The response of a damped anharmonic oscillator, being a model for many systems that undergo a bifurcation, is determined in the presence of small-amplitude parametric modulation of arbitrary dynamics and statistics. Inertia always stabilizes the trivial state and suppresses the growth of the bifurcating solution above the shifted stability threshold. The size of both effects depends on the modulation spectrum. Explicit formulas for threshold, bifurcating solution, and moments are given.

Within the framework of boson parquet-diagram summations in perturbation theory, we show analytically that several simple approximations lead inevitably to the radial distribution function $g\left(r\right)\mathrm{}$ which would be obtained with the Jastrow hypernetted-chain variational method. This is the first derivation of the Jastrow result from perturbation theory. Without mentioning pair-correlation functions, we have a clear interpretation of $g\left(r\right)\mathrm{}$ and the structure function, $S\left(k\right)\mathrm{}$, in terms of diagram sums.

The quantum behavior of two identical point vortices (e.g., in a superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ thin film) is studied. It is argued that this system obeys neither Bose nor Fermi statistics, but intermediate or $\theta$ statistics: We find that a single vortex in this system possesses quarter-fractional statistics (i.e., $\theta =\frac{\pi }{2} or 3\frac{\pi }{2}$). The source of the $\theta$ statistics is identified in the relative zero-point motion of the vortices.

Several statistical properties of the energy levels of a two-level atom interacting with one mode of quantized electromagnetic field are investigated in connection with possible "quantum-chaotic" behavior of the system.

We study an aggregation process which gives rise to compact clusters with no loops. A generalization of the node-counting theorem, applicable to such branched graphs (trees), is proved. This is used to determine the spectral dimension of these clusters on square and cubic embedding lattices by Monte Carlo simulations. The results are explained in terms of the geometric structure of these trees.

Using a simple example we show that the distribution for the energy levels for integrable systems is not the uncorrelated Poisson distribution as is commonly believed. In particular, the spectrum was found to be rather rigid. We conjecture that these are typical properties of the integrable quantum systems.

It is shown that very general features of the critical theory of the Yang-Lee edge singularity in two dimensions completely determine the way in which the theory realizes conformal invariance. This leads to the value $\sigma =-\frac{1}{6}$ for the edge exponent, and makes possible the calculation of the correlation functions.

We present a new method for treating classical (or quantal) wave equations in the shortwavelength (semiclassical) regime, based on a description of the wave in the ray phase space. The coherent-state representation is defined, and the equation which it obeys is given and solved under assumptions similar to those of conventional eikonal theory. As indicated by an example, the result is a smooth distribution on phase space which, when "projected" onto configuration space, yields a wave field with no caustic singularities.

We consider the two-dimensional Ising model on a rectangular lattice with one row of horizontal coupling strengths changed. We shall present our results for correlation functions with two energy-density operators in arbitrary positions, restricting ourselves here to the scaling limit. These exact results show an unexpected complexity.

Global scaling behavior for period-$n$ windows of chaotic dynamical systems is demonstrated. This behavior should be discernible in experiments.

A numerical simulation of electric breakdown is performed by means of a deterministic model of a two-dimensional network of resistors with nonlinear and irreversible characteristics. A breakdown of a resistor induces successive breakdowns of other resistors and finally the broken resistors form a percolation cluster. It has dendritic structure and is found to be a fractal with its dimension about 1.6.

The quantum continuum action for gauge theories in two dimensions is constructed in the axial gauge. For non-Abelian groups the action differs from the usual expectations by nonperturbative, noncoercive corrections, whose role is to ensure gauge invariance for the quantum field theory outside of perturbation theory. Consequently, ordinary perturbation theory and semiclassical approximations become inapplicable. Similar conclusions are expected to apply in more dimensions.

It appears possible to simulate the results of molecular-dynamics calculations of intramolecular and decomposition reactions by means of relatively simple one-dimensional maps.

Exploiting the solvability of the $d=2\mathrm{}$ nearest-neighbor Ising model we construct the critical surface in a thirteen-dimensional parameter space in the vicinity of the nearest-neighbor transition point. We see if the flows generated from this point by recent Monte Carlo renormalization-group transformations lie on this surface. We clarify and quantify the effects of truncation. We unearth a truncation scheme that should speed up the convergence to the fixed point even in $d=3\mathrm{}$.

The two-dimensional theory of gravity proposed by Jackiw is exactly quantized in the open case. The Wheeler-DeWitt equation is solved in closed form with and without cosmological constant. The theory has no degree of freedom and the unique wave functional reflects the classical aspects of the solution.

The renormalized trajectory for the O(3) Heisenberg model is found by use of a recently proposed microcanonical Monte Carlo method. On this trajectory, a remarkable improvement in the approach to the continuum limit is observed by evaluation of the mass gap.

It is shown that, in the context of an idealized ‘‘macroscopic quantum coherence’’ experiment, the predictions of quantum mechanics are incompatible with the conjunction of two general assumptions which are designated ‘‘macroscopic realism’’ and ‘‘noninvasive measurability at the macroscopic level.’’ The conditions under which quantum mechanics can be tested against these assumptions in a realistic experiment are discussed.

The quantum spectrum is studied in a corresponding classically nonintegrable triangular three-spin system with antiferromagnetic exchange coupling. A critical energy ${\mathrm{E}}_{\mathrm{c}}$ is identified numerically. Irregularly distributed (bandlike) levels and well-separated clusters of (localized) levels appear below and above ${\mathrm{E}}_{\mathrm{c}}$, respectively. The statistical average of second-order differences of eigenvalues shows a scaling behavior when the spin magnitude is increased towards +∞.

The temporal evolution of binary mixtures of hard rods in a ring is simulated in a computer with random initial velocities ±v. The time the system takes to reach a Maxwellian distribution dramatically diverges as the mass ratio ε→1 and it also increases, although rather slowly, when ε→∞. A negative ‘‘long-time tail,’’ i.e., a slow, power-law decay in the velocity autocorrelation function at large values of the time t, is observed whose behavior changes from ${\mathrm{t}}^{\mathrm{-}3}$ to ${\mathrm{t}}^{\mathrm{-}\mathrm{\delta }}$, δ≲1, as ε is increased from ε=1. .AE

Physical Processes are reviewed as computations, and the difficulty of answering questions about them is characterized in terms of the difficulty of performing the corresponding computations. Cellular automata are used to provide explicit examples of various formally undecidable and computationally intractable problems. It is suggested that such problems are common in physical models, and some other potential examples are discussed.