It is shown that anomalous diffusion (i.e., nonlinear growth of mean square displacements) can be caused by a specifically chaotic mechanism. It depends on deterministic diffusion and intermittency and gives rise to mean square displacements which grow asymptotically like $t^{\nu }$ with $0<\mathrm{}\nu <~1$ or like $\frac{t}{\ln t}$, depending on universality classes.

In the context of Einstein-Cartan-Sciama-Kibble theory, modified to include a time-dependent gravitational coupling $G\mathrm{}\left(t\right)\mathrm{}$, torsion is shown to be proportional to the variation $\frac{\dot{G}}{G}$. The magnitude of this effect based on recent experimental evidence is substantially greater than the usual spin-dependent torsion effects predicted by standard Einstein-Cartan-Sciama-Kibble theory. It is suggested that the detection of a time-varying $G$ could provide evidence for the existence of torsion.

The survival probability of a particle that performs a random walk on a linear chain with randomly distributed traps is considered. An asymptotically exact solution was found, valid for large step numbers $n$ and all trap concentrations $c$. The survival probability depends upon the combination $x={[-\pi \ln \mathrm{}(1-c\left)\right]}^{\frac{2}{3}}{n}^{\frac{1}{3}}$ of the variables $n$ and $c$ only and is in leading order equal to $\left(\frac{8}{\pi }\right){\left(\frac{2}{3}\pi \right)}^{\frac{1}{2}}{x}^{\frac{3}{2}}\exp (-\frac{3x}{2})$. The analytical result includes correction terms and is confirmed by Monte Carlo simulations.

We investigate stability of Freund-Rubin—type solutions of eleven-dimensional supergravity in which extra spaces are direct products of compact spaces of constant curvature. It is shown that the solutions of the form $\mathrm{Ad}{\mathrm{S}}_4\times B_7$ with $B_7=S^5\times S^2,S^4\times S^3,S^3\times C{P}^2,S^3\times S^2\times S^2$, etc., are unstable against relative dilatations between the subspaces.

We investigate nonrandom, half-random, and random variants of clusters generated by the "squig" process, using renormalization or Monte Carlo methods. Each variant is compared to two-dimensional percolation clusters at criticality from the viewpoints of the fractal dimensionalities of the whole, the backbone, the multiconnected parts, the ring hulls, the backbone links, the shortest paths, and the recurrences of diffusion, hence the fracton dimensionality. The random variant fits very well when its only parameter $s$ is a bit above 0.4.

We have performed Monte Carlo random-walk experiments on a one-dimensional periodic lattice with a trapping site using the logistic map as a generator of pseudorandom numbers. Comparison with analytical results indicates that, when it has sensitive dependence to the initial conditions, this map provides a true pseudorandom generator.

By numerically computing orbits for a chaotic, one-dimensional, many-body system placed between two thermal reservoirs, we verify directly that its energy transport obeys the Fourier heat law and we determine its thermal conductivity $K$. The same value of $K$ is independently obtained by use of the Green-Kubo formalism. These numerical studies verify that chaos is the essential ingredient of diffusive energy transport, and they validate the Green-Kubo formalism.

An explicit expression is derived for the quantum gravitational vacuum state by use of the saddle-point approximation and a conformally invariant action. It consists of a superposition of zero-energy classical vacua that differ in the topology both of the local Lorentz frames (leading to $\mathrm{CP}$ nonconservation) and of space itself. Space is found to have Planck-size wormholes and spinorial knots, and is surrounded by a vapor of disconnected submanifolds. The scattering matrix constructed on this vacuum is not unitary but the superscattering matrix conserves probability.

We consider a new Lie-algebraic framework for the Pöschl-Teller potential, in which a single [$\mathrm{sp}\mathrm{}(2,R)\mathrm{}$ Casimir] operator has both bound and scattering states. This approach allows the determination of the $S$ matrix by purely algebraic means.

We introduce a scalar field in eleven-dimensional supergravity through the dual transformation of the totally antisymmetric torsion. All supersymmetries are spontaneously broken by the scalar field and the remaining four-dimensional Einstein space corresponds to anti-de Sitter space. The holonomy group of the seven-dimensional compact manifold is found to be an SO(7) subgroup of the original SO(8).

A nonlinear stochastic process is presented that, for each realization and for large times, reproduces Lüder's projection postulate. The corresponding density operator undergoes a linear evolution reproducing von Neumann's projection postulate. The violation of the Bell inequality, for instance, is described with the two apparatus acting independently on the composed system.

The Hausdorff dimension $D_0$ of a strange attractor is argued to be the fixed point of a recursive relation, defined in terms of a suitable average of the smallest distances ${\delta }_i$ between points on the attractor. A fast numerical algorithm is developed to compute $D_0$. The spread $\lambda$ in the convergence rates towards zero of the distances ${\delta }_i$ (uniformity factor) as well as the stability of the fixed point are discussed in terms of the entropy of the ${\delta }_i$ distribution.

For a model quantum system the distributions of the spacings between adjacent levels in specified intervals of the energy spectrum are fitted by the Brody distribution. This analysis displays for the first time the expected transition from regularity to irregularity, in terms of a smooth increase of the parameter of the Brody distribution from 0 to 1.

The model of cluster growth by diffusion-limited aggregation of clusters is studied in two dimensions and the cluster size distribution $n_s\mathrm{}\left(t\right)\mathrm{}$ is determined as a function of the cluster size $s$ and the time $t$. A dynamic scaling function of the form $n_s\mathrm{}\left(t\right)\mathrm{}\sim t^{-w}{s}^{-\tau }f\left(\frac{s}{t^z}\right)$ is proposed and is shown to lead to the scaling relation $w=(2-\tau )z$ among the critical exponents. The simulation results support this scaling law.

The phase shift introduced by rotational motion of a two-crystal neutron interferometer has been measured and found to agree with prediction within 0.4%. This agreement is obtained without making the in-crystal phase corrections employed in a recent study of a linearly accelerated three-crystal interferometer.

The lengths of molecular trajectories in an argonlike liquid simulated by molecular dynamics have been found to be fractal over a substantial range of scales with a Richardson coefficient of $\alpha =0.65\mathrm{}$ (Mandelbrot dimensionality of $d=1.65\mathrm{}$). The results obey a simple empirical relation with three parameters. We believe $\alpha$ to be a useful new parameter to describe the form of trajectories of molecules in fluids where none has previously been available, and that it is a state function.

Conformal invariance and unitarity severely limit the possible values of critical exponents in two-dimensional systems.

The spin van der Waals model may be regarded as an infinite-lattice-dimensional limit of the spin-½ anisotropic Heisenberg model. By solution of the generalized Langevin equation, time-dependent behavior is obtained. The geometry of realized Hilbert spaces lends a simple interpretation for critical dynamics.

Quasiparticles at the fractional quantized Hall states obey quantization rules appropriate to particles of fractional statistics. Stable states at various rational filling factors may be constructed iteratively by adding quasiparticles or holes to lower-order states, and the corresponding energies have been estimated.

The quantum Hall effect is shown be equivalent to a chiral anomaly in quantum electrodynamics. The integers $n_1$ and $n_2$ in the rationally quantized Hall conductance $g=\left(\frac{e^2}{2\pi \hslash }\right)\left(\frac{n_1}{n_2}\right)$ arise from vortex excitations carrying $n_1$ electrons and $n_2$ flux quanta. It is found that $n_2=2j$, where $j=l+\frac{1}{2}$ is the angular momentum per electron in the vortex. This explains the odd-denominator rule $n_2=2l+1\mathrm{}$.