Collision Rules Research Articles

Size Matters: Logarithmic Space is Real Time

We show that all the problems solvable by a nondeterministic machine with logarithmic work space (NL) can be solved in real time by a parallel machine, no matter how tight the real-time constraints are. We also show that several other real-time problems are in effect solvable in nondeterministic logarithmic space once their real-time constraints are dropped and they become non-realtime. We thus conjecture that NL contains exactly all the problems that admit feasible real-time parallel algorithms. The issue of real-time optimization problems is also investigated. We identify the class of such problems that are solvable in real time. In the process, we determine the computational power of directed reconfigurable multiple bus machines (DRMBMs) with polynomially bounded resources and running in constant time, which is found to be the same as the power of directed reconfigurable networks with the same properties. We also show that write conflict resolution rules such as Priority or even Combining do not add computational power over the Collision rule, and that a bus of width 1 (a wire) suffices for any constant time computation on DRMBM.

Networks based on collisions among mobile agents

We investigate in detail a recent model of colliding mobile agents [M.C. González, P.G. Lind, H.J. Herrmann, Phys. Rev. Lett. 96 (2006) 088702. cond-mat/0602091], used as an alternative approach for constructing evolving networks of interactions formed by collisions governed by suitable dynamical rules. The system of mobile agents evolves towards a quasi-stationary state which is, apart from small fluctuations, well characterized by the density of the system and the residence time of the agents. The residence time defines a collision rate, and by varying this collision rate, the system percolates at a critical value, with the emergence of a giant cluster whose critical exponents are the ones of two-dimensional percolation. Further, the degree and clustering coefficient distributions, and the average path length, show that the network associated with such a system presents non-trivial features which, depending on the collision rules, enables one not only to recover the main properties of standard networks, such as exponential, random and scale-free networks, but also to obtain other topological structures. To illustrate, we show a specific example where the obtained structure has topological features which characterize the structure and evolution of social networks accurately in different contexts, ranging from networks of acquaintances to networks of sexual contacts.

Generic features of the wealth distribution in ideal-gas-like markets

We provide an exact solution to the ideal-gas-like models studied in econophysics to understand the microscopic origin of Pareto law. In these classes of models the key ingredient necessary for having a self-organized scale-free steady-state distribution is the trading or collision rule where agents or particles save a definite fraction of their wealth or energy and invest the rest for trading. Using a Gibbs ensemble approach we could obtain the exact distribution of wealth in this model. Moreover we show that in this model (a) good savers are always rich and (b) every agent poor or rich invests the same amount for trading. Nonlinear trading rules could alter the generic scenario observed here.

Microscopic dynamics of thin hard rods

We analyze the microscopic dynamics and transport properties of a gas of thin hard rods. Based on the collision rules for hard needles we derive a hydrodynamic equation that determines the coupled translational and rotational dynamics of a tagged thin rod in an ensemble of identical rods. Specifically, based on a pseudo-Liouville operator for binary collisions between rods, the Mori-Zwanzig projection formalism is used to derive a continued fraction representation for the correlation function of the tagged particle's density, specifying its position and orientation. Truncation of the continued fraction gives rise to a generalized Enskog equation, which can be compared to the phenomenological Perrin equation for anisotropic diffusion. Only for sufficiently large density do we observe anisotropic diffusion, as indicated by an anisotropic mean-square displacement, growing linearly with time. For lower densities, the Perrin equation is shown to be an insufficient hydrodynamic description for hard needles interacting via binary collisions. We compare our results to simulations and find excellent quantitative agreement for low densities and qualitative agreement for higher densities.

A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number

Standard methods of simulating the mean velocity and mean force in a rarefied gas flow suffer from statistical errors associated with the random thermal motion of the gas molecules. The ratio of the statistical error to the mean velocity or force diverges in the limit of small Mach number, which is relevant to applications in aerosol science and microdevices. We present a novel extension of the direct-simulation Monte Carlo method in which the deviation from equilibrium is captured by noninteger weightings of the simulation gas particles, so that the mean velocity and force associated with the deviation from equilibrium can be computed without contamination by the thermal noise in the equilibrium state. The collision rules between simulation particles reproduce the changes in the gas velocity distribution function that would result from hard sphere intermolecular collisions. Because the collision rules produce new particles, particle splitting and recombination steps, which do not alter the velocity distribution function, must be introduced to maintain a finite number of particles and a finite maximum weighting per particle. The method is validated by comparing simulation results for pressure-driven flow in a channel and flow past a single sphere to corresponding solutions of the linearized Boltzmann equation. The method is then applied to obtain new results for the resistance to the relative motion of two spherical particles along their line of centers.

Selection rules for laser-assisted electron-atom collisions with the laser field normal to the scattering plane

General selection rules for differential cross sections for laser-assisted electron-atom scattering are derived for the geometry in which the linearly polarized laser field is perpendicular to the scattering plane. Numerical calculations using the $R$-matrix Floquet theory are presented for electron-helium scattering in order to discuss the possibility of observing these selection rules experimentally.

Simulation of break-up behavior of immiscible droplet under shear field by means of continuous-velocity lattice gas cellular automata

Continuous-velocity lattice gas cellular automata method is applied to the simulation of the motion of a single droplet under a three-dimensional shear field. Immiscible binary fluid model with segregation intensity parameter is employed. A simple collision rule is introduced in order to describe the momentum transport from moving boundaries to the fluid. The calculations well simulate two types of the deformation and break-up behaviors, fracture and tipstreaming, of the droplet. The segregation intensity parameter of the fluid particles reasonably controls the surface tension of the droplet. The break-up behavior of the droplet is well correlated with the ratio of the shear strength and the segregation intensity parameter.

The inelastic hard dimer gas: A nonspherical model for granular matter

We study a two-dimensional gas of inelastic smooth hard dimers. Since the collisions between dimers are dissipative, being characterized by a coefficient of restitution alpha<1, and no external driving force is present, the energy of the system decreases in time and no stationary state is achieved. However, the resulting nonequilibrium state of the system displays several interesting properties in close analogy with systems of inelastic hard spheres, whose relaxational dynamics has been thoroughly explored. We generalize to inelastic systems a recently method introduced [G. Ciccotti and G. Kalibaeva, J. Stat. Phys. 115, 701 (2004)] to study the dynamics of rigid elastic bodies made up of different spheres held together by rigid bonds. Each dimer consists of two hard disks of diameter d, whose centers are separated by a fixed distance a. By describing the rigid bonds by means of holonomic constraints and deriving the appropriate collision rules between dimers, we reduce the dynamics to a set of equations which can be solved by means of event-driven simulation. After deriving the algorithm we study the decay of the total kinetic energy, and of the ratio between the rotational and the translational kinetic energy of inelastic dimers. We show numerically that the celebrated Haff's homogeneous cooling law t(-2), describing how the kinetic energy of an inelastic hard-sphere system with a constant coefficient of restitution decreases in time, holds even in the case of these nonspherical particles. We fully characterize this homogeneous decay process in terms of appropriate decay constants and confirm numerically the scaling behavior of the velocity distributions.

Stationary States and Energy Cascades in Inelastic Gases

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.

Development of New Simulation Method for Flow Behavior of Granular Materials in a Blast Furnace Using Cellular Automaton

A new simulation method that permits the large-scale of simulation of coke and granular flow in a blast furnace was developed using cellular automata. The automaton rule consists of the transition rule of constituent particles and interaction rule between particles. The interaction rule is divided into three rules; the collision rule, the static contact rule between particles, and propagation rule of impulsive force through the particle bed. The interaction rule was derived quantitatively on the basis of microscopic information on the interactive force between flowing particles under gravity obtained by DEM simulation. The evolution of granular flow with time can be simulated by defining the equivalent time to one step in automaton simulation, and the state variables in granular flow can be obtained by the proposed simulation method. The validity of the proposed powder cellular automaton method was confirmed by comparison of the simulated flow pattern of discharging flow from a hopper with experimental one. Granular flow of iron ore and coke particles in a blast furnace can be simulated with high speed by the proposed automaton method.

Repos ou mouvement conspirant : Leibniz et les articles 54 et 55 de la partie II des Principia philosophiae / Rest or conspiring motion : Leibniz and articles 54 and 55 of the second part of the Principia philosophiae

SUMMARY. — Leibniz' early annotations on Cartesian physics, as expounded in the Principia, were restricted to criticism of the rules of collision and do not take into account Descartes' elaborations on the state of matter or the qualities pertaining to the cohesion of bodies. It was only later that Leibniz associated the rejection of the rules of collision with a detailed refutation of the concepts of hardness and fluidity specific to Cartesian physics. This paper aims to examine Leibniz' critique as fully developed.

Collision Rules For High-Speed Craft

Re-visiting the requirement to introduce amended collision regulations to meet the developing threat in encounters between high-speed craft and slower, mainly smaller vessels. This paper looks at the requirements of different types of encounter and the ability to identify high-speed craft through AIS. It introduces proposed changes to the Colregs that could help to reduce the perceived threat to small, slow craft.

Modelling the dynamics of a minimal protocell container

This paper is a discussion on how reaction kinetics and three-dimensional (3D) lattice simulations can be used to elucidate the dynamical properties of micelles as a possible minimal protocell container. We start with a general discussion on the role of molecular self-assembly in prebiotic and contemporary biological systems. A simple reaction kinetic model of a micellation process of amphiphilic molecules in water is then presented and solved analytically. Amphiphilic molecules are polymers with hydrophobic (water-fearing), e.g. hydrocarbon tail groups, and hydrophilic (water-loving) head groups, e.g. fatty acids. By making a few simplifying assumptions an analytical expression for the size distribution of the resulting micelles can be derived. The main part of the paper presents and discusses a lattice gas technique for a more detailed 3D simulation of molecular self-assembly of amphiphilic polymers in aqueous environments. Water molecules, hydrocarbon tail groups and hydrophilic head groups are explicitly represented on a three-dimensional discrete lattice. Molecules move on the lattice proportional to their continuous momentum. Collision rules preserve momentum and kinetic energy. Potential energy from molecular interactions are also included explicitly. The non-trivial thermodynamics of large-scale and long-time dynamics are studied. In this paper we specifically demonstrate how, from a random initial distribution, micelles are formed and grow until they destabilize and can divide. Eventually a steady state of growing and dividing micelles is formed. Towards the end of the paper we discuss the relevance of the presented results to the design of a minimal artificial protocell.

Nuclear spin selection rules in chemical reactions by angular momentum algebra

The detailed selection rules for reactive collisions reported by Quack using molecular symmetry group are derived by using angular momentum algebra. Instead of the representations of the permutation–inversion group for both nuclear spin and rovibronic coordinate wavefunctions, those of the rotation group for nuclear spin wavefunction only are used. The method allows more straightforward derivation of Quack’s results and further extension of the calculation for separating elementary reactions and application to higher proton systems.

Asymptotic Properties of the Inelastic Kac Model

We discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distributed like stable laws, and attract initial densities which belong to the normal domain of attraction. If the initial density is assumed of finite energy, with higher moments bounded, it is shown that the solution converges for large-time to a profile with power law tails. These tails are heavily dependent of the collision rule.

Transport coefficients for stochastic rotation dynamics in three dimensions.

Explicit expressions for the transport coefficients of a recently introduced stochastic model for simulating fluctuating fluid dynamics are derived in three dimensions by means of Green-Kubo relations and simple kinetic arguments. The results are shown to be in excellent agreement with simulation data. Two collision rules are considered and their computational efficiency is compared.

Cellular Automata for Simulating Molecular Self-Assembly

We present a lattice gas technique for simulating molecular self-assembly of amphiphilic polymers in aqueous environments. Water molecules, hydrocarbons tail-groups and amphiphilic head-groups are explicitly represented on a three dimensional discrete lattice. Molecules move on the lattice proportional to their continuous momentum. Collision rules preserve momentum and kinetic energy. Potential energy from molecular interactions are also included explicitly. Non-trivial thermodynamics of large scale and long time dynamics are studied. In this paper we specifically demonstrate how, from a random initial distribution, micelles are formed, and grow until they destabilize and divide. Eventually a steady state of growing and dividing micelles is formed.

Towards the simplest hydrodynamic lattice-gas model

It has been known since 1986 that it is possible to construct simple lattice-gas cellular automata whose hydrodynamics are governed by the Navier-Stokes equations in two dimensions. The simplest such model heretofore known has six bits of state per site on a triangular lattice. In this work, we demonstrate that it is possible to construct a model with only five bits of state per site on a Kagome lattice. Moreover, the model has a simple, deterministic set of collision rules and is easily implemented on a computer. In this work, we derive the equilibrium distribution function for this lattice-gas automaton and carry out the Chapman-Enskog analysis to determine the form of the Navier-Stokes equations.

MODELS OF FREE COOLING GRANULAR GASES

We consider the free evolution of systems of granular particles whose dynamics is characterized by a collision rule which preserves the total momentum, but dissipates the kinetic energy. Starting from an inelastic version of a minimal model proposed by Ulam for a gas of Maxwell molecules, we introduce a new lattice model aimed at investigating the role of dynamical correlations and the onset of spatial order induced by the inelasticity of the interactions. We study, in one- and two-dimensional cases, the velocity distribution, the decay of the energy, the formation of spatial structures and topological defects. Finally, we relate our findings to other models known in other fields.

Burgers turbulence initialized by a regenerative impulse

In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the corresponding velocity field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a large deviation type. First, we estimate the probability that a typical particle is located at time 1 at distance at least D from its initial position, when D tends to infinity. Its behaviour is related to the left tail of the gap measure of the regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution. Second, we analyse the event that a given particle stands still at time T as T tends to infinity. The data to which we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the shock structure.