Simple Scaling Properties Research Articles

Measurements of scaling properties for acoustic propagation in a single pore

An experimental investigation of sound propagation in a single pore of circular cross section is described. By working at low frequencies and in various pressure ranges, viscous and thermal penetration depths which range from less than, to much greater than, the pore radius are achieved. The system is an experimental model for a pore in either sound absorbing material or the “stack” region of a thermoacoustic engine. The measured acoustic response compares favorably with theoretical predictions. Deviations are attributed to the cell geometry. In certain ranges of pressure and frequency, simple scaling properties may be utilized to determine the relative viscosity and thermal diffusivity of gases. Measurements of both quantities (relative to those of argon) for helium, neon, nitrogen, krypton, and water vapor (viscosity only) are presented and compared with known values. Practical applications of the experimental technique are discussed.

Exact Solution and Multifractal Analysis of a Multivariable Fragmentation Model

Most theoretical kinetic approaches proposed so far to describe fragmentation processes rely on the assumption that the fragments can be characterized by a unique relevant variable, for example their size or mass h. We investigate the consequences of introducing additional variables by considering a fragmentation model in which the fragments are described by two internal variables, h, w, and a breakup rate proportional to hw~. The exact solution. already partly presented in reference ill, is derived and discussed in details with an emphasis on the fragment distribution function. It is shown that~ because of the random multip)icative nature of the process, no reduction to an effective one-variable problem is usually possible. The fragment mass distribution has no simple scaling properties; it rather exhibits multiscaling, a feature that is interpreted in the framework of a multifractal formalism.

Thermodynamics of the QCD1+1 nonrelativistic baryon gas.

The nonrelativistic baryon gas of QCD[sub 1+1] for SU(2) color is studied in the low density regime using two complementary approaches: In the classical limit, the Gibbs free energy can be evaluated analytically, yielding an exact value for the second virial coefficient and a bound for the equation of state at higher densities. Certain thermodynamic observables can already be given for the entire range of densities due to simple scaling properties. On the other hand, a quantum mechanical baryon-baryon scattering calculation yields the behavior of the second virial coefficient away from the classical limit down to low temperatures.

Fractal properties of escape from a two-dimensional potential

This paper summarizes a numerical investigation of the escape of particles from the two-dimensional potential V(x, y) = 1 2 x 2 + 1 2 y 2 - ϵx 2y 2 , for variableϵ but fixed energy h = 0.12. For ϵ < ϵ 1 ≡ 1 (4h) ≈ 2.08 , escape is impossible, but for ϵ≥ϵ 1, particles will escape. For such larger ϵ, the final outcome, as characterized by the time and direction in which particles escape, is a smooth function of initial conditions in certain phase space regions. However, in other regions the evolution evidences an extremely sensitive dependence on initial conditions indicative of a complex microscopic evolution. For sufficiently large values of ϵ> ϵ 2≈4.90±0.01, these sensitive regions appear to exhibit a fractal structure with simple scaling properties implying that, at a coarse-grained level, the late time behavior is quite regular. Specifically, the evolution is asymptotically Markovian in the sense that, for particles that have not yet escaped, at late times the coarse-grained escape probabilities per unit time tend toward time-independent values p ∞ > 0 independent of the initial conditions. The asymptotic values p ∞ and the rate of convergence towards these values were examined as a function of ϵ and the scale of the coarse-graining, and were observed to exhibit simple scaling behaviour. At least for values of ϵ≤5.7, p ∞∞( ϵ− ϵ 2) α , with a critical exponent α≈0.49±0.05. For a coarse-graining corresponding to a linear scale r=0.05, the time T required for convergence towards this value scales as T( ϵ)∞( ϵ− ϵ 2 - β , with β≈0.39 +0.14 -0.06. The difference γ≡α-β=0. 09 +0.04 -0.02. The value p ∞ appears independent of r, but the convergence time scales as T( r)∞ r - δ , with δ=0.08±0.03, seemingly independent of ϵ. To within statistical uncertainties, α − β − δ ≈0.

The gravitational field of string matter when the dilaton is massive

We study numerically the gravitational field of a star made of massive and neutral string states for the case in which the dilaton is massive. The solution exhibits very simple scaling properties in the dilaton mass. There is no horizon and the singularity is surrounded by a halo (the physical size of which is inversely proportional to the dilaton mass) where the scalar curvature is very large and proportional to the square of the dilaton mass.

Scaling properties of a transformation defined on cellular automaton rules

A one-parameter family of transformations defined on the set of all one-dimensional cellular automata is studied. The class of a given cellular automaton is unchanged under the set of these transformations. For class 3 cellular automata, transformed statistical quantities satisfy simple scaling properties.

Studies on multiple electromagnetic excitation with fast heavy ions: scaling properties for the excitation of rotational-vibrational states

Multiple electromagnetic excitation of a rotational band coupled to high-lying giant dipole states is studied theoretically. The excitation of the rotational band is treated in the sudden approximation. With the help of rather simple scaling properties one can obtain insight into the physics of the process. The authors give explicit results for 16O-238U and 238U-238U collisions in the range of E/A approximately=100 MeV up to the ultrarelativistic limit ( gamma to infinity ).

Quantum statistical derivation of the hydrodynamics of a plasma model

We derive macroscopic electrohydrodynamical equations of Euler and Maxwell from the many-particle Schrödinger equation of the Jellium model, subject to viable initial conditions, together with certain simple assumptions of macroscopic regularity. The dynamics of the model becomes tractable, on a suitable macroscopic scale and in the limit where its size becomes infinite, because of simple scaling properties and the long range of the Coulomb forces. Our derivation of its phenomenological dynamics is obtained via that of the classical Vlasov equation.

Exact expressions for scalar modal eigenvalues and group delays in power-law optical fibers

Considerable use is made of power-law refractive-index profiles in fiber optics. These profiles have simple scaling properties that are exploited in this paper to obtain formulas for the exact modal propagation constants and group delays. When an infinite profile is assumed, the results of the exact, WKB, and geometric optics theories are all shown to agree. This conclusion remains valid when linear material dispersion is included. The effect of the cladding is discussed. The results are correct for noncircular fibers of power-law type.