Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

P-adic coverage method in fractal analysis of showers

Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/k parts of the space studied.     

Fractional hydrodynamic equations for fractal media

We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the “fractional” continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered.     

Pattern analysis of fractal properties in multilayer systems with metamaterials

Fractal pattern analysis is performed for aperiodicmultilayer systems with metamaterials. Morphological features of pattern formation in optical characteristics of structures with different geometry are revealed having regard to the manifestation of the phase-compensation effect and presence of metamaterial layers.     

Master equations for coupled systems

A formalism to cope with the problem of dynamically coupled systems is developed. A time-dependent projection operator of the type given by Willis-Picard and Grabert-Weidlich is used to derive a time-convolutionless master equation from the Liouville equation for the total composite system. A systematic perturbational expansion formula with respect to the interaction between systems is also given. Finally, the comparison with the usual non-Markoffian master equation is discussed.     

Method for Lippmann-Schwinger equations

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind is extended to singular equations of the Lippmann-Schwinger type. Numerical calculations of on- and off-shell t-matrix elements are carried out at positive and negative energies, for the ¹S₀ Reid soft-core nucleon-nucleon potential. Satisfactory convergence is achieved, even with the simplest version of the s     

Hamilton's equations for constrained dynamical systems

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.     

Reaction-diffusion equations for interacting particle systems

We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ^–2 the macroscopic density, defined on spatial scale ^–1, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.This work was supported by NSF Grant DMR81-14726-02.Partially supported by CNR.Partially supported by CNPq Grant No. 201682-83.     

Coarse-grained kinetic equations for quantum systems

The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τ_stoch of the stochastic process is much lower than the characteristic time τ_steady of the establishment of the system steady state populations, then on the time scale Δt ～ τ_steady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.     

Effective equations for disordered one-dimensional systems

Using the method of supersymmetry, effective equations are derived for the one-particle Green's function of various one-dimensional disordered models. As an example, explicit expressions for the density of states and the localization length are derived for the two-band model of a one-dimensional semiconductor.     

Solutions of Hartree-Fock equations for Coulomb systems

This paper deals with the existence of multiple solutions of Hartree-Fock equations for Coulomb systems and related equations such as the Thomas-Fermi-Dirac-Von Weizäcker equation.Partially supported by CEA-DAM     

Using the Exterior Matrix Method to find eigenfrequencies when the governing equations are systems of first-order equations

In this paper, we will apply the Exterior Matrix Method to the case where the governing equations for one piece of the structure is a system of 4 first-order equations. This generalizes a result when the governing equation was a single fourth-order equation. Ironically, the governing equations do not have to be solved in order to find the corresponding exterior matrix, and the exterior matrices can be used to find the eigenfrequencies of the system, even if there are dissipative joints added to the system. We will first look at the well-understood example of Euler–Bernoulli beams to illustrate the concept, and then move on to the more difficult inclined cable problem.     

Binomial moment equations for stochastic reaction systems

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations.     

Born-Green-Yvon equations for strongly interacting Fermi systems

A subset of the Fermi-Born-Green-Yvon equations for extended Fermi systems is investigated numerically. We present results for nuclear matter, neutron matter, and normal liquid³He at zero temperature. A comparison with the (alternative) Fermi-Hypernetted-Chain theory demonstrates the Fermi-Born-Green-Yvon equations to be of comparable accuracy.