Advancing Shannon entropy for measuring diversity in systems

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy H has been used to re-think diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon's approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a re-normalization of probability distributions based on the notion of case-based entropy Cc as a function of the cumulative probability c. Given a probability density p(x), Cc measures the diversity of the distribution up to a cumulative probability of c, by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of ^pc(x) up to cumulative probability c. We illustrate the utility of our approach by re-normalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions for energy of sub-atomic particles. The comparison shows that Cc is a vast improvement over H as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions

Seismic Earth Pressure Development in Sheet Pile Retaining Walls: A Numerical Study

The design of retaining walls requires the complete knowledge of the earth pressure distribution behind the wall. Due to the complex soil-structure effect, the estimation of earth pressure is not an easy task; even in the static case. The problem becomes even more complex for the dynamic (i.e., seismic) analysis and design of retaining walls. Several earth pressure models have been developed over the years to integrate the dynamic earth pressure with the static earth pressure and to improve the design of retaining wall in seismic regions. Among all the models, MononobeOkabe (M-O) method is commonly used to estimate the magnitude of seismic earth pressures in retaining walls and is adopted in design practices around the world (e.g., EuroCode and Australian Standards). However, the M-O method has several drawbacks and does not provide reliable estimate of the earth pressure in many instances. This study investigates the accuracy of the M-O method to predict the dynamic earth pressure in sheet pile wall. A 2D plane strain finite element model of the wall-soil system was developed in DIANA. The backfill soil was modelled with Mohr-Coulomb failure criterion while the wall was assumed behave elastically. The numerically predicted dynamic earth pressure was compared with the M-O model prediction. Further, the point of application of total dynamic force was determined and compared with the static case. Finally, the applicability of M-O methods to compute the seismic earth pressure was discussed

Exactly solvable PT\mathcal{PT}-symmetric models in two dimensions

Non-hermitian, $\mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $\mathcal{PT}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $\mathcal{PT}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $\mathcal{PT}$ breaking in two dimensions has a rich phase diagram, with multiple re-entrant $\mathcal{PT}$ symmetric phases.Comment: 6 pages, 6 figure