Hole probability for non-interacting fermions in a d-dimensional trap

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability P (R) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P (R) is described by a universal scaling function of kF R, for which we obtain an exact formula (kF being the local Fermi wave-vector). It exhibits a super exponential tail P (R) / e-κd(kF R)d+1 where κdis a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

On the Importance of Statistical Homogeneity to the Scaling of Rain

Scaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener–Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the “observed” power spectrum no longer represents a “universal” scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.

EXOTIC CARBON SYSTEMS IN TWO-NEUTRON HALO THREE-BODY MODELS

The general properties of exotic carbon systems, considered as a core with a two-neutron (n - n) halo, are described within a renormalized zero-range three-body model. In particular, it is addressed the cases with a core of 18C and 20C. In such a three-body framework, 20C has a bound subsystem (19C), whereas 22C has a Borromean structure with all subsystems unbound. 22C is also known as the heaviest carbon halo nucleus discovered. The spatial distributions of such weakly-bound three-body systems are studied in terms of a universal scaling function, which depends on the mass ratio of the particles, as well as on the nature of the subsystems.

An extended Cahn-Hilliard model for interfaces with cubic anisotropy

For studying systems with a cubic anisotropy in interfacial energy σ, we extend the CahnHilliard model by including in it a fourth rank term, which leads to an additional linear term in the evolution equation for the composition field. It also leads to an orientation-dependent effective fourth rank coeficient γ(hkl) in the governing equation for the one-dimensional composition profile across a planar interface. The main effect of a non-negative γ(hkl) is to increase both σ and interfacial width w, each of which, upon suitable scaling, is related to γ(hkl) through a universal scaling function. The anisotropy in the interfacial energy can be large enough to give rise to corners in the Wulff shapes in two dimensions. In particles of finite sizes, the corners get rounded, and their shapes tend towards the Wulff shape with increasing particle size. In the study of unmixing of concentrated alloys, the anisotropy nt only leads to non-spherical particle shapes, but also to strongly elongated morphologies.

An Extended Cahn-Hilliard Model for Interfaces with Cubic Anisotropy

ABSTRACTFor studying systems with a cubic anisotropy in interfacial energy σ, we extend the Cahn-Hilliard model by including in it a fourth rank term, which leads to an additional linear term in the evolution equation for the compositioneld. It also leads to an orientation-dependent effective fourth rank coeffcient γ(hkl) in the governing equation for the one-dimensional composition prole across a planar interface. The main effect of a non-negative γ(hkl) is to increase both σ and interfacial width w, each of which, upon suitable scaling, is related to γ(hkl) through a universal scaling function. The anisotropy in the interfacial energy can be large enough to give rise to corners in the Wul. shapes in two dimensions. In particles of finite sizes, the corners get rounded, and their shapes tend towards the Wul. shape with increasing particle size. In the study of unmixing of concentrated alloys, the anisotropy not only leads to non-spherical particle shapes, but also to strongly elongated morphologies.