The analytical forms of free-electron susceptibilities d (q) and their range functions d (r) are derived at nonzero temperature starting from the Green's function representation by properly evaluating the contributions from the poles of free-electron temperature Green's function for each dimension dϭ1, 2, and 3. The present formalism produces not only d (q) and d (r) which show more accurate temperature-dependent behavior than our previous results for dϭ1 and 3, but also temperature-dependent two-dimensional 2 (q) and 2 (r) for a wide range of temperature. Our analytical results show that irrespective of dimension, the singular behavior of d (q) at qϭϮ2k F becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors 2k 0 Ϯ , and this transition causes d (r) to be exponentially damped with common damping factor e Ϫ2 0 sin r k F r ϳe ϪTЈk F r for low enough temperature, where the exponent of the damping factor corresponds to an imaginary part of wave vectors 2k 0 Ϯ. We also show that the causality relation of the response function is essential in understanding the correct behavior of d (q) and d (r) for all dimensions.

The characteristic features of the collapse of the ground state in trapped one-component attractive Bose-Einstein condensates are studied by applying the catastrophe theory. From numerically obtained stable and unstable solutions of the Gross-Pitaevskii equation, we derive the catastrophe function defining the stability of the stationary points on the Gross-Pitaevskii energy functional. The bifurcation diagram and the universal scaling laws stemming from the catastrophe function show quantitative agreement with the numerical results.

A deuteron dipole glass Rb1−x(ND4)xD2PO4 (DRADP-x, x = 0.47) was studied on glassy freezing by low-frequency dielectric measurements at low temperatures of slow dynamic freezing. A very broad distribution of relaxation times (relaxation frequencies) was observed to show an apparent crossover behavior from the relaxation time distributions with a long-time (low-frequency) tail to the distributions with a short-time (high-frequency) tail as temperature was lowered to below 40 K. We show that this apparent crossover behavior may be originated by two competing relaxations in coexistence: one characterized by an Arrhenius law and the other by an inverse Arrhenius law temperature dependence, corresponding to the two slow-relaxation modes (C < 0 and C > 0) of the dynamically correlated domains (DCDs) model of Chamberlin et al.