Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy

The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^\prime$) generalizations of the logarithmic and exponential functions (noted respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (noted $S_{q,q^\prime}$). The remarkable properties of the $(q,q^\prime)$-generalized logarithmic function make the entropic form $S_{q,q^\prime} \equiv k \sum_{i=1}^W p_i \ln_{q,q^\prime}(1/p_i)$ to satisfy, for large regions of $(q,q^\prime)$, important properties such as {\it expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily {\it composability}.Comment: 9 pages, 4 figure

Locality of forces in molecular systems

Locality is a basic requirement for most modern methods for modelling systems of thousands of atoms or more. Despite its widespread use, the assumption of locality remains largely unfounded in the underlying quantum mechanical description. This work presents an algorithm for quantifying the locality of forces in a general molecular system. The algorithm was tested on linear hydrocarbon systems; it confirmed the locality of the tight-binding model DFTB and quantified the extent to which chemical changes, such as bond conjugation and oxygen addition, introduce long-range effects within the more accurate DFT model. The results motivated the development of an intramolecular hydrocarbon potential using the GAP machine learning method; the new potential is a promising alternative to existing models used in hydrocarbon simulation

Resistivity phase diagram of cuprates revisited

The phase diagram of the cuprate superconductors has posed a formidable scientific challenge for more than three decades. This challenge is perhaps best exemplified by the need to understand the normal-state charge transport as the system evolves from Mott insulator to Fermi-liquid metal with doping. Here we report a detailed analysis of the temperature (T) and doping (p) dependence of the planar resistivity of simple-tetragonal HgBa$_2$CuO$_{4+\delta}$ (Hg1201), the single-CuO$_2$-layer cuprate with the highest optimal $T_c$. The data allow us to test a recently proposed phenomenological model for the cuprate phase diagram that combines a universal transport scattering rate with spatially inhomogeneous (de)localization of the Mott-localized hole. We find that the model provides an excellent description of the data. We then extend this analysis to prior transport results for several other cuprates, including the Hall number in the overdoped part of the phase diagram, and find little compound-to-compound variation in (de)localization gap scale. The results point to a robust, universal structural origin of the inherent gap inhomogeneity that is unrelated to doping-related disorder. They are inconsistent with the notion that much of the phase diagram is controlled by a quantum critical point, and instead indicate that the unusual electronic properties exhibited by the cuprates are fundamentally related to strong nonlinearities associated with subtle nanoscale inhomogeneity.Comment: 22 pages, 5 figure

Dense packing crystal structures of physical tetrahedra

We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of the parameter, providing an atlas of crystal structures which might be observed in systems of nano-particles with tetrahedral symmetry