Generalized Scaling Laws Research Articles

Reaction dynamics and statistical theory for the growth of hydrogen bonding clusters

The similarities between the formation of hydrogen bonds and polycondensation reactions are stated from the statistical viewpoint, and then taking the hydrogen bonding system of AaDd type as an example, the growing process of hydrogen bonding clusters is investigated in terms of the theory of reaction dynamics and statistical theory for polymeric reactions. The two methods lead to the same conclusions, stating that the statistical theory for polymerization is applicable to the hydrogen bonding systems. Based on this consideration, the explicit relationship between the conversions of proton-donors and proton-acceptors and the Gibbs free energy of the system under study is given. Furthermore, the sol-gel phase transition is predicted to take place in some hydrogen bonding systems, and the corresponding generalized scaling laws describing this kind of phase transition are obtained.

Generalization and consolidation of scaling laws of potential formation and associated effects in the GAMMA 10 tandem mirror

Generalized scaling laws for the formation of plasma confiningpotentials and the associated effectiveness of the potentials produced aresystematically investigated to find the physics essentials common to therepresentative tandem mirror operational modes of GAMMA 10, and toexplore novel extended operational modes from the scaling bases constructed.(a) The potential formation scalings are generalized using a novel finding ofwider validity of Cohen's strong ECH theorycovering the representative modes. (b) The potentials produced, in turn,provide a favourable novel scaling of the increase in the central cell electrontemperatures Te with increasing thermal barrier potentials ϕb, limited by theavailable ECH power. The scaling of Te with ϕb is well interpreted in terms ofthe generalized Pastukhov theory of plasma potential confinement. A detailedcomparison of the results from several related modified theories is also made.(c) Consolidation of the two major scalings of (a) and (b) in a tandem mirror iscarried out by the use of an electron energy balance equation for the first time.In addition, (d) an empirical scaling of ϕc with ECH power in the plug regionand the central cell densities are studied to discover whether there is the possibility of extending these theoretically well interpreted scaling data to parametersin the future scalable regime. There is also a discussion about numerical scalings in thethree dimensional parameter spaces.

Hopping in the glass configuration space: Subaging and generalized scaling laws

Aging dynamics in glassy systems is investigated by considering the hopping motion in a rugged energy landscape whose deep minima are characterized by an exponential density of states $\ensuremath{\rho}{(E)=T}_{g}^{\ensuremath{-}1}\mathrm{exp}{(E/T}_{g}),$ $\ensuremath{-}\ensuremath{\infty}<E<~0.$ In particular we explore the behavior of a generic two-time correlation function $\ensuremath{\Pi}{(t}_{w}{+t,t}_{w})$ below the glass transition temperature ${T}_{g}$ when both the observation time t and the waiting time ${t}_{w}$ become large. We show the occurrence of ordinary scaling behavior, $\ensuremath{\Pi}{(t}_{w}{+t,t}_{w})\ensuremath{\sim}{F}_{1}{(t/t}_{w}^{{\ensuremath{\mu}}_{1}}),$ where ${\ensuremath{\mu}}_{1}=1$ (normal aging) or ${\ensuremath{\mu}}_{1}<1$ (subaging), and the possible simultaneous occurrence of generalized scaling behavior, ${t}_{w}^{\ensuremath{\gamma}}[1\ensuremath{-}\ensuremath{\Pi}{(t}_{w}{+t,t}_{w})]\ensuremath{\sim}{F}_{2}{(t/t}_{w}^{{\ensuremath{\mu}}_{2}})$ with ${\ensuremath{\mu}}_{2}<{\ensuremath{\mu}}_{1}$ (subaging). Which situation occurs depends on the form of the effective transition rates between the low-lying states. Employing a ``partial equilibrium concept,'' the exponents ${\ensuremath{\mu}}_{1,2}$ and the asymptotic form of the scaling functions are obtained both by simple scaling arguments and by analytical calculations. The predicted scaling properties compare well with Monte Carlo simulations in dimensions $d=1\ensuremath{-}1000$ and it is argued that a mean-field-type treatment of the hopping motion fails to describe the aging dynamics in any dimension. Implications for more general situations involving different forms of transition rates and the occurrence of many scaling regimes in the $t\ensuremath{-}{t}_{w}$ plane are pointed out.

Generalized Scaling Laws of the Formation and Effects of Plasma-Confining Potentials for Tandem-Mirror Operations in GAMMA 10

The main operations from 1979 to 2000 in the GAMMA 10 tandem-mirror, characterized in terms of the high-potential mode having kV-order plasma-confining potentials and the hot-ion mode yielding fusion neutrons with 10-20 keV bulk-ion temperatures, are summarized and generalized as a result of scalings of the formation and the effects of the potentials. The wide validity of potential-formation physics from Cohen's theory and the validity of the generalized Pastukhov's theory for the effects of thermal-barrier potentials on electron confinement are verified and consolidated through electron-energy balance.

High Rayleigh Number Turbulent Convection in a Gas near the Gas-Liquid Critical Point

${\mathrm{SF}}_{6}$ in the vicinity of its critical point was used to study turbulent convection up to exceptionally high Rayleigh numbers, Ra (up to $5\ifmmode\times\else\texttimes\fi{}{10}^{14}$) and to verify for the first time the generalized scaling laws for the heat transport and the large scale circulation velocity as a function of Ra and the Prandtl number, Pr, in a very wide range of these parameters. Both scaling laws obtained are consistent with theoretical predictions by Shraiman and Siggia [Phys. Rev. A 42, 3650 (1990)].

The spherical pinch: Generalized scaling laws and experimental verification of the stability of imploding shock waves in spherical geometry

The Spherical Pinch (SP) is a particular variation of the inertial confinement scheme, in that the two criteria required for plasma fusion breakeven condition, namely the attainment of a sufficiently high plasma temperature and of a product nτ in excess of 1014 cm-3. sec, are satisfied through two separate mechanisms. The advantage of this approach is that the scaling laws for breakeven are easier to satisfy than in the classical inertial confinement scheme. The first derivation of the scaling laws for spherical pinch experiments (Panarella & Savic 1983; Panarella 1987) was obtained under the simplifying assumption of an infinitely small central plasma in a sphere reaching the required high temperature for fusion, and of an undefined time delay Δt between the creation of such central plasma and the launching of the peripheral shock waves used to contain the central plasma. The analysis reported in this paper removes these assumptions and derives scaling laws which are generalized in their parameters, in that they explicitly contain the radius Rp of the central plasma and the time delay Δt.

Coronal Loops in Solar Active Regions

AbstractWe discuss the appearance of solar active regions as seen in EUV line radiation in terms of a system of simple static coronal loop models. The loop models are based on those of Rosner et al. (1978) and may be understood in terms of the scaling laws introduced by these authors. We present here some generalized scaling laws which demonstrate that the global properties of loops in fixed length are insensitive to everything but the overall heating rate or the basal pressure. The model and scaling laws appear to describe hot loops (T< 106K) well. Empirically determined temperatures and densities in cooler loops are not in agreement with coronal loop models.We have computed a simple model of an active region using a dipole magnetic structure. The individual loop structures were calculated on the basis of the two different assumptions of constant basal pressure and of constant heating rate. This paper presents a preliminary discussion of the results and a comparison with observation.

Renormalization group treatment of excluded volume effects in a polyelectrolyte chain in the weak electrostatic coupling limit. II. Decomposition of interactions and calculation of properties

A theoretical description of excluded volume in polyelectrolytes is considered in the flexible chain limit. The approach generalizes the work of Kholodenko and Freed and removes some deficiencies, present in their work, which become apparent in further extensions of the theory. Renormalization group calculations are pursued to order ε=4−d for the mean square end-to-end distance 〈R2〉, the radius of gyration 〈S2〉, and the second virial coefficient A2. The importance of including both a short range excluded volume interaction as well as a long range electrostatic one is stressed. The electrostatic interaction is effectively treated as a perturbation, so that the results are valid only in the weak electrostatic coupling region. The theory shows that polyelectrolyte excluded volume effects cannot generally be described in a scaling limit, and generalized scaling laws are derived which contain several parameters including the two interaction parameters. A qualitative comparison of the theory with available experimental data is included, and a program for evaluating effective charge distributions on polyelectrolytes is proposed based upon comparison of experiments with extensions of the theory to higher concentrations.

Generalized Scaling Laws in Threshold Pkenomena of Non-Equilibrium Systems

A general discussion on threshold phenomena, namely exponent behaviors of abrupt transitions between steady states near a threshold for a non-equilibrium system satisfying potential condition and having the arbitrary values of z and c, both characteristic parameters of the system, is given. It is shown that the scaling hypothesis in general homogeneous function form holds for threshold phenomena. The expressions of the threshold exponents, and of the threshold amplitudes, and , and the generalized scaling laws obeyed by them are all obtained. These laws reduce to the same as the scaling laws in critical phenomena when . The results support, in respect of exponent behaviors of transitions, the statement on the great similarity between the phase transitions in equilibrium systems and the abrupt transitions of steady states in non-equilibrium systems.

Critical Phenomenon Analogies in Single Mode One-Photon Laser System with Saturable Absorber

Beginning with a more complete quantum-mechanical description of the single mode one-photon laser system with a saturable absorber and using the catastrophe theory, we discuss critical phenomenon analogies, namely exponent behaviors of the abrupt transitions between steady states near a threshold, in the system. An isolated threshold with order is found. Four threshold exponents and , and four threshold amplitudes and , in two various spaces of control parameters, are obtained. These threshold exponents obey the generalized scaling laws, e.g., , of or depending on two various choices of the space of control parameters. The source for this departure of exponent behaviors from the usual scaling laws is also discussed.

Concentration and excluded volume dependence of coherent scattering functions for polymers: Chain conformational space renormalization group

The concentration dependent chain conformational space renormalization group theory is extended to a consideration of the explicit dependence on the strength of the excluded volume interaction in the nonscaling, ‘‘cross-over,’’ region between the single chain Gaussian and self-avoiding limits. Generalized ‘‘scaling laws’’ in this cross-over domain are derived without the use of perturbation theory or ε expansions. In the dilute and semidilute region these generalized scaling laws have distances scaled by the infinite dilution radius of gyration RG(ζ) where ζ is a scaling variable which vanishes in the Gaussian chain limit and tends to infinity in the good solvent limit. Hence, concentrations c are scaled by the overlap concentration c*ζ evaluated from the cross-over RG(ζ). Quantities like the coherent scattering function are then functions of kRG(ζ), c/c*ζ, and ζ with k the momentum transfer vector. The explicit ζ dependence persists through the cross-over regime for all c and is eliminated in the simple scaling limits ζ→0 or ∞. We use the theory to evaluate the coherent scattering functions for a single tagged polymer chain, for a set of tagged chains at concentration cT<c, as well as the full polymer solution. The concentration and excluded volume dependences of a tagged chain radius of gyration and the full solution correlation length are evaluated, and the former is studied as a function of the ratio of the tagged chain molecular weight to that for an average solution chain.

Exponent Behaviors of Single Mode Laser System with Saturable Absorber Near Threshold

In this letter we discuss the exponent behaviors near a threshold of single mode laser system in the presence of a saturable absorber. Four threshold exponents, and , and four threshold amplitudes and are obtained. They obey the generalized scaling laws of .

Final state hadrons in inclusive electroproduction processes

Inclusive deep inelastic electroproduction processes, in which one (or more) hadrons in the final state are observed in coincidence with the electron, are investigated using Regge-Mueller language, via the inclusive virtual photoproduction processes. The hadron distributions and generalized scaling laws are obtained. Assuming that Pomeron is the dominant Regge trajectory, it is a simple pole, and therefore factorizes at high energies. The contribution of the isospin-carrying secondary trajectories cause charge asymmetries in the central and current fragmentation regions, and these increase with Q 2, for fixed total energy. Average multiplicity grows logarithmically with energy, as in the hadronic case. However, in the Bjorken limit, the major contribution comes from the current fragmentation region, contrary to the hadronic case; and the contribution of central region scales, and increases with increasing ω −1. Finally, we show that the average transverse momentum of the produced particles is limited, and this limit depends on the density of the particles in the phase space (rate of increase of average multiplicity with ln s), and ω. All the predictions are consistent with the data.

On the π+/π- asymmetry in inclusive electroproduction of hadronsasymmetry in inclusive electroproduction of hadrons

Mueller's Regge analysis, combined with the so-called generalized scaling laws of the parton model, predicts an increase in the π+/π- asymmetry in the photon fragmentation region for increasingQ2 when all other relevant variables are kept fixed.

Critical behaviour of the anisotropic Ising model

The first eleven terms of the high temperature susceptibility series for the anisotropic Ising model on a simple cubic lattice are calculated. The series is analysed to obtain estimates of the critical temperature and critical exponent for various values of the anisotropy parameter eta . Strong evidence is found for an apparent continuous variation of the exponent with eta which is in contradiction with the discontinuous change at eta =0 which has been generally believed to occur. The detailed variation of the critical temperature with eta is also in disagreement with predictions from generalized scaling laws.

Generalized Scaling Laws for the Electroproduction of Hadrons

The light-cone structure suggested by the quark model is used to propose a generalized scaling law in electroproduction with detection of one final hadron. At fixed momentum transfer between initial and final hadrons, the cross section scales in two variables: the usual Bjorken variable and the fractional longitudinal momentum of the final hadron.

Scale Invariance and Current Commutators Near the Light Cone

Consequences of the assumption that current commutators have power-type singularities near the light cone are derived. An extension of Wilson's short-distance expansion of commutators is suggested for almost lightlike distances. Consequences for power-type singularities are derived for processes involving currents. Especially, generalized scaling laws are obtained for inelastic electron-nucleon scattering. The connection with Regge behavior is discussed. An application to electron-positron annihilation into hadrons is also made.