Asymptotic Slope Research Articles

The inner structure of ΛCDM haloes - III. Universality and asymptotic slopes

We investigate the mass profile of cold dark matter (ΛCDM) haloes using a suite of numerical simulations spanning five decades in halo mass, from dwarf galaxies to rich galaxy clusters. These haloes typically have a few million particles within the virial radius (r200), allowing robust mass profile estimates down to radii <1 per cent of r200. Our analysis confirms the proposal of Navarro, Frenk & White (NFW) that the shape of the ΛCDM halo mass profiles differs strongly from a power law and depends little on mass. The logarithmic slope of the spherically averaged density profile, as measured by β=−d ln ρ/d ln r, decreases monotonically towards the centre and becomes shallower than isothermal (β< 2) inside a characteristic radius, r2. The fitting formula proposed by NFW provides a reasonably good approximation to the density and circular velocity profiles of individual haloes; circular velocities typically deviate from NFW best fits by <10 per cent over the radial range that is numerically well resolved. Alternatively, systematic deviations from the NFW best fits are also noticeable. Inside r2, the profile of simulated haloes becomes shallower with radius more gradually than predicted and, as a result, NFW fits tend to underestimate the dark matter density in these regions. This discrepancy has been interpreted as indicating a steeply divergent cusp with asymptotic inner slope, β0≡β(r = 0) 1.5. Our results suggest a different interpretation. We use the density and enclosed mass at our innermost resolved radii to place strong constraints on β0: density cusps as steep as r1.5 are inconsistent with most of our simulations, although β0= 1 is still consistent with our data. Our density profiles show no sign of converging to a well-defined asymptotic inner power law. We propose a simple formula that reproduces the radial dependence of the slope better than the NFW profile, and so may minimize errors when extrapolating our results inward to radii not yet reliably probed by numerical simulations.

Velocity Dispersion Profiles in Dark Matter Halos

Numerous numerical studies indicate that dark matter halos show an almost universal radial density profile. The origin of the profile is still under debate. We investigate this topic and pay particular attention to the velocity dispersion profile. To this end, we have performed high-resolution simulations with two independent codes, ART and GADGET. The radial velocity dispersion can be approximated as a function of the potential by σ = a(Φ/Φout)κ(Φout - Φ), where Φout is the outer potential of the halo. For the parameters a and κ, we find that a = 0.29 ± 0.04 and κ = 0.41 ± 0.03. We find that the power-law asymptote, σ2 ∝ Φκ, is valid out to much larger distances from the halo center than any power-law asymptote for the density profile ρ ∝ r-n. The asymptotic slope n(r → 0) of the density profile is related to the exponent κ via n = 2κ/(1 + κ). Thus, the value obtained for κ from the available simulation data can be used to obtain an estimate of the density profile below currently resolved scales. We predict a continuously decreasing n toward the halo center with the asymptotic value n ≲ 0.58 at r = 0.

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.

Computer simulation study of bulk atactic polystyrene in the vicinity of the glass transition

AbstractMolecular dynamics (MD) simulations of bulk atactic polystyrene have been performed in a temperature range from 100 K to 650 K at atmospheric pressure. Local translational mobility has been investigated by measuring the mean square translational displacements of monomers. The long‐time asymptotic slope of these dependencies is 0.54 at T&gt;Tg, showing Rouse behavior. Cross‐over from motion in the cage to Rouse like dynamics has been studied at T&gt;Tg with a characteristic crossover time follows a power law behavior as a function of T, as predicted by mode‐coupling theory (MCT). Local orientational mobility has been studied via the orientational autocorrelation functions, ACFs, (Legendre polynomials of the first and second, order) of both the main‐chain and side‐group bonds. The relaxation times of the orientational α‐relaxation follow the same power law (γ∼2.9) as the characteristic translational diffusion time. Below T&gt;Tg both types of dynamics are described by the same activated law. The ACFs time‐distribution functions reveal the existence of activated local rearrangements already above T&gt;Tg.

Dark Halo Cusp: Asymptotic Convergence

We propose a model for how the buildup of dark halos by merging satellites produces a characteristic inner cusp, with a density profile ρ ∝ r, where αin → αas 1, as seen in cosmological N-body simulations of hierarchical clustering scenarios. Dekel, Devor, & Hetzroni argue that a flat core of αin 1. Using merger N-body simulations, we learn that this cusp is stable under a sequence of mergers and derive a practical tidal mass transfer recipe in regions where the local slope of the halo profile is α > 1. According to this recipe, the ratio of mean densities of the halo and initial satellite within the tidal radius equals a given function ψ(α), which is significantly smaller than unity (compared to being ~1 according to crude resonance criteria) and is a decreasing function of α. This decrease makes the tidal mass transfer relatively more efficient at larger α, which means steepening when α is small and flattening when α is large, thus causing convergence to a stable solution. Given this mass transfer recipe, linear perturbation analysis, supported by toy simulations, shows that a sequence of cosmological mergers with homologous satellites slowly leads to a fixed-point cusp with an asymptotic slope αas > 1. The slope depends only weakly on the fluctuation power spectrum, in agreement with cosmological simulations. During a long interim period the profile has an NFW-like shape, with a cusp of 1 < αin < αas. Thus, a cusp is enforced if enough compact satellite remnants make it intact into the inner halo. In order to maintain a flat core, satellites must be disrupted outside the core, possibly as a result of a modest puffing up due to baryonic feedback.

Planetary bow shocks: Asymptotic MHD Mach cones

A direct approach for determining the asymptotic MHD Mach cone is formulated and solved. An implicit analytical solution enables the calculation of the asymptotic downstream slope of MHD Mach cones at any clock angle for arbitrary Ms, Ma, and ϑbv. The solution obtained includes all previously known symmetric cases. The elongation and shift of the asymptotic fast mode shock cross-section are studied for a wide range of upstream plasma parameters as well as its unusual ‘chopped’ shape under certain conditions. Our results may be useful for planetary shock modeling and MHD numerical codes verification.

On the dual cascade in two-dimensional turbulence

We study the dual-cascade scenario for two-dimensional (2D) turbulence driven by a spectrally localized forcing applied over a finite wavenumber range [ k min, k max] (with k min>0) such that the respective energy and enstrophy injection rates ϵ and η satisfy k min 2 ϵ≤ η≤ k max 2 ϵ. The classical Kraichnan–Leith–Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scale-selectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For 2D turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the across-channel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy β>5 and α+ β≥8, where − α (− β) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analysed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.

Correlated Segmental Dynamics in Amorphous Atactic Polystyrene: A Molecular Dynamics Simulation Study

Molecular dynamics simulations of bulk atactic polystyrene have been performed in a temperature range from 100 to 650 K at atmospheric pressure. Local translational mobility has been investigated by measuring the mean square translational displacements of monomers. The long-time asymptotic slope of these dependencies is 0.54 at T > Tg, showing Rouse behavior. Crossover from motion in the cage to Rouse-like dynamics has been studied at T > Tg with a characteristic crossover time follows a power law behavior as a function of T, as predicted by mode-coupling theory (MCT). We studied the coherent intermediate scattering function and show the agreement with scaling predictions of MCT. Local orientational mobility has been studied via the orientational autocorrelation functions, ACFs (Legendre polynomials of the first and second order), of both the main-chain and side-group bonds. The ACF relaxation times analysis shows strong coupling between the phenyl groups and the backbone relaxation. The relaxation times ...

A code-matched interleaver design for turbo codes

A code-matched interleaver design for turbo codes in which a particular interleaver is constructed to match the code weight distribution is proposed. The design method is based on the code distance spectrum. The low weight paths in the code trellis which give large contributions to the error probability in the signal-to-noise ratio region of interest for practical communication systems are eliminated so that they do not appear in the overall code trellis after interleaving. The proposed interleaver improves the code error performance at moderate to high signal-to-noise ratio and considerably increases the asymptotic slope of the error probability curves.

Large-scale computer simulation of local segmental dynamics in amorphous atactic polystyrene

Abstract Molecular dynamics computer simulations of amorphous atactic polystyrene have been performed for chains of up to 320 monomer units in a temperature range from 100 to 600 K and in a pressure range from 0.1 to 1000 MPa. The MD-determined values of the glass transition temperature T g are in a good agreement with experimental PVT data at different values of applied pressure. Local translational mobility has been investigated by measuring the mean square translational displacements of monomers as a function of time. The long-time asymptotic slope of these dependencies is close to 0.6 at T > T g , showing diffusive behaviour. The cage effect when local translational motions are essentially frozen in the glassy state has been studied. Local orientational mobility is studied with the help of Legendre polynomials of the first, P 1 ( t ), and second, P 2 ( t ), order for the bonds in the main chain and along the phenyl side groups. Temperature dependencies of the orientational relaxation times at different values of pressure are well described by the Vogel–Fulcher law.

Molecular Dynamics Simulation of Bulk Atactic Polystyrene in the Vicinity of Tg

Molecular dynamics (MD) simulations of bulk atactic polystyrene have been performed for chains up to 320 monomer units in a temperature range from 100 to 600 K and in a broad pressure range from 0.1 to 1000 MPa. The MD-determined specific volume vs temperature curves are in a good agreement with experimental PVT data at different values of applied pressure, but the measured glass-transition temperature, Tg, is displaced to somewhat higher temperature than the longer time experimental value. Local translational mobility has been investigated by measuring the mean-square translational displacements of monomers as a function of time. The long-time asymptotic slope of these dependencies is close to 0.6 at T > Tg, showing diffusive behavior. The cage effect, when local translational motions are essentially frozen in the glassy state, has been studied. The characteristic time of cage release does not depend on molecular weight, but the duration of the crossover to the diffusive regime increases almost linearly with increasing chain molecular weight, for both the backbone monomers and phenyl side groups.

Search for the lepton family number violating process nu(mu)e(-) --> mu(-)nu(e).

The NuTeV experiment at Fermilab has used a sign-selected neutrino beam to perform a search for the lepton number violating process nu(mu)e(-)-->mu(-)nu(e), and to measure the cross section of the standard model inverse muon decay process nu(mu)e(-)-->mu(-)nu(e). NuTeV measures the inverse muon decay asymptotic cross-section slope sigma/E to be (13.8 +/- 1.2 +/- 1.4) x 10(-42) cm(2)/GeV. The experiment also observes no evidence for lepton number violation and places one of the most restrictive limits on the cross-section ratio sigma(nu(mu)e(-)-->mu(-)nu(e))/sigma(nu(mu)e(-)-->mu(-)nu(e)) < or = 1.7% at 90% C.L. for V-A couplings and < or = 0.6% for scalar couplings.

Code characteristic matching for iterative decoding of serially concatenated codes

The design of serially concatenated codes has yet been dominated by optimizing asymptotic slopes of error probability curves. We propose mutual information transfer characteristics for soft in/soft out decoders to design serially concatenated codes based on the convergence behavior of iterative decoding. The exchange of extrinsic information is visualized as a decoding trajectory in the Extrinsic Information Transfer Chart (exit chart). By finding matching pairs of inner and outer decoder transfer characteristics we are able to construct serially concatenated codes whose iterative decoder converges towards low bit error rate at signal- to- noise ratios close to the theoretical limits.

Limiting performance of block-fading channels with multiple antennas

We derive the performance limits of a radio system consisting of a transmitter with t antennas and a receiver with r antennas, a block-fading channel with additive white Gaussian noise (AWGN), delay and transmit-power constraints, and perfect channel-state information available at both the transmitter and the receiver. Because of a delay constraint, the transmission of a codeword is assumed to span a finite (and typically small) number M of independent channel realizations; therefore, the relevant performance limits are the information outage probability and the (or nonergodic) capacity. We derive the coding scheme that minimizes the information outage probability. This scheme can be interpreted as the concatenation of an optimal code for the AWGN channel without fading to an optimal beamformer. For this optimal scheme, we evaluate minimum-outage probability and delay-limited capacity. Among other results, we prove that, for the fairly general class of regular fading channels, the asymptotic delay-limited capacity slope, expressed in bits per second per hertz (b/s/Hz) per decibel of transmit signal-to-noise ratio (SNR), is proportional to min (t,r) and independent of the number of fading blocks M. Since M is a measure of the time diversity (induced by interleaving) or of the frequency diversity of the system, this result shows that, if channel-state information is available also to the transmitter, very high rates with asymptotically small error probabilities are achievable without the need of deep interleaving or high-frequency diversity. Moreover, for a large number of antennas, delay-limited capacity approaches ergodic capacity.

Calibration of the calcium dissociation constant of Rhod2in the perfused mouse heartusing manganese quenching

Both theoretical and experimental results are presented for in vivo calibration of the dissociation constant KCadof the calcium-sensitive fluorescent dye Rhod2in the perfused mouse heart, using manganese quenching of fluorescence transients. An analytical model is derived, based on the biochemical equilibrium of manganese competition with calcium for Rhod2binding. Expressing the differential of the changes between systole and diastole in fluorescence transient (δΔ Fsys-dia). δΔFsys-diain a beating heart as a function of the perfusate manganese concentration [Mn2+]pallows correlation of the measured differential transient changes δΔ Fsys-diawith the calcium dissociation constant KCadof Rhod2and the calcium concentration in the heart. Numerical modeling indicates that the KCadpredominantly affects the asymptotic slope of the δΔ Fsys-diaversus [Mn2+]pcurve at certain manganese concentrations, which suggests that the KCadcan be inversely calculated by partially fitting the δΔ Fsys-diadistribution as a function of the perfusate manganese concentration. The feasibility of this approach is confirmed by quenching of calcium transients by manganese infusion into isolated perfused beating mouse hearts. The resulting calculated dissociation constant KCadof Rhod2is 720nM. Using the same approach, we are able to also estimate intracellular calcium concentrations of 700nM at peak systole and 300nM in diastole. This is in good agreement with values obtained by calibration of fluorescence values with a calcium saturation tetanization procedure in the same perfused mouse heart model.

Fluid analysis of arrival routing

In Hajek's arrival routing problem (1984), customers are routed to one of n queues to minimize average holding cost. Interarrival and service times are exponentially distributed. We solve the associated fluid model. The optimal fluid policy tells us the asymptotic slopes of the switching surfaces in the original problem when the queues are large. If these slopes are nonzero, then numerical tests indicate that the fluid policy performs well in the original stochastic network. The fluid policy also indicates the approximate path that will be taken to recover from large queues: Routing only switches to queues with larger holding cost and once a large queue empties it will remain approximately empty.

Dynamic contact angle for etting of a surface by a van der waals fluid

We consider the creeping motion of a thin layer of a nonvolatile viscous fluid spreading due to capillary forces over a rigid surface covered by a thin homogeneous film (microfilm). The influence of van der Waals forces on the asymptotic slope of the free boundary of the layer is studied in the region of large thickness, where capillary forces dominate. A solution of the problem of the slope angle is obtained for the entire possible range of the microfilm thickness. In the limit of small thickness of the microfilm, this solution is in agreement with the well-known solution of the problem of the dynamics of wetting of a dry surface in the presence of a precursory film and van der Waals forces. The role of the condition at the end of the precursory film is studied.

Cold collapse and the core catastrophe

We show that a universe dominated by cold dark matter fails to reproduce the rotation curves of dark matter dominated galaxies, one of the key problems that it was designed to resolve. We perform numerical simulations of the formation of dark matter haloes, each containing ≳106 particles and resolved to 0.003 times the virial radius, allowing an accurate comparison with rotation curve data. A good fit to both Galactic and cluster-sized haloes can be achieved using the density profile ρ(r)∝[(rrs)1.5(1+(rrs)1.5)]−1, where rs is a scale radius. This profile has a steeper asymptotic slope, ρ(r)∝r−1.5, and a sharper turn-over than found by lower resolution studies. The central structure of relaxed haloes that form within a hierarchical universe has a remarkably small scatter. We compare the results with a sample of dark matter dominated, low surface brightness (LSB) galaxies with circular velocities in the range 100–300 km s−1. The rotation curves of discs within cold dark matter haloes rise too steeply to match these data, which require a constant mass density in the central regions. The effects of Ωmass and Λ cannot reconcile the cold dark matter (CDM) model with data - even if we leave the concentration as a free parameter, we are unable to reproduce the observations with such a steep central density profile. It is important to confirm these results using stellar rather than H i rotation curves for LSB galaxies. We test the effects of introducing a cut-off in the power spectrum that may occur in a universe dominated by warm dark matter. In this case, haloes form by a monolithic collapse but the final density profile barely changes, demonstrating that the merger history does not play a role in determining the halo structure.

Aubry-Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ1. It is well-known that its smooth invariant curves correspond to smooth Z2-periodic solutions of the PDE ut + H(x, t, u)x = 0. In this paper, we establish a connection between the Aubry-Mather theory of invariant sets of the Hamiltonian system and Z2-periodic weak solutions of this PDE by realizing the Aubry-Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry-Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry-Mather set, defined in (2.24). The graph itself is a backward-invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry-Mather theory into the characteristic fields of the above PDE. This is done by making use of one- and two-sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z2-periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two-sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.

Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters

In this work, we present the results from direct numerical simulations of the fully turbulent channel flow of a polymer solution. Using constitutive equations derived from kinetic and network theories, in particular the FEN E-P and the Giesekus models, we predict drag reduction for a variety of rheological parameters, extending substantially previous calculations, [Sureshkumar et al., Phys. Fluids, 9 (1997) 743-755]. The simulation algorithm is based on a semi-implicit, time-splitting technique which uses spectral approximations in the spatial domain. The computations were carried out on a CRAY T3E-900 parallel supercomputer, under fully turbulent conditions. In this work, we demonstrate the existence of a critical range of the Weissenberg number, where the onset of drag reduction occurs, which is independent of the model and also remains the same as the chain extensibility is increased. By allowing for higher extensibility of the polymer chains, we also observed an almost triple in magnitude increase in drag reduction from previous and reported results. The simulations show that the polymer induces several changes in the turbulent flow characteristics, all of them consistent with available experimental results. In addition to decreased fluctuations in the streamwise vorticity and increased streak spacing, we have seen changes, such as the increase of the slope of the logarithmic layer asymptote for the mean velocity profile, which are consistent with high magnitude of drag reduction, as well as with the behaviour of more concentrated systems. This is more consistent with the use of the Giesekus model, which is well suited for concentrated systems, suggesting that there is potential with that model for capturing quite subtle changes in the structure of the turbulent flow field. Results for different contributions of molecular extensibility, L, and solvent viscosity ratio, β, indicate that for the FENE-P model the phenomena are determined almost exclusively by the extensional viscosity and Weissenberg number. However, results obtained with the Giesekus model, for the same extensional viscosity, demonstrate a further drag reducing effect which can be attributed to the non-zero second normal stress coefficient. All results point to a mechanism for drag reduction where a partial inhibition of eddies within the buffer layer by the macromolecules. The simulation results are consistent with the hypothesis that one of the prerequisites for the phenomenon of drag reduction is sufficiently enhanced extensional viscosity, corresponding to the level of intensity and duration of extensional rates typically encountered during the turbulent flow, as has been proposed by various investigators in the past.