Dynamics of many-particle fragmentation in a cellular automaton model

Evolution of power law distributions in science and society

Power law distributions have been observed in numerous physical and social systems; for example, the size distributions of particles, aerosols, corporations, and cities are often power laws. Each system is an ensemble of clusters, comprising units that combine with or dissociate from the cluster. Constructing models and investigating their properties are needed to understand how such clusters evolve. To describe the growth of clusters, we hypothesize that a distribution obeys a governing population dynamics equation based on a reversible association-dissociation process. The rate coefficients are considered to depend on the cluster size as power expressions, thus providing an explanation for the asymptotic evolution of power law distributions.

Thermodynamics from three-dimensional many-body fragmentation simulations on a cellular automaton model

The thermal equilibrium of many-body systems subject to finite range interactions is investigated numerically, by means of a multipurpose 3D cellular automaton dynamic model developed by the authors. The numerical experiments, carried out at fixed number of bodies, volume and energy, demonstrate the formation of an equilibrium among 3D aggregates of bodies. The distribution of the aggregates against size obeys a power law of (negative) exponent tau approximately 2.2 (against 1.3 in 2D). Our experiments, indicating that the exponent is insensitive to the precise parameter values and the precise parametrization of the interactions, are consistent with the idea of the existence of a universality class corresponding to the thermal equilibrium. The numerical value for the exponent tau is in agreement with the theoretical thermal equilibrium analyses based on various other approaches, numerical and semianalytical, indicating that the cellular automaton approach provides an adequate methodology to investigate thermal equilibria. In this paper, as an illustration of this method, we refer to the problem of formation of clusters of nucleons in heavy ion collisions of nuclei leading on to fragmentation. The theoretical tau value, however, corresponding to the thermal equilibrium among the aggregation clusters, is 15 percent lower than the empirical value ( approximately 2.6 ) , as measured in laboratory nuclear fragmentation experiments induced by collision. There is then only a very approximate correspondence between the experimental and the thermal equilibrium value. On the basis of the results of this paper and of a previous paper of this series, we conjecture that the approximate agreement is due to a partial establishment of a thermodynamic equilibrium during the collision of the nuclei. The thermal equilibrium gives the main contribution to the observed tau value; the deviation from this possibly universal value is largely the consequence of the lack of full thermal equilibrium in actual laboratory experiments. This conjecture is extended to interpret the observed ubiquity of power laws of exponents exceeding 2.2, which refer to the distribution of various types of matter in 3D space.