Dynamical Degrees Of Freedom Research Articles

Lower-dimensional Hořava–Lifshitz gravity

We consider Horava-Lifshitz gravity in both 1+1 and 2+1 dimensions. These lower-dimensional versions of Horava-Lifshitz gravity are simple enough to be explicitly tractable, but still complex enough to be interesting. We write the most general (non-projectable) action for each case and discuss the resulting dynamics. In the 1+1 case we utilize the equivalence with 2-dimensional Einstein-aether theory to argue that, even though non-trivial, the theory does not have any local degrees of freedom. In the 2+1 case we show that the only dynamical degree of freedom is a scalar, which qualitatively has the same dynamical behaviour as the scalar mode in (non-projectable) Horava-Lifshitz gravity in 3+1 dimensions. We discuss the suitability of these lower-dimensional theories as simpler playgrounds that could help us gain insight into the 3+1 theory. As special cases we also discuss the projectable limit of these theories. Finally, we present an algorithm that extends the equivalence with (higher order) Einstein-aether theory to full Horava-Lifshitz gravity (instead of just the low energy limit), and we use this extension to comment on the apparent naturalness of the covariant formulation of the latter.

Invariant relationships deriving from classical scaling transformations

Because scaling symmetries of the Euler–Lagrange equations are generally not variational symmetries of the action, they do not lead to conservation laws. Instead, an extension of Noether's theorem reduces the equations of motion to evolutionary laws that prove useful, even if the transformations are not symmetries of the equations of motion. In the case of scaling, symmetry leads to a scaling evolutionary law, a first-order equation in terms of scale invariants, linearly relating kinematic and dynamic degrees of freedom. This scaling evolutionary law appears in dynamical and in static systems. Applied to dynamical central-force systems, the scaling evolutionary equation leads to generalized virial laws, which linearly connect the kinetic and potential energies. Applied to barotropic hydrostatic spheres, the scaling evolutionary equation linearly connects the gravitational and internal energy densities. This implies well-known properties of polytropes, describing degenerate stars and chemically homogeneous nondegenerate stellar cores.

Cyclic cosmology from Lagrange-multiplier modified gravity

We investigate cyclic and singularity-free evolutions in a universe governed by Lagrange-multiplier modified gravity, in both scalar-field cosmology and the f(R) one. In the scalar case, cyclicity can be induced by a suitably reconstructed simple potential, and the matter content of the Universe can be successfully incorporated. In the case of f(R)-gravity, cyclicity can be induced by a suitably reconstructed second function f2(R) of a very simple form; however, the matter evolution cannot be analytically handled. Furthermore, we study the evolution of cosmological perturbations for the two scenarios. For the scalar case the system possesses no wavelike modes due to a dust-like sound speed, while for the f(R) case there exists an oscillation mode of perturbations which indicates a dynamical degree of freedom. Both scenarios allow for stable parameter spaces of cosmological perturbations through the bouncing point.

HIGHER CURVATURE THEORIES OF GRAVITY IN THE ADM CANONICAL FORMALISM

We present a canonical formulation for higher-curvature theories of gravity whose action is a generic function of Riemann tensor. The Arnowitt–Deser–Misner canonical formalism is employed to identify the extra gravitational dynamical degrees of freedom other than metric. We also find a surface term that gives Dirichlet boundary conditions for the dynamical degrees of freedom.

Motion Planning of Nonholonomic Systems with Tracking Control Lyapunov Function

A common approach to motion planning of robots and vehicles involves finding suitable trajectories for the positions of each configuration variable, and then using feedback to regulate the system to these trajectories. However, when the system has less actuator than dynamical degrees of freedom, it is not always possible to do this arbitrarily. In this paper a tracking control Lyapunov function (TCLF) is proposed to guarantee that the trajectory generation is convergent and executable under nonholonomic constraint, and the simulation result conducted on surface vehicle shows its effectiveness.

Ferromagnetic phases in spin-Fermion systems

Spin-Fermion systems which obtain their magnetic properties from a system of localized magnetic moments being coupled to conducting electrons are considered. The dynamical degrees of freedom are spin-$s$ operators of localized spins and spin-1/2 Fermi operators of itinerant electrons. Renormalized spin-wave theory, which accounts for the magnon-magnon interaction, and its extension are developed to describe the two ferrimagnetic phases in the system: low temperature phase $0<T<T^{*}$, where all electrons contribute the ordered ferromagnetic moment, and high temperature phase $T^{*}<T<T_C$, where only localized spins form magnetic moment. The magnetization as a function of temperature is calculated. The theoretical predictions are utilize to interpret the experimentally measured magnetization-temperature curves of $UGe_2$..

Structural Analysis of Process Models Using Their Representation Graph

A graph-theoretical method for the structural analysis of dynamic lumped process models described by differential and algebraic equations (DAEs) is applied in this paper in order to determine the most important solvability properties (degree of freedom, structural solvability, model decomposition, dynamic degree of freedom, differential index, e.g.) of these models by using the so-called dynamic representation graph. The structure of the dynamic representation graph is suitable for the determination of the mentioned solvability properties. The most common methods in the modelling practice for the construction of models of complex systems are the union of submodels and hierarchical modelling. Our goal is to investigate the effect of the model union to the solvability properties, especially to the differential index. We show how the representation graph of a complex model can be built up from the representation graphs of submodels. The effect of the structure of submodels and their joining points to the structure of the complex graph and the conclusions drawn from the complex graph structure to the solvability properties are also investigated. The representation graph of the complex model can be easily built up from the representation graphs of the simple models according to the linking of the technological subsystems. If one of the submodels has greater than one differential index then the under and overspecified subgraphs referring to this higher index can be found in the representation graph of the complex model, too. The change in the relative position of the underspecified and the overspecified subgraphs has an effect to the value of differential index. If these subgraphs move further from their original positions the value of the differential index increases. If their relative positions do not change during the built up process then the value of the differential index of the complex system is equal to the value of the differential index of the subsystem having the higher value.

Extensive chaotic dynamics in neural networks in the balanced state

The irregular activity of neurons in the cortex [1] is thought to arise from strong input fluctuations [2], which result from a balance between excitatory and inhibitory synaptic inputs [3]. This is called the balanced state of cortical networks. However, recent studies of the underlying dynamics of the balanced state have led to contrary results, strongly depending on the used single neuron models [3, 4]. Here, we study the network dynamics of sparsely coupled theta neurons in the balanced state. The theta neuron model has an active spike generating mechanism and is the standard form of type I neurons [5]. In a random matrix approximation of the Jacobian, we derive an expression for the mean Lyapunov exponent. By analyzing the full set of Lyapunov exponents and the maximal Lyapunov vector in a numerically exact way, we reveal extensive spatio-temporal deterministic chaos. The studied networks exhibit high-dimensional chaotic attractors, giving rise to many dynamical degrees of freedom to encode information. At the same time, the intrinsic entropy production is surprisingly high, limiting information processing to the immediate stimulus response. Purely inhibitorily coupled networks

Field-induced quantum criticality of a spin-1/2 planar ferromagnet

The low-temperature critical properties and crossovers of a spin-½ planar ferromagnet in a longitudinal magnetic field are explored in terms of an anisotropic bosonic action, suitable to describe the spin model in the low-temperature regime. This is performed adopting a procedure which combines an averaging over dynamic degrees of freedom and the classical Wilson renormalization group transformation. Within this framework we get the phase boundary, ending in a quantum critical point, and general expressions for the correlation length and susceptibility as functions of the temperature and the applied magnetic field within the disordered phase. In particular, two crossovers occur decreasing the temperature with the magnetic field fixed at its quantum critical point value, which might be actually observable in complex magnetic compounds, as suggested by recent experiments.

Large Portfolio Risk Management with Dynamic Copulas

Modeling the dynamic high-dimensional multivariate distribution is very useful for active risk management and optimal portfolio allocation; however, available dynamic models are not easily applied for high-dimensional problems due to the curse of dimensionality. In the light of the recent development of multivariate GARCH techniques for a large number of underlying securities, I extend the framework of the Dynamic Conditional Correlation/Equicorrelation (DCC/DECO) (Engle, 2002 and Engle and Kelly, 2008) and an extreme value approach (McNeil and Frey, 2000) into a series of Dynamic Conditional Elliptical Copulas. By constructing portfolios of 89 stocks from CDX-listed …rms between 1995 and 2005, I examine Value at Risk (VaR) and Expected Shortfall (ES) by Monte Carlo simulation for passive portfolios and dynamic optimal portfolios through Mean-Variance and ES criteria. I …nd: (1) Modeling the marginal distribution is important for the dynamic high-dimensional multivariate models. (2) Neglecting the dynamic dependence in the copula causes over-aggressive risk management. (3) The DCC/DECO Gaussian copula and t-copula work very well for both VaR and ES. The DCC copula is necessary for value-weighted portfolios. For equally-weighted portfolios, the DECO copula performs about as well as the DCC copula. (4) Grouped t-copula and t-copula with dynamic degrees of freedom further match the fat tail. (5) Correctly modeling dependence structure makes an improvement in portfolio optimization. The optimal portfolio by ES does a good job against the tail risk in both DECO and DCC copulas. (6) Since portfolio optimization induces statistical error maximization, the assumption of multivariate t innovations with exogenously given degrees of freedom provides a ‡exible and applicable method for optimal portfolio risk management.

Crack detection in existing structures using noise-contaminated dynamic responses

Crack detection is existing structures using measured dynamic response information is presented in this paper. The general area of health assessment of existing structures in the presence of uncertainty in modeling the structure, characteristics of sensors, and the dynamic response information collected by the sensors is emphasized. A time domain system identification-based procedure is proposed to assess health of a structure at the finite element level in the presence of all major sources of uncertainty. It is denoted as the GILS-EKF-UI method. The method does not require information on excitation that caused the response and the noise-contaminated response information needs not be available at all dynamic degrees of freedom. The method is verified using computer generated analytical and actual measured response information emphasizing three items: (1) identification of the defect-free frame, (2) detection of location of a crack accurately within a defective element, and (3) detection of a crack using limited response information using the GILS-EKF-UI method.

Coupled quasicrystals

We predict that under certain experimental conditions external radiation with frequency half of that for the alternating Josephson effect can appear when a constant voltage is applied to a quasicrystal–quasicrystal tunnelling junction 1. The background for this possibility is the phase coherence of the wavefunctions across the barrier and the absence of scattering on the dynamical degrees of freedom (phonons).

Homeokinetics/Homeodynamics: A Physical Heuristic for Life and Complexity

This essay addresses the puzzlement, the missing piece, sensed when attempts are made to build a bridge from the synchronic, informational genotype to the diachronic, dynamic phenotype—a regular mapping that seems to be extraphysical. There is no formal, dynamic foundation for the bridge. Albert Einstein, Max Delbrück, and Erwin Schrödinger all expressed acute awareness of limitations of contemporary physics when considering biology because physics addresses much simpler sysems. As a proposed remedy, a new physical heuristic, homeokinetics, developed by Arthur Iberall and Harry Soodak (and later recast for biology by me as homeodynamics) is introduced here as a foundation for comprehending energy flows and transformations in complex systems, including those in metabolic networks of living systems. Their individual dynamic stability is flexible and marginal—it must allow for adaptations and changes in physiological and behavioral states to occur in an orderly fashion as external circumstances change. At the population level, stability must allow for evolvability of chemical networks that have energized terrestrial living systems for about 3.9 billion years. Homeokinetics/homeodynamics emphasizes that persistent, marginally stable metabolic networks, as open thermodynamic systems, necessarily organize energy processing as cyclic, physical action modes. Conceptually, that organization is under 2 kinds of biological time pressure—time as a cycle that daily closes the thermodynamic books and time as an arrow orthogonally pressing the cyles into the future, creating joint time as a helix. In most animals, after maturity, the helix is additionally shaped into a tapered ellipsoid by a senesence process that gains influence as dynamic degrees of freedom are frozen out by the constructions of development.

THE ROLE OF TRANSLATIONAL INVARIANCE IN NONLINEAR GAUGE THEORIES OF GRAVITY

The internal structure of the tetrads in a Poincaré nonlinear gauge theory of gravity is considered. Minkowskian coordinates becomes dynamical degrees of freedom playing the role of Goldstone bosons of the translations. A critical length allowing a covariant expansion similar to the weak field approach is deduced, the zeroth order metric being maximally symmetric (Minkowskian in some cases).

On the consistency relation of the three-point function in single-field inflation

The consistency relation for the three-point function of the cosmic microwave backgroundis a very powerful observational signature which is believed to be true for every inflationarymodel in which there is only one dynamical degree of freedom. Its importancerelies on the fact that deviations from it might be detected in next generationexperiments, allowing us to rule out all single-field inflationary models. Aftermaking more precise the already existing proof of the consistency relation, we use arecently developed effective field theory for inflationary perturbations to provide analternative and very explicit proof valid at leading non-trivial order in slow rollparameters.

Two ferromagnetic phases in spin-Fermion systems

We consider spin-fermion systems which get their magnetic properties from a system of localized magnetic moments being coupled to conducting electrons. The dynamical degrees of freedom are spin-$s$ operators of localized spins and spin-1/2 fermi operators of itinerant electrons. We develop modified spin-wave theory and obtain that system has two ferromagnetic phases. At the characteristic temperature T* the magnetization of itinerant electrons becomes zero, and high temperature ferromagnetic phase (T*<T<T_C) is a phase where only localized electrons give contribution to the magnetization of the system. An anomalous increasing of magnetization below T* is obtained in a good agrement with experimental measurements of the ferromagnetic phase of UGe2.

An analytic perturbation approach for classical spinning particle dynamics

A perturbation method to analytically describe the dynamics of a classical spinning particle, based on the Mathisson-Papapetrou-Dixon (MPD) equations of motion, is presented. By a power series expansion with respect to the particle's spin magnitude, it is shown how to obtain in general form an analytic representation of the particle's kinematic and dynamical degrees of freedom that is formally applicable to infinite order in the expansion. Within this formalism, it is possible to identify a classical analogue of radiative corrections to the particle's mass and spin due to spin-gravity interaction. The robustness of this approach is demonstrated by showing how to explicitly compute the first-order momentum and spin tensor components for arbitrary particle motion in a general space-time background. Potentially interesting applications based on this perturbation approach are outlined.

Internal Variables and Dynamic Degrees of Freedom

Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable. This duality provides the corresponding evolution equations depending on whether the Onsager–Casimir reciprocity relations are satisfied or not.

Thermoelasticity and entropy flow

The well-known theories of thermoelasticity are based on thermal expansion, and recently the theory of heat conduction has been looked into for improvements. The reason is that heat conduction is described with scalar and vector variables but elasticity with second order tensors, and in linear order there is no direct coupling between second order tensors and vectors or scalars if the material is isotropic. The observed deviations from the present theories urge that new tracks for research be sought. Such a new track can be opened by Onsager’s thermodynamics supplemented with dynamic degrees of freedom. This theory is usually referred to as extended thermodynamics. The key moment is in the general form of the entropy current out of local equilibrium, which leads to the formal introduction of the transport of the dynamic degrees of freedom. The skeleton of the possible theories is based on the introduction of one or more vectorial dynamic variables. They can be coupled to the current density of the heat flow, while their ‘diffusion’ intensities are second order tensors coupled directly in linear order to the stress tensor even if the material is isotropic. The possibilities are demonstrated on an example with one dynamic degree of freedom.

CANONICAL FORMULATION OF A BOSONIC MATTER FIELD IN (1 + 1)-DIMENSIONAL CURVED SPACE

We study a bosonic scalar in (1 + 1)-dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein–Hilbert action [Formula: see text] when written in first-order form. After applying the Dirac constraint formalism to the Einstein–Hilbert action and the action of the bosonic scalar field separately, we apply it to these actions when they are combined. Only in the latter case does a dynamical degree of freedom emerge.