We consider intermittency in inverted-pitchfork bifurcation of a 1D dissipative map and of a 2D conservative map. Exact solutions to respective renormalization-group equations are constructed and scaling ratios obtained. The effect of noise is considered and scaling laws in the presence of noise are deduced. Results for the saddle-node bifurcation in 2D area-preserving maps are presented.

We discover a low-energy equivalence between two apparently unrelated Lagrangians with fractional statistics. Exploiting this equivalence, we are able to study the quantum disordered phase of the nonlinear σ model with Hopf term. We find that the quasiparticles in the disordered phase also have fractional statistics. There appears to be a dual relationship between the ordered and disordered phases.

The Julia set associated with the Newtonian map for the solution of $z^3$-1=0 in the complex plane is investigated by purely analytic means. The hierarchical organization of this strange repeller is explained from basic principles. The family of generalized fractal dimensions and the equivalent spectrum of scaling indices is evaluated using an extension of the Hentschel-Procaccia renormalization scheme. Our quantitative results are in excellent agreement with direct numerical computations.

This paper gives a new classification of the eigenphase shifts caused by a noncentral potential in the Schrödinger equation and generalizes the well-known low-energy behavior of phase shifts for central potentials. It also presents a new Levinson theorem for each eigenphase shift.

Motivated by recent models of Bak, Tang, and Wiesenfeld we study dissipative transport in open systems. A simple continuum equation is constructed to describe fluctuations around a steady state in a flowing ‘‘sandpile.’’ The principle of scale invariance and self-similarity is understood in terms of a conservation law in dynamics. A dynamic renormalization-group calculation allows us to determine various critical exponents exactly in all dimensions.

The influence of the Lyapunov instability on a trajectory in phase space is investigated numerically for a Lennard-Jones fluid under a constant shear rate γ. The time dependence of the distance d(t,γ;Γ) between two individual trajectories, starting at the same phase point Γ, but moving either with or without an imposed shear, can be described by d(t,γ;Γ)=Aγ exp(t/${\mathrm{\tau }}_{\mathrm{L}}$), where ${\mathrm{\tau }}_{\mathrm{L}}$ is the inverse of the largest Lyapunov exponent of the system without shear. A possible explanation of this behavior and the consequences for the validity of linear-response theory are discussed.

We use the exact renormalization procedure, introduced by Machta [Phys. Rev. B 24, 5260 (1981)], to evaluate the waiting time density for a walker to jump to one of its nearest neighbors on a decimated lattice, in terms of the original waiting time densities on the nondecimated lattice. We explicitly identify all the fixed points and their corresponding domain of attraction for the one-dimensional case and for the Sierpinsky gasket.

The direct determination of the f(α) singularity spectrum from experimental data is a difficult problem. This Letter introduces a simple method for computing f(α) based on the theorems of Shannon, Eggelston, and Billingsley which is markedly superior to other recently proposed methods, especially when dealing with experimental data from low-dimensional chaotic systems where the underlying dynamics are unknown.

We present a generalization of Hamilton’s geometric theory of turns, originally invented for SU(2), to the noncompact group Sp(2,R) relevant in a variety of physical applications.

We describe a novel feature of certain arrays of N-coupled nonlinear oscillators. Specifically, the number of stable limit cycles scales as (N-1)! To accomodate this huge multiplicity of attractors, the basins of attraction crowd ever more tightly in phase space with increasing N. Our simulations show that for large enough N, even minute levels of noise cause the system to hop freely among the many coexisting stable attractors.

Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager’s critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d¯=(187/96 and a percolative susceptibility exponent γ=(91/48, in good agreement with existing numerical estimates. This d¯ is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent ${\mathrm{y}}_{\mathrm{J}}$=(13/24 controlling the crossover between clusters and droplets.

For a dilute gas of fermions interacting via an arbitrary pair potential in d=2 dimensions, we show that the many-body ground state is unstable to s-wave pairing at T=0 if and only if a two-body bound state exists. We further obtain, within a variational pairing Ansatz, a smooth crossover from a Cooper-paired state (${\xi }_0$${\mathrm{k}}_{\mathrm{f}}$≫1) to a Bose condensed state of tightly bound pairs (${\xi }_0$${\mathrm{k}}_{\mathrm{f}}$≪1). We briefly discuss non–s-wave superconductors. Insofar as this model is appli- cable to the high-${\mathrm{T}}_{\mathrm{c}}$ materials, they are in the interesting regime with the coherence length ${\xi }_0$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{-}1}$.

The ratio $\frac{h}{m^{\prime }}$ of Panck's constant to the observable Cooper-pair mass ($\sim 2m_e$) is determined directly using a rotating superconducting ring by measuring the London moment flux $\left(\frac{2m^{\prime }c}{e^{*}}\right)\omega S$, where $S$ is the area bounded by the ring, in units of the flux quantum ($\frac{\mathrm{hc}}{e^{*}}$). The flux in the ring is zero for a set of rotation frequencies which are equally spaced by $\Delta \nu$, and $\frac{h}{m^{\prime }}=2S\Delta \nu$. We have measured $\Delta \nu$ and $S$ with a precision of 5 ppm and an accuracy of 21 ppm. Our result for niobium, ${\left(\frac{m^{\prime }}{2m_e}\right)}_{\mathrm{Nb}}=1.000\mathrm{}084\left(21\right)\mathrm{}$, disagrees with theoretical predictions of 0.999 992.

We give an approximate formula for the Lyapunov exponent of the product of $N$ independent random matrices $M_k$ all distributed according to the same probability distribution. We consider the case of a Bernoullian distribution, $M_k=A$ with probability $p$ and $M_k=B$ otherwise. The method holds fixed the number of matrices of a given type, thus reducing fluctuations. We compare our analytic result to numerical calculations of the Lyapunov exponent for matrices $A$ and $B$ related to localization in the one-dimensional Anderson model where the approximation appears very precise for a wide range of energies. We give the fluctuation spectrum for the Lyapunov exponent also in terms of our approximation.

The behavior of solitary waves has been studied both experimentally and numerically in a strongly dissipative Toda lattice. In the absence of dissipation the system is an integrable soliton system, but due to dissipation the solitary wave decreases, and develops a tail. Numerical simulation of the head-on collision of two solitons with different amplitudes indicates that the scattering is nevertheless purely elastic. This suggests that certain parts of soliton theory should be applicable in systems where the wave itself is not conserved.

The effective interaction between interfaces or membranes is renormalized by thermally excited shape fluctuations of these surfaces. For a large class of interactions, this renormalization leads to a complex phase diagram which is governed by an unusual renormalization-group flow. This flow exhibits a line of renormalization-group fixed points and leads to essential singularities and nonuniversal critical exponents; it must, however, be distinguished from the well-known Kosterlitz-Thouless flow since it has a parabolic character.

The proper classical limit $\hslash \to 0$ of the quantum Bethe-Ansatz thermodynamics gives a new set of integral equations for the thermodynamics of the classical massive-Thirring-sine-Gordon model. The theory contains the thermodynamic information about breathers and solitons, and reproduces the transfer-matrix-method result for arbitrary coupling constant.

We point out difficulties with our proposed local quantum field theory of small violations of the Pauli principle and give a qualitative rationale for the surprising rigidity of the exclusion principle.

It is proposed to set stringent limits on possible nonlinear corrections to ordinary quantum mechanics by searching for the detuning of resonant transitions. A suggested nonlinear generalization of quantum mechanics is used to show that such detuning would be expected in the rf transition in ${}_{}{}^9{}_{}{}^{}\mathrm{Be}_{}^{+}{}_{}{}^{}$ ions that is used to set frequency standards. Measurements at the National Bureau of Standards already set limits of order ${10}^{-21\mathrm{}}$ on the fraction of the energy of the ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ nucleus that could be due to nonlinear corrections to quantum mechanics, with good prospects of improving this by 2-3 orders of magnitude.

We interpret the exchange-correlation potential of Kohn-Sham density-functional theory to be the work required to move an electron against the electric field of its Fermi-Coulomb hole charge distribution. We substantiate this interpretation in the exchange-only approximation by proving the satisfaction of a necessary condition as well as by application to few-electron atomic and many-electron metallic surface systems. This interpretation also provides insights into the exact Slater exchange potential.