Corrections to scaling for period doubling

We have studied a multiple scaling which describes corrections to scaling. For the period doubling in one-dimensional dissipative maps, two-dimensional areapreserving maps, and four-dimensional symplectic maps, the multiple scaling is seen to be well-obeyed, and new scaling factors have been found. The multiple scaling is also seen to be a very powerful tool for searching for scaling behavior.     

Approximate scaling of period doubling windows

An experimental test is made of a prediction that subharmonic windows which display period doubling should exhibit global scaling. It is shown that this is approximately true in a multidimensional system.     

The trajectory scaling function for period doubling bifurcations in flows

We extend the trajectory scaling function as defined for maps to flows whose dynamics is governed by ordinary differential equations. The results are obtained for the Duffing oscillator and are expected to be the same for other dissipative flows as well.     

Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes

This paper is a review of the present status of studies relating to occurrence of deterministic chaos and its characterization in one-dimensional maps. As our primary aim is to introduce the nonspecialists into this fascinating world of chaos we start from very elementary concepts and give sufficient arguments for clarity of ideas. The two main scenarios during onset of chaos viz. the period doubling and intermittency are dealt with in detail. Although the logistic map is often discussed by way of illustration, a few more interesting maps are mentioned towards the end.     

Logarithmic corrections to scaling in turbulent thermal convection

We use a simplified model of turbulent convection to compute analytically heat transport in a horizontal layer heated from below, as a function of the Rayleigh and the Prandtl number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we recover power classical scaling regimes of the Nusselt versus Rayleigh number, with exponent 1/3 or 1/4. At larger Reynolds number, velocity and temperature fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the Nusselt versus Rayleigh or Prandtl. Instead, we obtain logarithmic corrections to the classical soft (exponent 1/3) or ultra-hard (exponent 1/2) regimes, in a way consistent with the most accurate experimental measurements available nowadays. This sets a need for the search of new measurable quantities that are less prone to dimensional theories.     

Mass corrections to scaling in deep inelastic processes

We study subasymptotic hadron target and quark-parton mass corrections to scaling in deep inelastic scattering, ignoring interactions. The results can be summarized using a modified scaling variable common to parton, light-cone and short-distance operator product expansion formalisms, but with model-dependent spectral conditions. The analysis is expected to break down near the kinematic boundaries because of the bound state nature of hadrons. Related effects probably also dominate mass corrections due to very light quarks, but the analysis should be applicable to the production of new heavy quarks in neutrino production. Experimental deviations from scaling in deep inelastic electroproduction do not seem to be describable in terms of mass corrections alone, suggesting that interaction effects may be important at large momentum transfers as suggested by the renormalization group.     

Finite-size corrections to scaling behavior in sorted cell aggregates

Cell sorting is a widespread phenomenon pivotal to the early development of multicellular organisms. In vitro cell sorting studies have been instrumental in revealing the cellular properties driving this process. However, these studies have as yet been limited to two-dimensional analysis of three-dimensional cell sorting events. Here we describe a method to record the sorting of primary zebrafish ectoderm and mesoderm germ layer progenitor cells in three dimensions over time, and quantitatively analyze their sorting behavior using an order parameter related to heterotypic interface length. We investigate the cell population size dependence of sorted aggregates and find that the germ layer progenitor cells engulfed in the final configuration display a relationship between total interfacial length and system size according to a simple geometrical argument, subject to a finite-size effect.     

On the corrections to scaling in three-dimensional Ising models

The leading correction-to-scaling amplitudes for the spin-1/2, nearest-neighbor sc, bcc, and fee Ising models are considered with the particular aim of determining their signs. On the basis of previous two-variable series analyses by Chen, Fisher, and Nickel and renormalization group=4–d expansions, it is concluded that the correction amplitudes for the susceptibility, correlation length, specific heat, and spontaneous magnetization arenegative for all three lattices. Thus, for example, the effective exponent _eff(T) asymptotically approaches the true susceptibility exponent fromabove. Other earlier and more recent high-temperature series and field-theoretic analyses are seen to be consistent with this result. However, the usual nonasymptotic, perturbative field-theoretic approaches are essentially committed to positive correction amplitudes. The question of the signs therefore relates directly to the applicability of these non-asymptotic field-theoretic calculations to three-dimensional Ising models as well as to different experimental systems.     

Gauge invariance and corrections to deep inelastic scaling

The asymptotic behavior of the correction terms to scaling for the deep inelastic structure functions is analysed in the covariant parton model. It is shown by consideration of the ω→ 1 limit that the non-canonical secondary terms from the “hand-bag” diagram cannot be made gauge invariant. The large ω behavior of the terms is also considered, and it is concluded that scale breaking effects are likely to be significant at large values of ω.     

Leading corrections to finite-size scaling for mixed-spin chains

The leading corrections to finite-size scaling relations for the correlation length and twist order parameter of three mixed-spin quantum spin chains for the critical feature that develops at ϑ = π, corresponding to a change in the topological realization of the ground states, are identified. The text was submitted by the authors in English.     

Binary tree approach to scaling in unimodal maps

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.     

Logarithmic corrections to finite-size scaling in the four-state Potts model

The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz equations for equivalent eigenstates of a modifiedXXZ chain. Scaled gaps are found to behave for large chain lengthL asx+dL+0[(lnL)^–1], wherex is the anomalous dimension of the associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudesd are in agreement with predictions of conformai invariance. The implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x andd are found to be extremely difficult even with data available from large lattices,L500.     

Analytics of period doubling

Approximate analytic solutions for periodic orbits of the quadratic mapxrx(1–x) are developed using algebraic methods. These solutions form the basis of an exact algorithm which predicts the quantitative order of periodic points characteristic of the Feigenbaum scenario. The algorithm holds for any one dimensional unimodal map. A general procedure is developed which permits calculation of period doubling parameters for large period orbits from those of low period to any desired degree of accuracy. Explicit equations are given through second order.     

Universality,revisions of and corrections to scaling in fluids

PVT and internal energy data in the critical region of steam are accurately described by a thermodynamic potential based on renormalization-group calculations for Ising-like spin systems. The potential includes both “corrections-to-scaling” and “mixing-of-variables” terms.     

High-order perturbed corrections with scaling transformation

The performance of the so-called superconvergent quantum perturbation theory （Wenhua Hal et al2000 Phys. Rev. A 61 052105） is investigated for the case of the ground-state energy of the helium-like ions. The scaling transformation τ → τ/Z applied to the Hamiltonian of a two-electron atomic ion with a nuclear charge Z （in atomic units）. Using the improved Rayleigh-SchrSdinger perturbation theory based on the integral equation to helium-like ions in the ground states and treating the electron correlations as perturbations, we have performed a third-order perturbation calculation and obtained the second-order corrected wavefunctions consisting of a few terms and third-order energy corrections. We find that third-order and higher-order energy corrections are improved with decreasing nuclear charge. This result means that the former is quadratically integrable and the latter is physically meaningful. The improved quantum perturbation theory fits the higher-order perturbation case. This work shows that it is a development on the quantum perturbation problem of helium-like systems.     

Virial scaling corrections to Koopmans' ionization potentials in atoms

Relaxation corrections to Koopmans' approximation for atomic ionization potentials are derived on the basis of the virial theorem. They require no information in addition to that available in the Hartree-Fock computation for the neutral system, and considerably improve the agreement with more accurate ionization potentials.