 380. W. H. White, ]. Colloid Interf. Sci. 87 (1982), 204. H. Hayakawa, submitted to ]. of Phys. A. 1)

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the noise can be represented by a linear combination of the standard Gaussian noises. They showed that from the initial flat velocity field a number of ·shocks saturated to a certain value after some numerical steps. Therefore, our model can be regarded as an idealized one for the forced Burgers system where the configuration of waves is not a perfect trains of random triangular shocks. The second example is the dynamic growth of random surface. Karder et al. 10 ) gave a simple nonlinear Langevin equation for a local growth of the profile with a Gaussian noise. .They showed that the equation can be mapped to two familiar forms which are the diffusion equation and the Burgers equation with a noise. If their Gaussian noise is replaced by injected noises satisfying Eq. (1) and gaps of gradients of surfaces by masses, we can regard our model as that of the growth of random surface. Although we do not obtain any theoretical explanations for our models, in mean field theory, we have known that steady solutions of Smoluchowski equation with source term obey power laws.ll) Therefore, we believe that the condition of Eq. (1) leads to power law for mass distribution in general cases.