We numerically study the excitation mechanism of the hydrogen atom in a microwave field and show that quantum mechanics imposes limitations to the classical chaotic diffusion. In particular a multiphoton resonance pattern has been found. We suggest that a direct laboratory experimental verification of these phenomena should be possible.

A condensed-matter analog of (2+1)-dimensional electrodynamics is constructed, and the consequences of a recently discovered anomaly in such systems are discussed.

We consider a grand canonical ensemble of self-avoiding walks and study its properties on a Bethe lattice of coordination number $q=3\mathrm{}$ and in one dimension, without using the $n\to 0$ trick. The study enables us to identify the ordering field as the activity for the walk ends. We also identify the order parameter of the problem for the first time: The order parameter is the probability of obtaining an infinite self-avoiding walk (reaching the "boundary"), having started at the origin in some direction.

Dynamic light-scattering measurements on fractal aggregates of colloidal silica show that the mean decay rate (Rayleigh linewidth) of the intensity autocorrelation function scales like $q^{2.7}$, where $q$ is the momentum transfer. This result is compared to the $q^3$ prediction for percolation clusters and the more complex results derived here for cluster-cluster aggregation.

A new technique is described for direct measurement of mass changes of a conducting polymer electrode during electrochemical cycling. A quartz microbalance is used, with one of its contacts serving as the working electrode. Its high sensitivity permits resolution of mass changes less than 100 ng/${\mathrm{cm}}^2$. The results demonstrate that on reduction of poly(pyrrole perchlorate), charge compensation is achieved by diffusion of the lithium cation into the polymer and not the perchlorate anion into solution.

The Dirac propagator ${(\gamma p-m)}^{-1\mathrm{}}$ has been calculated as a path integral over a recently proposed classical action. Thus a new formulation is given to the longstanding problem of extending the path integration to discrete quantum spin as an integral over a continuous variable.

The results of microcanonical simulation of an Ising model defined on a truncated, lattice-embedded, fractal, Sierpinsky carpet of dimension ∼ 1.86 are presented. Estimates for a variety of exponents give, via hyperscaling at criticality and via the behavior of the spin-spin correlation away from criticality, consistently, a dimension ∼ 1.66±0.10.

The diffusion-limited-aggregation model is generalized in order to take into account the surface effects playing an essential role during most of the growth processes. With variation of a parameter of the model the geometry of the clusters generated in the Monte Carlo simulations gradually changes from the randomly branched diffusion-limited-aggregation clusters into compact, nearly regular, snowflakelike patterns. The deposition of particles along a line results in patterns similar to those observed in the experiments on directional solidification.

We show that supersymmetric quantum mechanics may be used to interrelate the spectra of different atoms and ions. This supersymmetry is broken by electron-electron interactions.

The dynamics of a time-dependent quantum system can be qualitatively different from that of its classical counterpart when the latter is chaotic. It is shown that small noise can strongly alter this situation.

We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical $d$-dimensional system is mapped onto the statics of an associated quantum system and of its classical ($d+1\mathrm{}$)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\Delta$ and the static susceptibility exponents of three models.

The estimation problem of the mean-free-path parameter in the case of limited statistics and censored exponential distributions is outlined. A new estimator is suggested which, although biased, has desirable properties and could preferably be used for analyzing nuclear emulsion data as well as data from CR-39 etched track detectors in the search for anomalous behavior of heavy-ion fragments.

A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, $f(x,y)=\Sigma c_{\mathrm{ij}}{x}^i{y}^j$, our approach uses two one-variable series in $x$, $f$ and $\frac{\partial f}{\partial y}$, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are $\gamma =1.2378\mathrm{}\left(6\right)\mathrm{}$, $2\nu =1.2623\mathrm{}\left(6\right)\mathrm{}$, and $\eta =0.0375\mathrm{}\left(5\right)\mathrm{}$; correction-to-scaling exponents are ${\theta }_{\chi }=0.52\mathrm{}\left(3\right)\mathrm{}$ and ${\theta }_{{\xi }^2}=0.49\mathrm{}\left(4\right)\mathrm{}$; and the subdominant critical amplitude ratio is $\frac{a_{\xi }}{a_{\chi }}=0.83\mathrm{}\left(5\right)\mathrm{}$.

The $N$-color Ashkin-Teller model is solved exactly in the large-$N$ limit in two dimensions. The phase diagram is found. It is shown that for positive four-spin coupling the transition is first order while for negative four-spin coupling the transition is continuous and Ising type. The specific heat near the second-order transition is calculated and it is found to be finite at the transition because of large corrections to scaling. The latent heat and correlation length at the first-order transition are also calculated.

Results from the solution of the inverse-scattering problem in three dimensions are used to obtain representations of the potential and of the zero-energy wave function in terms of the scattering amplitude, the bound states, and the positive-energy wave functions. Some simple consequences are derived.

The linear stability analysis for solitary waves is revisited. It is shown that the generally ignored defective degeneracy of the Goldstone mode is associated with a perturbation with algebraic time dependence generating a solitary wave moving with infinitesimally changed velocity. If damping is switched on, the defective degeneracy of the Goldstone mode is lifted by the occurrence of a universal relaxation mode. Similarly, soft-instability modes of solitary waves are also defectively degenerate and thus accompanied by an algebraic mode.

The Einstein quadrupole formula, in the version "news function equals third time derivative of transverse quadrupole moment," is established for a specific quasi-Newtonian system by a method based upon curved-space null cones.

A well defined distribution-theory Lagrangian for a classical charged point particle is constructed which reveals the connection between the Lorentz-Dirac equation and a conserved energy tensor. No renormalization is required. Everything is finite.

The depolarization of rotating spins that perform a random walk on a linear chain is investigated. For a large number of steps $n$ the decay of polarization is given by $\exp (-\mathrm{const}\times {\sigma }^{\gamma }n)$, where $\sigma$ is the width of the frequency distribution and $\gamma \approx \frac{4}{3}$. This behavior follows from the distribution of the spans of a one-dimensional random walk; it is derived in an approximation and verified by numerical simulations. A lower bound on the polarization decay is also given which confirms the dependence on $n^1$.

Consider a macroscopic system rendered unstable by both thermal fluctuation and quantum tunneling. Kramer's classical theory of the rate of decay by thermal activation is extended to lower temperatures where quantum tunneling prevails. By means of a functional-integral approach, a general formula for the decay rate is derived which describes the transition between the high- and low-temperature regimes. The influence of dissipation on the decay rate is emphasized.