Rescaling Factor Research Articles

Energy in the spacetime field of the charged rotating BTZ black hole via approximate Lie symmetries

Using Lie approximate symmetry methods for differential equations, second-order approximate symmetries of the geodesic equations for the charged rotating Bañados, Teitelboim and Zanelli black hole are investigated. For consistency of the trivial second-order approximate symmetries of the geodesic equations, a rescaling factor for energy in this spacetime is obtained.

The small world effect on the coalescing time of random walks

A small world is obtained from the d -dimensional torus of size 2 L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time T L of two random walks moving on this small world and compare it with the result on the torus. On the torus, in order to have convergence, we have to rescale T L by a factor C 1 L 2 if d = 1 , by C 2 L 2 log L if d = 2 and C d L d if d ≥ 3 . We prove that on the small world the rescaling factor is C d ′ L d and identify the constant C d ′ , proving that the walks always meet faster on the small world than on the torus if d ≤ 2 , while if d ≥ 3 this depends on the probability of moving along the random connection. As an application, we obtain results on the hitting time to the origin of a single walk and on the convergence of coalescing random walk systems on the small world.

Energy of the Bardeen model using an approximate symmetry method

In this paper, we investigate the energy problem in general relativity using approximate Lie symmetry methods for differential equations. This procedure is applied to the Bardeen model (the regular black-hole solution). Here, we are forced to evaluate the third-order approximate symmetries of the orbital and geodesic equations. It is shown that energy must be re-scaled by some factor in the third-order approximation. We discuss the insights into this re-scaling factor.

Nearest-neighbor graphs on the cantor set

Let be a collection of n uniform, independent, and identically distributed points on the Cantor ternary set. We consider the asymptotics for the expected total edge length of the directed and undirected nearest-neighbor graph on We prove convergence to a constant of the rescaled expected total edge length of this random graph. The rescaling factor is a function of the fractal dimension and has a log-periodic, nonconstant behavior.

Nearest-neighbor graphs on the cantor set

Let be a collection of n uniform, independent, and identically distributed points on the Cantor ternary set. We consider the asymptotics for the expected total edge length of the directed and undirected nearest-neighbor graph on We prove convergence to a constant of the rescaled expected total edge length of this random graph. The rescaling factor is a function of the fractal dimension and has a log-periodic, nonconstant behavior.

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non- trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is r-dependent. This re-scaling factor is compared with that for the Reissner–Nordstrom metric.

Traveling waves and impact-parameter correlations

It is usually assumed that the high-energy evolution of partons in QCD remains local in coordinate space. In particular, fixed impact-parameter scattering is thought to be in the universality class of one-dimensional reaction-diffusion processes as if the evolutions at different points in the transverse plane became uncorrelated through rapidity evolution. We check this assumption by numerically comparing a toy model with QCD-like impact-parameter dependence to its exact counterpart with uniform evolution in impact-parameter space. We find quantitative differences, but which seem to amount to a mere rescaling of the strong coupling constant. Since the rescaling factor does not show any strong ${\ensuremath{\alpha}}_{s}$ dependence, we conclude that locality is well verified, up to subleading terms at small ${\ensuremath{\alpha}}_{s}$.

Two-dimensional skew scattering in the vicinity of and away from the resonant scattering condition

We studied the energy dependence of the 2D skew scattering from strong potential, for which the Born approximation is not applicable. Since the skew scattering cross section is zero both at low and at high energies, it exhibits a maximum as a function of energy of incident electron. We found analytically the shape of the maximum for an exactly solvable model of circular-barrier potential. Within a rescaling factor, this shape is universal for strong potentials. If the repulsive potential has an attractive core, the discrete levels of the core become quasilocal due to degeneracy with continuum. For energy of incident electron close to the quasilocal state with zero angular momentum, the enhancement of the net cross section is accompanied by resonant enhancement of the skew scattering. By contrast, near the resonance with quasilocal states having momenta $\pm 1$, the skew scattering cross section is an odd function of energy deviation from the resonance, and passes through zero, i.e., it exhibits a sign reversal. In the latter case, in the presence of the Fermi sea, the Kondo resonance manifests itself in strong temperature dependence of the skew scattering.

Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $\partial_t u - \Delta u + |\nabla u|^q = 0$ is investigated for the critical exponent q=(N+2)/(N+1). Convergence towards a rescaled self-similar solution to the linear heat equation is shown, the rescaling factor being $(\ln{t})^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.

Remote characterisation of shallow seafloor using airborne TEM

Helicopter-borne time-domain electromagnetic (EM) systems combine the advantages of portability with large transmitter moments. In order to demonstrate the potential of helicopter TEM for shallow bathymetric mapping, a HoistEM survey was conducted near the entrance of Port Jackson (Sydney Harbour) in March 2002. 1D inversion of rescaled HoistEM yielded accurate (± 1m) water depth predictions to depths of about 55m. The seawater depths inferred from HoistEM were in good agreement with the depths obtained from accurate single-beam and multi-beam sonar soundings. It was necessary to rescale the HoistEM data in order to account for improper calibration. Re-scaling was possible because a considerable amount of ``ground truth' was available. Synthetic TEM decays were computed for numerous models representing shallow water, sediment and basement at a number of locations within the survey area. Measured and calculated data were compared at each delay time, and were found to be linearly related. A rescaling factor was then applied to each decay time, irrespective of water depth. Given the successful bathymetric mapping demonstration, the potential of airborne EM for seafloor characterisation has been investigated. Accordingly, the sediment thickness inferred from 1D inversion of HoistEM has been compared with estimates based on marine seismic studies. Generally, the sediment thickness, and hence depth to resistive bedrock, inferred from HoistEM was in reasonable agreement with depths estimated from marine seismic data when the seawater was less than about 20 m deep and the sediment was less than about 40 m thick.

Recursive renormalization-group calculation for the eddy viscosity and thermal eddy diffusivity of incompressible turbulence

The recursive renormalization-group method with an asymptotically free conditional average is conducted for calculating the effective eddy viscosity and thermal eddy diffusivity of incompressible turbulence. For different values of spatial rescaling factor Λ, the dependence of renormalized eddy viscosity on wave number is examined at the zeroth- and the first-order truncations of expansion of subgrid-scale energy spectrum density in Taylor series, respectively. A strong cusp-like behavior is observed near the supergrid–subgrid cutoff for the first order. This is in agreement with the result of testing-field model. However, no cusp behavior is found for the zeroth order. Calculation of Kolmogorov constant C K indicates a special region where C K is insensitive to Λ. The first order shows a range much larger than that for the zeroth order, and gives C K =1.56±0.04 in the range of 0.4⩽ Λ⩽0.8, consistent with the generally accepted experimental values (1.2–2.2). Furthermore, the wave-number dependence of renormalized thermal eddy diffusivity is investigated, and no cusp behavior is observed around the supergrid–subgrid cutoff at the first-order truncation of expansion of subgrid-scale energy spectrum density, different from that of renormalized eddy viscosity. The Batchelor constant C B is also found insensitive to Λ in the range of 0.4⩽ Λ⩽0.8, and is estimated to be 1.15±0.05, in good agreement with the result of ε-renormalization group approach.

Canonical solution of classical magnetic models with long-range couplings

We study the canonical solution of a family of classical $n-vector$ spin models on a generic $d$-dimensional lattice; the couplings between two spins decay as the inverse of their distance raised to the power $\alpha$, with $\alpha<d$. The control of the thermodynamic limit requires the introduction of a rescaling factor in the potential energy, which makes the model extensive but not additive. A detailed analysis of the asymptotic spectral properties of the matrix of couplings was necessary to justify the saddle point method applied to the integration of functions depending on a diverging number of variables. The properties of a class of functions related to the modified Bessel functions had to be investigated. For given $n$, and for any $\alpha$, $d$ and lattice geometry, the solution is equivalent to that of the $\alpha=0$ model, where the dimensionality $d$ and the geometry of the lattice are irrelevant.

Efficient Near Infrared Si/Ge Quantum Dot Photo-Detector Based on a Heterojunction Bipolar Transistor

AbstractGe dots embedded in Si offer the possibility of Si-based light detection at 1.3-1.55 μm. In this communication, we report a very efficient photo-detector based on a Si/SiGe heterojunction bipolar transistor structure with 10 Ge dot layers (8 ML Ge each) incorporated in the basecollector junction. The device structures were grown using low-temperature molecular beam epitaxy, and fabricated for both normal and edge incidence with no electrical contact to the base. The processed Ge-dot transistor detectors revealed a rather low dark current density, 0.01 mA/cm2 at -2 V. Photoconductivity measurements were performed at room temperature. At 1.31 μm, responsivity values of 50 mA/W at normal incidence have been directly measured at Vce = -4 V, without involving any rescaling factor due to light coupling. This value is a ∼250-fold increase compared to a reference p-i-n diode with the same dot layer structure, due to the current amplification function of the transistor. For a rib waveguide device, a very high responsivity value of about 470 mA/W (Vce = -4V) has been obtained at 1.31 μm. Measurements were also performed at 1.55 μm, and the photo-response of the waveguide phototransistor was 25 mA/W, which is again a large improvement compared with the reference waveguide photodiode (∼1 mA/W). Moreover, time-resolved photoconductivity measurements have been carried out. The results have indicated that the device frequency performance is primarily limited by the emitterbase junction capacitance.

A Comparison of Breeding and Ensemble Transform Kalman Filter Ensemble Forecast Schemes

Abstract The ensemble transform Kalman filter (ETKF) ensemble forecast scheme is introduced and compared with both a simple and a masked breeding scheme. Instead of directly multiplying each forecast perturbation with a constant or regional rescaling factor as in the simple form of breeding and the masked breeding schemes, the ETKF transforms forecast perturbations into analysis perturbations by multiplying by a transformation matrix. This matrix is chosen to ensure that the ensemble-based analysis error covariance matrix would be equal to the true analysis error covariance if the covariance matrix of the raw forecast perturbations were equal to the true forecast error covariance matrix and the data assimilation scheme were optimal. For small ensembles (∼100), the computational expense of the ETKF ensemble generation is only slightly greater than that of the masked breeding scheme. Version 3 of the Community Climate Model (CCM3) developed at National Center for Atmospheric Research (NCAR) is used to test ...

Gibbs solution of the van der Waals–Maxwell problem and universality of the liquid–gas coexistence curve

The conventional statements of the scaling theory are: (i) the real fluids belong to the lattice gas universality class; (ii) the coexistence curves of real fluids, as well as of the van der Waals (vdW) model do not possess any apparent symmetry and therefore corrections to the asymptotic power laws and asymmetry must be taken into account in the extended critical region. We demonstrate that scaling and classical basic models, applied to the extended critical region have a common line of symmetry—the continuation of the critical isochore ρc by the line of the rectilinear diameter ρc = (ρl + ρg)/2 if a new parameter for the distance from the critical point is used instead of temperature. This is the reduced difference of molar entropies: x = (sg − sl)/2R which can be obtained from the measurable latent heat rs(T)–data along the coexistence curve. The parametric solution of the van der Waals–Maxwell problem proposed by Gibbs in terms of an unspecified parameter demonstrates similar symmetry if the parameter is identified as x. Nearly ideal linearity between the reduced densities ρl/ρc, ρg/ρc, (ρl − ρg)/ρc and parameter x has been found for a set of well-studied fluids: Ar, C2H4, CO2, H2O. The symmetry of the vdW-model differs from that of real fluids by the value of critical slopes for ρl,g(x)-functions expressed in terms of dimensionless variables. The slope is close to ±1/2 for real fluids, ±2/3 for the vdW-model, and ±∞ for the lattice gas model. We conclude that the symmetry in real fluids is much more similar to the vdW-model than to the lattice gas model. Therefore, to achieve an adequate description of real fluids in the extended critical region (ρ = ρc ± 0,3ρc, x ≤ 0,5), a combination of background (vdW-like) and scaling terms should be taken into account up to the critical point. With the introduction of constant rescaling factor for each above-named fluid a novel coexistence curve model can be obtained providing a high level of prediction accuracy on the basis of the parametric solution of the vdW–Maxwell problem.

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is \(\) for ν = 1 and (ln T)\(\), for ν=2; for the covariance it is \(\), ν = 1,2.

Random surfaces from hierarchical deposition of debris with alternating rescaling factors

An analytical study of surface profiles that result from hierarchical random impact of debris on the line is performed in terms of logarithmic fractals theory. The hierarchical random deposition model is extended for the case of time-dependent probabilities P (for positioning a hill on the surface) and Q (for digging a hole) and spatial rescaling factor λ. The periodic deposition model is solved exactly, and the logarithmic fractal roughness of the surface profile is found to be robust with respect to time-dependent perturbations. The fractal amplitudes associated with the proliferation of the surface length are compared with those calculated in the static regime and are shown to have a nontrivial interaction. It is verified that amplitude repulsion, attraction, neutrality, and auto-repulsion take place. The transient regime is also studied and is shown to have exponential decay towards the asymptotic regime. Special attention is devoted to the case of alternating rescaling factors, for which new results are derived.

Renormalized expression for the turbulent energy dissipation rate.

Conditional elimination of degrees of freedom is shown to lead to an exact expression for the rate of turbulent energy dissipation in terms of a renormalized viscosity and a correction. The correction is neglected on the basis of a previous hypothesis [W. D. McComb and C. Johnston, J. Phys. A 33, L15 (2000)] that there is a range of parameters for which a quasistochastic estimate is a good approximation to the exact conditional average. This hypothesis was tested by a perturbative calculation to second order in the local Reynolds number, and the Kolmogorov prefactor (taken as a measure of the renormalized dissipation rate) was found to reach a fixed point which was insensitive to initial values of the kinematic viscosity and to values of the spatial rescaling factor h in the range 0.4< or =h< or =0.8.

Non-Gaussian numerical errors versus mass hierarchy

We probe the numerical errors made in renormalization group calculations by varying slightly the rescaling factor of the fields and rescaling back in order to get the same (if there were no round-off errors) zero momentum 2-point function (magnetic susceptibility). The actual calculations were performed with Dyson's hierarchical model and a simplified version of it. We compare the distributions of numerical values obtained from a large sample of rescaling factors with the (Gaussian by design) distribution of a random number generator and find a significant departure from the Gaussian behavior. In addition, the average value differs (robustly) from the exact answer by a quantity which is of the same order as the standard deviation. We provide a simple model in which the errors made at shorter distance have a larger weight than those made at larger distance. This model explains in part the non-Gaussian features and why the central-limit theorem does not apply.

Exotic marginal fractals from hierarchical random deposition

We study marginal forms on the borderline between Euclidean shapes and fractals. The fractal dimension equals the topological dimension. The fractal measure features a logarithmic correction factor, leading to a linear divergence of the area or length upon increasing the resolution instead of the usual exponential law for fractals. We discuss a physical model of random deposition of particles or fragments obeying a strict size hierarchy. The landscapes resulting from this deposition process resemble modern cities and satisfy the logarithmic fractal law. The roughness of the shapes is characterized by a fractal amplitude, which can be calculated exactly as a function of the (time-dependent) probabilities and rescaling factor of the random deposition process. Besides possible realizations in particle or cluster deposition physics and colloid physical chemistry, an application to the science of porous media in the plane is described.