We report the dependence of the percolation threshold of the three-dimensional sticks systems on aspect ratio and on macroscopic anisotropy. This Monte Carlo study is the first determination of percolation thresholds for randomly oriented objects in three-space. The results show that the above dependence is determined by the excluded volume of the sticks. However, the total excluded volume of randomly oriented objects is lower than that of the same objects in parallel alignment.

A rigorous theory for low-energy scattering in quantum systems in $n$ dimensions, $n>~2$, with general, possibly long-range, interactions is presented. It gives new insights into lowenergy phenomena; e.g., for $n=2\mathrm{}$ the controversy around the existence of scattering parameters is resolved. Furthermore, the important role of a parametrization of scattering observables in the study of spin-polarized quantum systems is illustrated. For $n=3\mathrm{}$, an analytic expansion of the phase shift and scattering length for charged, finite-size particle scattering, around their neutral counterparts, is derived.

The need for inducing a probability distribution from partial data and the complementary problem of the analysis of an observed distribution in terms of fewer relevant variables occur in many branches of physics. For reproducible experiments, consistency conditions which must be satisfied by any algorithm for inferring a discrete probability distribution with given averages are formulated. The only consistent algorithm is the one leading to the distribution of maximal entropy subject to the given constraints.

A new analytic method for the determination of quasinormal oscillations of a black hole is presented. It is based on a connection between the quasinormal modes and the bound states of the inverted black-hole potentials. With use of this method, the quasinormal frequencies of a Schwarzschild, a Reissner-Nordström, and a slowly rotating Kerr black hole are determined in the eikonal approximation, Moreover, we provide evidence for the stability of an extreme Kerr black hole.

It is shown how to obtain the Dirac commutator, $\mathrm{}[x,p]=i\hslash$, from the London-Weyl formalism.

An effective-spin model for finite-temperature QCD is derived by use of a variant of the Migdal-Kadanoff renormalization group. Mean field theory and [for SU(3)] Monte Carlo simulation are applied to the model in order to find the deconfining phase transition. The results are in very good agreement with Monte Carlo simulations of the full theory. Mean field theory predicts that the deconfining transition in pure $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ lattice gauge theory is first order for all $N>~3$.

To describe the irreversible growth of linear polymers, we introduce a new type of correlated walk, related to the zero—initiator-concentration limit of the kinetic gelation model. Our model simulates real polymer growth by permitting the walker to form the next bond from the unsaturated monomers at the neighboring sites of its present location. A heuristic kinetic argument of the Flory type suggests a fractal dimension $d_f=\frac{\mathrm{}(d+1\mathrm{})\mathrm{}}{2}$, in agreement with our Monte Carlo and series expansion results (including a logarithmic correction at the upper critical dimension $d_c=3\mathrm{}$).

Classical diffusion is studied on random chains with nearest-neighbor transfer rates having Gaussian distributions about sufficiently large systematic rates. Both the cases of static and dynamical white-noise randomness are considered, for symmetric as well as for asymmetric transfer rates. Probability densities for displacements along the chains are determined exactly in the continuum limit valid at long times, and are found to be Gaussian. The variance increases linearly in time for dynamical disorder and shows a qualitatively new quadratic growth for static disorder.

We construct arbitrary matrix elements of the quantum evolution operator for a wide class of self-adjoint canonical Hamiltonians, including those which are polynomial in the Heisenberg operators, as the limit of well defined path integrals involving Wiener measure on phase space, as the diffusion constant diverges. A related construction achieves a similar result for an arbitrary spin Hamiltonian.

A practical method is presented for calculating many-parameter renormalization-group trajectories for both spin systems and lattice gauge theories from Monte Carlo computer simulations. It provides a direct check on the assumption of short-range renormalized interactions, and preliminary work indicates improvement of the convergence in the calculation of critical exponents.

A local entropy density for a fluid obeying the revised Enskog equation is defined and the entropy balance equation is derived. The source term is nonnegative for arbitrarily strong inhomogeneities.

A Euclidean lattice formulation is presented of the general fourth-order gravitational action involving $R_{\mu \nu }^2$ terms. This lattice theory is bounded and reflection positive. The existence of a positive Hilbert space and Hamiltonian, and hence unitarity, follow then from the standard nonperturbative Osterwalder-Schrader construction.

A real-space renormalization-group method is used to investigate particle motion on percolation lattices in the presence of an external force $\overrightarrow{\mathrm{K}}$. At the percolation threshold $p_c$ the particle position, $〈R〉$, varies as $〈R〉\propto t^{\chi }$, where $\chi <1\mathrm{}$ and $t^{\chi }$ gives the mean square displacement. Near $p_c$, the linear response theory is found to hold only asymptotically under the condition $\gamma \equiv \frac{\mathrm{Kl}}{k_{\mathrm{B}}T}\lesssim {\xi }^{-1\mathrm{}}$ or $\gamma \lesssim {\mathrm{}\left(\mathrm{wt}\right)\mathrm{}}^{-\frac{\chi }{2}}$, where $l$ is a lattice constant, $\xi$ is a coherence length, and $w$ is a transition rate.

We present a new NMR method for the investigation of rotational Brownian motion based on the appearance of dipolar relaxation as the particles rotate. Our experimental results obtained on submicrometer polystyrene spheres in water confirm quantitatively the basic laws of rotational Brownian motion. The effects of hydrodynamic interactions and long-time tails are discussed.

This paper presents three two-dimensional integrable Hamiltonians, whose second invariant $I_2$ is rational or transcendental in momenta. A third invariant $I_3$ (canonically conjugate to $I_2$) is also found for them, and the motion of the particle explicitly solved. The original Painlevé test does not work very well for these models; however, expansions with four resonances are found in each case.

Transitions are considered which are effected by two simultaneous interaction mechanisms—one electromagnetic and the other nonelectromagnetic. The intent is to compare the effectiveness of plane-wave versus quasistatic electric fields. If most of the transition energy is electromagnetic, then low-frequency plane-wave and quasistatic electric fields are equivalent. If most of the transition energy is nonelectromagnetic, then plane-wave transitions are far more probable than quasistatic electric transitions.

An operational formula is derived for a positive phase-space distribution function in quantum mechanics. It is shown that this expression, which is based on a realistic detection mechanism, has many attractive features. The possibility of an application of such a probability distribution to a nonlinear wave mechanics is also briefly mentioned.

Starting with an expression for the fractal dimension $d_u$ of the unscreened perimeter of an arbitrary fractal of dimension $d_f$, there are derived for the random superconducting network the results $\tilde{s}=(2-d)+d_u$, from which follow ${\tilde{\varphi }}_s=d_u$ and $d_w=d-d_u$. Here $\tilde{s}$ is the conductivity exponent, ${\tilde{\varphi }}_s$ the conductance exponent, and $d_w$ the fractal dimension of a random walk on the network. For $d=2\mathrm{}$, these results differ from the Alexander-Orbach conjecture by 0.3%.

A scaling theory is developed to describe the time evolution of the irreversible diffusive recombination process $A+B\to \mathrm{inert}$. Fluctuations are shown to alter radically the decay laws predicted from the rate-equation approach. For unequal initial densities, the minority species is predicted to decay as $t^{-\alpha }$ for short times, crossing over to an $\exp (-A{t}^{\alpha })$ decay for long times, with $\alpha =\frac{d}{4}$ and $\alpha =\frac{\mathrm{}(d+1\mathrm{})\mathrm{}}{4}$ for unbiased and biased diffusion, respectively.

The exact transition temperature, as well as some of the critical exponents, of certain three-dimensional Ising and Potts models is presented. With the analytic results supplemented by assumptions concerning the validity of universality, the critical point is identified as a Liftshitz point. On the basis of hyperscaling, the dynamic exponent $z$ for two-dimensional Ising and Potts model is predicted to be given by $z\nu =2+\alpha$.