Simple Scaling Theory Research Articles

Critical behavior of a three-dimensional kinetic gelation model.

We present extensive results from a computer-simulation study of a kinetic growth model for radical-initiated irreversible gelation. Lattices as large as 100\ifmmode\times\else\texttimes\fi{}100\ifmmode\times\else\texttimes\fi{}100 were used to examine polymerization of a system of tetrafunctional and bifunctional monomers with initiator concentrations ${c}_{I}$, ranging from 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}2}$ to 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}6}$. The cluster-size distribution shows unexpected oscillations which become increasingly pronounced as ${c}_{I}$\ensuremath{\rightarrow}0. The scaling properties of the cluster-size distribution cannot be described by simple droplet scaling theory, and we propose a generalized form for the scaling. The bulk properties show critical exponents which are independent of ${c}_{I}$ and identical to percolation values within the errors. The amplitude ratio ${C}^{\mathrm{\ensuremath{-}}}$/${C}^{+}$ is not independent of ${c}_{I}$. The backbone of the largest cluster at the gel transition is also investigated and its fractal dimension is found to be distinctly larger than that of a random percolation cluster at ${p}_{c}$.

Relationship between path-integral and scaling theories of small polarons.

The detailed correspondence between the Emin-Holstein scaling theory and Feynman path-integral method for small-polaron formation is presented. In particular it is shown that the first-cumulant approximation to the path integral using a harmonic trial action in the adiabatic limit is identical to the use of a Gaussian trial wave function in the Emin-Holstein one-parameter scaling theory for the ground-state energy with polaron radius. This is then generalized to a simple two-parameter scaling theory involving both the polaron radius as well as nonadiabaticity of the electron-phonon interaction.

Scaling theory for classical anisotropic Heisenberg chain

The classical anisotropic Heisenberg chain (with Jperpendicular to /J/sub ///= alpha ) is treated by a simple scaling theory valid in the low-temperature limit (K identical to J/sub ////KBT>>1) for any alpha between 0 (Ising) and 1 (isotropic case). The correlation lengths are given in terms of universal single-variable scaling functions Phi /sub //, perpendicular to /: 1/ xi /sub //, perpendicular to /=(cosh-1(1/ alpha )) Phi /sub //, perpendicular to /(K(1- alpha 2)12/+F( alpha )). The term F( alpha ), given in the text, is negligible except when alpha approximately 0. The asymptotic forms of the functions are: Phi /sub ///(x) approximately Phi perpendicular to (x) approximately A/x for x >1 (where A, B and C are constants), giving the characteristic 'Heisenberg-like' and 'Ising-like' behaviours occurring on either side of the crossover which takes place at x approximately 1. Exact critical and crossover exponents are deduced.