Difference equations in statistical mechanics. I. Cluster statistics models

A review and some new results are presented for several cluster statistics models, solutions of which can be reduced to difference equations. Mathematical techniques suitable for solving these equations are surveyed.     

Restoration of universality for the rod-to-coil transition scaling in the infinite-dimensionality limit: Exact results for directed walks

We derive scaling forms for the thermodynamic and correlation quantities for the turn-weighted fully and partially directed self-avoiding walks on the hypercubic lattices ind2. In the grand canonical (fixed fugacity per step) ensemble, the conformational rod-to-coil transition sets up in the regimew¯N=O(1), wherew is the weight of each 90° turn and¯N is the (fugacity-dependent) average number of steps. Contrary to the conventional critical phenomena wisdom, the scaling functions for the two different walk models, directed and partially directed, become universal only in the limitd.     

The elliptical gaussian wave transformation due to diffraction by an elliptical hologram

Realized as an interferogram of a spherical and a cylindrical wave, the elliptical hologram is treated as a plane diffracting grating which produces Fresnel diffraction of a simple astigmatic gaussian incident wave. It is shown that if the principal axes of the incident beam coincide with the principal axes of the hologram, the diffracted wave field is composed of three different astigmatic gaussian waves, with their waists situated in parallel but distinct planes. The diffraction pattern, observed on a transverse screen, is the result of the interference of the three diffracted wave components. It consists of three systems of overlapped second-order curves, whose shape depends on the distance of the observation screen from the hologram, as well as on the parameters of the incident wave beam and the hologram. The results are specialized for gratings in the form of circular and linear holograms and for the case of a stigmatic gaussian incident wave, as well as for the normal plane-wave incidence on the three mentioned types of hologram.     

Density of states for intermediate valence and Kondo systems

We calculate the density of states for the nondegenerate Anderson model for various values ofu=U/ andn _f using the perturbation theory withu as the expansion parameter. Summing all the -independent self-energy diagrams, we use the Friedel sum rule and Ward identities to express the physical quantities in terms of the remaining -dependent part of the self-energy, which we evaluate to the 2nd order. The results for the spin and charge susceptibilities obtained in such a way compare rather well with the Bethe-ansatz results. The density of states exhibits different features in different parts of the parameter space. In Kondo region (u>1,n _f 1, i.e., – _f ~U/2), we obtain a many-body resonance (half-width T _K ) around the Fermi level and two broad peaks () at about _f +n _f U and _f +U. In the VF region (u>1, and | _f |) we obtain only two peaks (), one at about _f and one between _f +n _f U and _f +U. The consequences regarding the shape of the photoemission and inverse photoemission spectra of Ce intermetallics are discussed.     

Renormalisation group calculation of the interfacial tension for the three-dimensional Ising model

The interfacial tension of a three-dimensional Ising model in the whole temperature region is calculated by the real-space renormalisation group methods. The cumulant and the finite-cluster approximations with different projection rules are used. The results are consistent with expected low-temperature and global behaviour of the interfacial tension in the three-dimensional model.     

A wave-packet formulation of multiparticle production at high energies

Starting from a free boson gas in a box, we formulate a statistical approach to multi-particle production. The specific feature of the small box quantization is the fact that one-particle wave functions are wave packets instead of plane waves. In this context we investigate the jet model with Bose-Einstein (BE) statistics. The transverse cut-off function is naturally related to the size of the box (production region). We compare our results with experimental data.     

Three Electrons in a Two-Dimensional Parabolic Trap: The Relative Motion Solution

In agreement with the Kohn theorem the relative motion (rel) of three electrons in a two-dimensional parabolic trap separates from the centre-of-mass (CM) motion. By introducing new coordinates the Hamiltonian for relative motion in the approximation of non-interacting electrons can be taken to the normal form. The eigenstates of the normalized Hamiltonian are products of the Fock-Darwin states for normal modes. The energy levels for relative motion are obtained by diagonalizing the exact Hamiltonian in the eigenbasis for the non-interacting case. In this basis the interaction matrix elements can be obtained in the analytical form. Since the rank of the Hamiltonian matrix is significantly reduced, the calculations are faster and more accurate than those for the full (CM + rel) motion. This advantage is especially important for the calculations of excited states and the analysis of energy spectra.     

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.     

Experimental indications for the response of the spectators to the participant blast

Precise momentum distributions of identified projectile fragments, formed in the reactions 238U+Pb and 238U+Ti at 1A GeV, are measured with a high-resolution magnetic spectrometer. With increasing mass loss, the velocities first decrease as expected from previously established systematics, then level off, and finally increase again. Light fragments are on the average even faster than the projectiles. This finding is interpreted as the response of the spectators to the participant blast. The reacceleration of projectile spectators is sensitive to the nuclear mean field and provides a new tool for investigating the equation of state of nuclear matter.