Using computer algebra to run Einstein's equations "backward", from field to source rather than from source to field, we design an artificial gravity field for a spaceship. Everywhere inside astronauts experience normal Earth gravity, while outside they float freely. The stress-energy that generates the field contains exotic matter of negative energy density but also relies importantly on shears and pressures. The same techniques can be readily used to design other interesting spacetimes and thereby elucidate the connection between the source and field in general relativity.
The quest for quantum gravity has been long and difficult. Causal Dynamical Triangulation is a new and straightforward approach to quantum gravity that recovers classical spacetime at large scales by enforcing causality at small scales. CDT combines quantum physics with general relativity in a Feynman sum-over-geometries and converts the sum into a discrete statistical physics problem. We solve this problem using a new Monte Carlo simulation to compute the spatial fluctuations of an empty universe with one space and one time dimensions. Our results compare favorably with theory and provide an accessible but detailed introduction to quantum gravity via a simulation that runs on a laptop computer.
One-way or unidirectional coupling is a striking example of how topological considerations — the parity of an array of multistable elements combined with periodic boundary conditions — can qualitatively influence dynamics. Here we introduce a simple electronic model of one-way coupling in one and two dimensions and experimentally compare it to an improved mechanical model and an ideal mathematical model. In two dimensions, computation and experiment reveal richer one-way coupling phenomenology: in media where two-way coupling would dissipate all excitations, one-way coupling enables soliton-like waves to propagate in different directions with different speeds.
We describe the theory, design, and construction of simple electromechanical devices that automatically and continually track celestial objects. As Earth rotates and revolves, a star tracker always points at a star or other object fixed to the celestial sphere, such as the center of the Milky Way galaxy. A planet tracker can fixate on any celestial object, including planets, the Sun, or the Moon. A sidereal clock mechanism drives the star tracker, and software which encoding astronomical algorithms controls an inexpensive robot that drives the planet tracker. The star tracker acts like a gyroscope, rigidly oriented in space, despite Earth's motion. Both trackers indicate the passing of time just like clocks and calendars. The resulting lecture, hallway, or museum displays promote awareness of and excitement about our place in the universe.
We study the classical dynamics of two bodies, a massive line segment or slash (/) and a massive point or dot (.), interacting gravitationally. For this slashdot (/.) body problem, we derive algebraic expressions for the force and torque on the slash, which greatly facilitate analysis. The diverse dynamics include a stable synchronous orbit, generic chaotic orbits, sequences of unstable periodic orbits, spin stabilized orbits, and spin-orbit coupling that can unbind the slash and dot. The extension of the slash provides an extra degree of freedom that enables the interplay between rotation and revolution. In this way, the slashdot body problem exhibits some of the richness of the three body problem with only two bodies and serves as a valuable prototype for more realistic systems. Applications include the dynamics of asteroid-moonlet pairs and asteroid rotation and escape rates.
Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum world. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its surroundings. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the latter case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.
We have experimentally realized unidirectional or one-way coupling in a mechanical array by powering the coupling with flowing water. In cyclic arrays with an even number of elements, soliton-like waves spontaneously form but eventually annihilate in pairs, leaving a spatially alternating static attractor. In cyclic arrays with an odd number of elements, this alternating attractor is topologically impossible, and a single soliton always remains to propagate indefinitely. Our experiments with 14 and 15-element arrays highlight the dynamical importance of both noise and disorder and are further elucidated by our computer simulations.
A surprising number of physics problems are well suited to "embarrassingly parallel" computations that do not require complicated software algorithms or specialized hardware. As faculty and students at small institutions, we are readily incorporating parallel computing in diverse levels of our curricula, and we are embracing the opportunity to utilize high performance computing to attack contemporary research problems in summer research, senior theses, and course work. This article describes how we do this in three significant examples: spatiotemporal patterns of one-way coupled oscillators, ray-tracing in curved spacetime, and solar escape as a three-body problem.
We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.
Recent work has demonstrated that undriven and overdamped bistable systems, which are normally quiescent, can oscillate if unidirectionally coupled into arrays with cyclic boundary conditions. Here, we understand such oscillations as corresponding to the propagation of soliton-like waves. Further, in large arrays, we demonstrate how noise and coupling, together, mediate the resulting complex spatiotemporal dynamics.
Recently, we have shown the emergence of oscillations in overdamped undriven nonlinear dynamic systems subject to carefully crafted coupling schemes and operating conditions. Here, we present experimental results obtained on a system of N = 3 coupled ferromagnetic cores, the underpinning of a "coupled-core fluxgate magnetometer" (CCFM); the oscillatory behavior is triggered when the coupling constant exceeds a threshold value (bifurcation point), and the oscillation frequency exhibits a characteristic scaling behavior with the "separation" of the coupling constant from its threshold value, as well as with an external target DC magnetic flux signal. The oscillations, which can be induced at frequencies ranging from a few Hz to high-kHz, afford a novel new detection scheme for weak target magnetic signals; in addition, the coupled-core system offers the possibility of enhanced responsivity over its conventional (single fluxgate) counterpart. We also present the first (numerical) results on noisy cooperative behavior in this system.
Noise and coupling can optimize the response of arrays of nonlinear elements to periodic signals. We analyze such array-enhanced stochastic resonance (AESR) using finite-state transition rate models. We simply derive the transition rate matrices from the underlying potential energy function of the corresponding Langevin problem. Our implementation exploits Floquet theory and provides useful theoretical and numerical tools. Our framework both facilitates analysis and elucidates the mechanism of AESR. In particular, we show how sub-linear coupling diminishes AESR, but super-linear coupling enhances it.
In transducing mechanical stimuli into electrical signals, at least some hair cells in vertebrate auditory and vestibular systems respond optimally to weak periodic signals at natural, non-zero noise intensities. We understand this stochastic resonance by constructing a faithful mechanical model reflecting the hair cell geometry and described by a nonlinear stochastic differential equation. This Langevin description elucidates the mechanism of hair cell stochastic resonance while supporting the hypothesis that noise plays a functional role in hearing.
Josephson junction arrays provide an ideal physical realization for studying the complex dynamics of the sort found in sandpile models. They provide a means of separately investigating the dual physical effects of nonlinearity and disorder, and hold promise as an example for establishing a rigorous connection between the governing differential equations and the corresponding cellular automaton.
We investigate generalized seeding of the attracting states of Abelian sandpile automata and find there exists a class of global perturbations of such automata that are completely removed by the natural local dynamics. We derive a general form for such self-erasing perturbations and demonstrate that they can be highly nontrivial. This phenomenon provides a new conceptual framework for studying such automata and suggests possible applications for data protection and encryption.
We study a cellular automaton derived from the phenomenon of magnetic flux creep in two-dimensional granular superconductors. We model the superconductor as an array of Josephson junctions evolving according to a set of coupled ordinary differential equations. In the limit of slowly increasing magnetic field, we reduce these equations to a simple cellular automaton. The resulting discrete dynamics, a stylized version of the continuous dynamics of the differential equations, is equivalent to the dynamics of a gradient sand pile automaton. We study the dynamics as we vary the symmetry of the underlying lattice and the shape of its boundary. We find that the "simplest" realization of the automaton, on a square lattice with commensurate boundaries, results in especially simple dynamics, while "generic" realizations exhibit more complicated dynamics characterized by statistics with broad distributions, even in the absence of noise or disorder.
We analyze solar escape as a special case of the restricted three-body problem. We systematically vary the parameters of our model solar system to show how optimal launch angle and minimum escape speed depend on the mass and size of Earth. In some cases, it is best to launch near the direction of Earth's motion, but slightly outward; in other cases, it is best to launch near the perpendicular to Earth's motion, but inward, toward Sun (so as to obtain a solar gravity assist). Between direct escapes for high launch speeds and trapped trajectories for low launch speeds is an irregular band of chaotic orbits that reveals something of the true complexity of solar escape and the three-body problem.
As dynamical models, cellular automata sometimes provide compelling alternatives to differential equations. In addition to rapid simulations, their stylized dynamics may elucidate the essence of the underlying physics. In this letter, we demonstrate the efficacy with which cellular automata can model spatiotemporal nonlinear dynamics. We explicitly construct and carefully test an automaton that faithfully simulates an array of slowly torqued, heavily damped pendulums. This example is representative of a variety of systems having two time scales (slow driving followed by fast readjustment) that naturally discretize in the singular limit where the ratio of the time scales vanishes.
We present a simple nonlinear system that exhibits multiple distinct stochastic resonances. By adjusting the noise and coupling of an array of underdamped, monostable oscillators, we modify the array's natural frequencies so that the spectral response of a typical oscillator in an array of N oscillators exhibits N - 1 different stochastic resonances. Such families of resonances may elucidate and facilitate a variety of noise-mediated cooperative phenomena, such as Noise Enhanced Propagation, in a broad class of similar nonlinear systems.
External feedback can enhance (or depress) the response of a noisy bistable system to monochromatic signals, significantly magnifying its natural stochastic resonance. We compare and contrast a variety of such feedback strategies, using both numerical simulations and analog electronic experiments. These noninvasive control techniques are especially valuable for noisy bistable systems that are difficult or impossible to modify internally.
By adding constant-amplitude pulses to a noisy bistable system, we enhance its response to monochromatic signals, significantly magnifying its unpulsed stochastic resonance. We observe the enhancement in both numerical simulations and in analog electronic experiments. This simple noninvasive control technique should be especially useful in noisy bistable systems that are difficult or impossible to modify internally.
Recently, Sinha and Ditto [Phys. Rev. Lett. 81, (1998) 2156] demonstrated the computational possibilities of an array of coupled maps. We generalize this nonlinear dynamical system to improve its computational usefulness. We then consider a second nonlinear system, a parameterized map, and use it to illustrate why logic requires nonlinearity.
We designed and constructed an array of ten forced damped nonlinear pendulums. We drove the pivot of the pendulums in a circle and torsionally coupled them with springs. We analyzed the motion using digitized videotape. The behavior of the real array closely mirrored the behavior of its computer simulation. For a homogeneous array of identical pendulums, the spatiotemporal dynamics was chaotic; for a heterogeneous array of nonidentical pendulums, the spatiotemporal dynamics was periodic. Such temporally fixed but spatially varying chaos control has been called "disorder taming chaos".
We use noise to extend signal propagation in one and two-dimensional arrays of two-way coupled bistable oscillators. In a numerical model, we sinusoidally force one end of a chain of noisy oscillators. We record a signal-to-noise ratio at each oscillator. We demonstrate that moderate noise significantly extends the propagation of the sinusoidal input. Both the optimal noise and the maximum propagation length scale like the square root of the coupling. We obtain similar results with two-dimensional arrays. The simplicity of the model suggests the generality of the phenomenon.
We collide rods of different lengths and infer the vibrational motion of the longer rod by a spectral analysis of the resulting sound. Collisions of rods of square and circular cross section are audibly different. While longitudinal modes of vibration do not discriminate between different cross sectional shapes, flexural modes do, and these enable us to hear the shapes of the rods. We use a microphone, an amplifier, and a spectrum analyzer to observe the longitudinal and flexural modes of the ringing rod. With an accessible mathematical model and a simple apparatus, we obtain good agreement between theory and experiment.
Recent studies suggesting evidence for determinism in the stochastic activity of the heart and brain have sparked an important scientific debate: Do biological systems exploit chaos or are they merely noisy? Here, we analyze the spike interval statistics of a simple integrate-and-fire model neuron to investigate how a real neuron might process noise and chaos, and possibly differentiate between the two. In some cases, our model neuron readily distinguishes noise from chaos, even discriminating among chaos characterized by different Lyapunov exponents. However, in other cases, the model neuron does not decisively differentiate noise from chaos. In these cases, the spectral content of the input signal may be more significant than its phase space structure, and higher-order spectral characterizations may be necessary to distinguish its response to chaotic or noisy inputs.
We study a coupled array of torqued damped nonlinear pendulums. Disordering this system can convert chaotic spatiotemporal evolution into periodic motion. Here, in numerical experiments, we elucidate and quantify this phenomenon. For each of several types of disorder, we find an optimal magnitude of disorder which minimizes the system's largest Lyapunov exponent.
Recently, the synchronization and signal processing ability of a locally and linearly coupled array of bistable elements was enhanced by the addition of uncorrelated noise [J.F. Lindner, B.K. Meadows, W.L. Ditto, M.E. Inchiosa, and A.R. Bulsara, Phys. Rev. Lett. 75 , 3 (1995)]. Here, we detail the performance of such an array as a function of both coupling and noise. Simple theoretical arguements, grounded in extensive numercial studies, suggest how to "tune" the array for best synchronization and signal-to-noise ratio. Specifically, we propose that, for large array size N, the optimal coupling scales like N2 and the optical noise variance scales like N. This scaling matches the coupling-induced correlation length to the array length and the noise-generated mean hopping time to the modulation period, thereby creating a stochastic resonance in space and time.
Techniques to remove, suppress, and control the chaotic behavior of nonlinear systems are reviewed. Analysis of a forced damped nonlinear oscillator provides a brief overview of the relevant nonlinear dynamics of dissipative systems. Various techniques for suppression and control of chaos are then outlined, compared and contrasted. A unified mathematical notation facilitates the comparison. The successes of each strategy in numerical simulations and physical experiments are carefully noted. Their strengths and weaknesses are analyzed, and they are evaluated according to whether they employ feedback, are goal-oriented, are model-based, merely remove chaos — or truly exploit it. An elementary derivation of the important OGY control equation is supplied. Critical references provide an entry into the literature. It is argued that nonlinearity can be a real-world advantage, and it is hoped that this review will serve as a summary of, and invitation to, the nascent field of nonlinear design.
Disorder and noise in physical systems usually tend to destroy spatial and temporal regularity, but recent research into nonlinear systems provides intriguing couter-examples. In the phenomenon of stochastic resonance, for example, the presence of noise improves the abiliity of some nonlinear systems to transfer information reliably. Noise can also remove chaos in a model oscillator, and facilitate synchronization in an extended array of bistable elements. Here we explore the use of disorder as a means to control spatiotemporal chaos in coupled arrays of forced, damped, nonlinear oscillators. Chaotic behaviour in spatially extended systems, especially in biology and physiology, might be amenable to control, as occurs in low-dimensional temporally chaotic systems. In our numerical experiments, one-and two-dimensional arrays of identical oscillators behave chaotically, but the introduction of slight, uncorrelated differences between the oscillators induces ordered motion characterized by complex but regular spatiotemporal patterns.
The concepts of chaos and its control are reviewed. Both are discussed from an experimental as well as a theoretical viewpoint. Examples are then given of the control of chaos in a diverse set of experimental systems. Current and future applications are discussed.
We enhance the response of a "stochastic resonator" by coupling it into a chain of identical resonators. Specifically, we show via numerical simulation that local linear coupling of overdamped nonlinear oscillators significantly enhances the signal-to-noise ratio of the response of a single oscillator to a time-periodic signal and noise. We relate this array enhanced stochastic resonance to the global spatiotemporal dynamics of the array and show how noise, coupling, and bistable potential cooperate to organize spatial order, temporal periodicity, and peak signal-to-noise ratio.
In 1987, Bak, Tang, and Wiesenfeld introduced the notion of Self-Organized Criticality (SOC) in the guise of a computer simulation: a "sand pile Cellullar Automaton Machine". They supposed that a real, many-bodied, physical system in an external field assembles itself into a critical state. The system then relaxes about the critical state creating spatial and temporal self-similarities which give rise to fractal objects and 1/f noise. Their computer modeling was of a system like a sand pile at its critical angle of repose. This area provides a new paradigm for many-body dynamics. Understanding SOC may well allow substantial strides to occur in the theory of flow and transport. The simplest model system, one for which computer simulations and corresponding real experiments are feasible, is a "sand pile" near its critical angle of repose. The size and duration of avalanches occurring as subsequent "sand" grains are added can provide detailed information about the "sand pile" as a model of SOC, and for SOC as a basis for many-body dynamics. This article describes a fairly simple, junior-level experiment in this new field involving the measurement of the distribution of avalanche sizes which occur as grains of sand are added to a "sand pile". The universality of the phenomenon can be observed and a power law relationship can be deduced.
Stochastic Resonance is a statistical phenomenon that has been observed in periodically modulated, noise-driven, bistable systems. The characteristic signatures of the effect include an increase in the signal-to-noise of the output as noise is added to the system, and exponentially decreasing peaks in the probabilbity density as a function of residence times in one state. Presented are the results of a numerical simulation where these same signatures where observed by adding a chaotic driving term instead of a white noise term. Although the probability distributions of the noise and chaos inputs were significantly different, the stochastic and chaotic resonances were equal within the experimental error.