In some simple quantum mechanical systems, the degeneracy of typical energy levels grows as a power of the energy or size. We ask whether, after dividing out this average growth, there is a well defined probability distribution of scaled degeneracies in the limit of large size or energy. The answer is yes, for a free particle in a sphere or cube. For the sphere, the distribution of scaled degeneracies is shown to follow a circular law. For the cube, a numerical study shows that the distribution rises linearly for low values of the scaled degeneracy and decays exponentially for large values.

The systematics of photon absorption cross sections in nuclei and small metal particles are examined as a function of the number of constituent fermionsA. It is pointed out that the shell-structure-linked oscillations in the full width at half maximum (FWHM) of the photoneutron cross section in nuclei, earlier recognized forA>63, in fact persist down to the lightest nuclei. Averaging over the oscillations or focusing on the lower envelope of the oscillating curve (magic nuclei), the FWHM is seen to generally decrease with increasingA, consistent withA^−1/3, a dependence which was earlier known to hold in metal particle systems. If the FWHMs are scaled by the respective Fermi energies and the inverse radii by the Fermi wave vectors, the two data sets become comparable in magnitude. A schematic theoretical description of the systematics is presented.

When the dynamics of a system partitions the phase space of configurations into very many disjoint sectors, we are faced with an assignment problem: Given a configuration, how can we tell which sector it belongs to? We study this problem in connection with the dynamics of deposition and evaporation ofk particles at a time, from a lattice substrate. Fork ≥ 3, the system shows complex behaviour: (a) The number of disjoint sectors in phase space grows exponentially with the size. (b) The asymptotic time dependence of the autocorrelation function shows slow decays, with power laws which depend on the sector. Both (a) and (b) are explained in terms of a nonlocal construct known as the irreducible string (IS), formed from a particle configuration by applying a deletion algorithm. The IS provides a label for sectors; the multiplicity of possible IS’s accounts for (a), and let us determine sector numbers and sizes. The elements of the IS are conserved; thus their motion is responsible for the slow modes of the system, and accounts for (b) as well.

We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site k hops to its nearest neighbor with a quenched rate w(k). These rates are chosen randomly from the probability distribution f(w) ∼ (w − c)ⁿ, where c is the lower cutoff. For n>0, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the latter case, it is interesting to ask how the system locates the site with globally minimum rate. We use an argument based on the local equilibrium, supported by Monte Carlo simulations, to describe the approach to the steady state. We find that at large enough time, regions with a smooth density profile are described by a diffusion equation with site-dependent rates, while the isolated points where the mass distribution is singular act as the boundaries of these regions. Our argument implies that the relaxation time scales with the system size L as L^z with z = 2 + 1/(n + 1) for n>1 and suggests a different behavior for n<1.