arXiv subjects

Bruce N. Miller

Publications and source records attributed to Bruce N. Miller.

At least 19 recordsLinked to original sources

arXiv2020

Structure formation in a Dirac-Milne universe: comparison with the standard cosmological model

The presence of complex hierarchical gravitational structures is one of the main features of the observed universe. Here, structure formation is studied both for the standard ($Λ\rm CDM$) cosmological model and for the Dirac-Milne universe, a matter-antimatter symmetric universe that was {\gm recently} proposed as an alternative "coasting" cosmological scenario. One-dimensional numerical simulations reveal the analogies and differences between the two models. Although structure formation is faster in the Dirac-Milne universe, both models predict that it ends shortly before the present epoch, at cosmological redshift $z \approx 3$ for the Dirac-Milne cosmology, and at $z \approx 0.5$ for the $Λ\rm CDM$ universe. The present results suggest that the matter power spectrum observed by the Sloan Digital Sky Survey might be entirely due to the nonlinear evolution of matter and antimatter domains of relatively small initial dimensions, of the order of a few tens of parsecs comoving at cosmological redshift $z =1080$.

astro-ph.CO↗

arXiv2016

Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions

We study the thermodynamical properties of a one-dimensional gas with one-dimensional gravitational interactions, and placed in a uniform mass background. Periodic boundary conditions are implemented as a modification of the potential consisting of a sum over mirror images (Ewald sum), regularized with an exponential cut-off. The system has a phase transition at a critical temperature. Above the critical temperature the gas density is uniform, while below the critical point the system becomes inhomogeneous. Numerical simulations of the model confirms the existence of the phase transition, and are in good agreement with the theoretical results.

cond-mat.stat-mech↗

arXiv2016

Lyapunov spectra of Coulombic and gravitational periodic systems

We compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact time evolution of tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov-entropy density for each system at different degrees of freedom. Our approach forms an effective and approximation-free tool toward studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in large versions of the spatially periodic systems.

nlin.CD↗

arXiv2016

Dynamics of Coulombic and Gravitational Periodic Systems

We study the dynamics and the phase-space structures of Coulombic and self-gravitating versions of the classical one-dimensional 3-body system with periodic boundary conditions. We demonstrate that such a 3-body system may be reduced isomorphically to a spatially periodic system of a single particle experiencing a two-dimensional potential on a rhombic plane. For the case of both Coulombic and gravitational versions, exact expressions of the Hamiltonian have been derived in rhombic coordinates. We simulate the phase-space evolution through an event-driven algorithm that utilizes analytic solutions to the equations of motion. The simulation results show that the motion exhibits chaotic, quasiperiodic, and periodic behaviors in segmented regions of the phase space. While there is no evidence of global chaos in either the Coulombic or the gravitational system, the former exhibits a transition from a completely non-chaotic phase space at low energies to a mixed behavior. Gradual yet striking transitions from mild to intense chaos are indicated with changing energy, a behavior that differentiates the spatially periodic systems studied in this paper from the well-understood free-boundary versions of the 3-body problem. Our treatment of the 3-body systems is the first one of its kind and opens avenues for analysis of the dynamical properties exhibited by spatially periodic versions of various classes of systems studied in plasma and gravitational physics as well as in cosmology.

nlin.CD↗

arXiv2015

Cosmology in One Dimension: The Two Component Universe

We investigate structure formation in a one dimensional model of a matter-dominated universe using a quasi-newtonian formulation. In addition to dark matter, luminous matter is introduced to examine the potential bias in the distributions. We use multifractal analysis techniques to identify structures, including clusters and voids. Both dark matter and luminous matter exhibit fractal geometry as the universe evolves over a finite range. We present the results for the generalized dimensions computed on various scales for each matter distribution. We compare and contrast the fractal dimensions of two types of matter for the first time and show how dynamical considerations cause them to differ.

astro-ph.CO↗

arXiv2015

A Three Dimensional Gravitational Billiard in a Cone

Billiard systems offer a simple setting to study regular and chaotic dynamics. Gravitational billiards are generalizations of these classical billiards which are amenable to both analytical and experimental investigations. Most previous work on gravitational billiards has been concerned with two dimensional boundaries. In particular the case of linear boundaries, also known as the wedge billiard, has been widely studied. In this work, we introduce a three dimensional version of the wedge; that is, we study the nonlinear dynamics of a billiard in a constant gravitational field colliding elastically with a linear cone of half angle $θ$. We derive a two-dimensional Poincaré map with two parameters, the half angle of the cone and $\ell$, the $z$-component of the billiard's angular momentum. Although this map is sufficient to determine the future motion of the billiard, the three-dimensional nature of the physical trajectory means that a periodic orbit of the mapping does not always correspond to a periodic trajectory in coordinate space. We demonstrate several integrable cases of the parameter values, and analytically compute the system's fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of $\ell$ the conic billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase $\ell$ the dynamics becomes on the whole less chaotic, and the correspondence with the wedge billiard is lost.

nlin.CD↗

arXiv2015

Fractal Dimension Computation From Equal Mass Partitions

While the numerical methods which utilizes partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets, two mass- oriented methods are investigated by applying them to the one-dimensional generalized Cantor set. We show that both mass-oriented methods generate relatively good results for generalized dimensions for important cases where the box-counting method is known to fail. Both the strengths and limitations of the methods are also discussed.

physics.data-an↗

arXiv2015

Cosmology in one dimension: Symmetry role in dynamics, mass oriented approaches to fractal analysis

The distribution of visible matter in the universe, such as galaxies and galaxy clusters, has its origin in the week fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the universe, with its galaxies, clusters and super-clusters of galaxies indicates the absence of a natural length scale. In the Newtonian formulation, numerical simulations of a one-dimensional system permit us to precisely follow the evolution of an ensemble of particles starting with an initial perturbation in the Hubble flow. The limitation of the investigation to one dimension removes the necessity to make approximations in calculating the gravitational field and, on the whole, the system dynamics. It is then possible to accurately follow the trajectories of particles for a long time. The simulations show the emergence of a self-similar hierarchical structure in both the phase space and the configuration space and invites the implementation of a multifractal analysis. Here, after showing that symmetry considerations leads to the construction of a family of equations of motion of the one-dimensional gravitational system, we apply four different methods for computing generalized dimensions $D_q$ of the distribution of particles in configuration space. We first employ the conventional box counting and correlation integral methods based on partitions of equal size and then the less familiar nearest-neighbor and k-neighbor methods based on partitions of equal mass. We show that the latter are superior for computing generalized dimensions for indices $q<-1$ which characterize regions of low density.

astro-ph.CO↗

arXiv2014

Regular and Chaotic Dynamics of a Piecewise Smooth Bouncer

The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time i.e., the phenomenon of inelastic collapse. We address these so-called "sticking solutions" and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.

nlin.CD↗

arXiv2014

Chaotic Dynamics of One-Dimensional Systems with Periodic Boundary Conditions

We provide appropriate tools for the analysis of dynamics and chaos for one-dimensional systems with periodic boundary conditions. Our approach allows for the investigation of the dependence of the largest Lyapunov exponent on various initial conditions of the system. The method employs an effective approach for defining the phase-space distance appropriate for systems with periodic boundary and allows for an unambiguous test-orbit rescaling in the phase space required to calculate the Lyapunov exponents. We elucidate our technique by applying it to investigate the chaotic dynamics of a one-dimensional plasma with periodic boundary. Exact analytic expressions are derived for the electric field and potential using Ewald sums thereby making it possible to follow the time-evolution of the plasma in simulation without any special treatment of the boundary. By employing a set of event-driven algorithms, we calculate the largest Lyapunov exponent, the radial distribution function and the kinetic pressure by following the evolution of the system in phase space without resorting to numerical manipulation of the equations of motion. Simulation results are presented and analyzed for the one-dimensional plasma with a view to examining the dynamical and chaotic behavior exhibited by small and large versions of the system.

nlin.CD↗

arXiv2014

Path integral Monte Carlo on a lattice: extended states

The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the quantum particle and atoms are restricted to the sites of a one-dimensional lattice. A path-integral formalism is developed within the context of the canonical ensemble in which the quantum particle is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential $ε$, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.

quant-ph↗

arXiv2011

Dynamics of an Inelastic Gravitational Billiard with Rotation

The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola; i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The value of the coefficient of restitution employed in the model resulted in good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.

nlin.CD↗

arXiv2011

Dynamic Modeling and Simulation of a Real World Billiard

Gravitational billiards provide an experimentally accessible arena for testing formulations of nonlinear dynamics. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. Direct comparisons are made between the model's predictions and previously published experimental data. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.

nlin.CD↗

arXiv2010

Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation

Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so, over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position-velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. The methodology employed includes: 1) The derivation of explicit solutions for the gravitational potential and field for a one-dimensional system with periodic boundary conditions (Ewald sums for one dimension); 2) The development of a procedure for obtaining scale-free initial conditions for the growing mode in phase space for an arbitrary power-law index; 3) The evaluation of power spectra, correlation functions, and generalized fractal dimensions at different stages of the system evolution. It is shown that a simple analytic representation of the power spectra captures the main features of the evolution, including the correct time dependence of the crossover from the linear to nonlinear regime and the transition from regular to fractal geometry. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.

astro-ph.CO↗

arXiv2010

Ewald Sums for One Dimension

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the system evolution and show that it's possible to construct an efficient algorithm for carrying out simulations. In the cosmological setting we show that two approaches for satisfying periodic boundary conditions, one overly specified and the other completely general, provide a nearly identical clustering evolution until the number of clusters becomes small, at which time the influence of any size-dependent boundary cannot be ignored. Finally we compare the results with other recent work with the hope of providing clarification over differences these issues have induced. We explain that modern formulations of physics require a well defined potential which is not available if the forces are screened directly.

math-ph↗

arXiv2010

Solution of the Quantum Initial Value Problem with Transparent Boundary Conditions

Physicists have used billiards to understand and explore both classical and quantum chaos. Recently, in 2001, a group at the University of Texas introduced an experimental set up for modeling the wedge billiard geometry called optical billiard in two dimensions. It is worth mentioning that this experiment is more closely related with classical rather than quantum chaos. The motivation for the present work was born from the idea of laying the foundations of a quantum treatment for optical billiards, named "The Escape Problem", by presenting the concept of a Transparent Boundary Condition. We consider a gas of particles initially confined to a one dimensional box of length L, that are permitted to escape. We find the solution of a Quantum Initial Value Problem using a numerical method developed and entirely checked with an exact, analytic method. The numerical method introduces a novel way to solve a Diffusion Type Equation by implementing Discrete Transparent Boundary Conditions recently developed by mathematicians.

physics.comp-ph↗

arXiv2008

A Minimal Model for the Study of Polychronous Groups

A minimal model of polychronous groups in neural networks is presented. The model is computationally efficient and allows the study of polychronous groups independent of specific neuron models. Computational experiments were performed with the model in one- and two-dimensional neural architectures to determine the dependence of the number of polychronous groups on various connectivity options. The possibility of using polychronous groups as computational elements is also discussed.

cond-mat.dis-nn↗

arXiv2007

Development of Fractal Geometry in a 1+1 Dimensional Universe

Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of a group of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the $μ$ (position, velocity) space. The observed properties are simply a shadow (projection) of higher dimensional structure.

math-ph↗