Local thermal energy as a structural indicator in glasses

Identifying heterogeneous structures in glasses --- such as localized soft spots --- and understanding structure-dynamics relations in these systems remain major scientific challenges. Here we derive an exact expression for the local thermal energy of interacting particles (the mean local potential energy change due to thermal fluctuations) in glassy systems by a systematic low-temperature expansion. We show that the local thermal energy can attain anomalously large values, inversely related to the degree of softness of localized structures in a glass, determined by a coupling between internal stresses --- an intrinsic signature of glassy frustration ---, anharmonicity and low-frequency vibrational modes. These anomalously large values follow a fat-tailed distribution, with a universal exponent related to the recently observed universal $\omega^4$ density of states of quasi-localized low-frequency vibrational modes. When the spatial thermal energy field --- a `softness field' --- is considered, this power-law tail manifests itself by highly localized spots which are significantly softer than their surroundings. These soft spots are shown to be susceptible to plastic rearrangements under external driving forces, having predictive powers that surpass those of the normal-modes-based approach. These results offer a general, system/model-independent, physical-observable-based approach to identify structural properties of quiescent glasses and to relate them to glassy dynamics.Comment: 8 pages, 4 figures + Supporting Information, shorter title, minor textual change

Does local financial development matter?

We study the effects of differences in local financial development within an integrated financial market. We construct a new indicator of financial development by estimating a regional effect on the probability that, ceteris paribus, a household is shut off from the credit market. By using this indicator, we find that financial development enhances the probability an individual starts his own business, favors entry of new firms, increases competition, and promotes growth. As predicted by theory, these effects are weaker for larger firms, which can more easily raise funds outside of the local area. These effects are present even when we instrument our indicator with the structure of the local banking markets in 1936, which, because of regulatory reasons, affected the supply of credit in the following 50 years. Overall, the results suggest local financial development is an important determinant of the economic success of an area even in an environment where there are no frictions to capital movements

Relation Between Local Temperature Gradients and the Direction of Heat Flow in Quantum Driven Systems

We introduce thermometers to define the local temperature of an electronic system driven out-of-equilibrium by local ac fields. We discuss the behavior of the local temperature along the sample, showing that it exhibits spatial fluctuations following an oscillatory pattern. We show explicitly that the local temperature is the correct indicator for heat flow.Comment: 3 pages, 2 figure

An adaptive GMsFEM for high-contrast flow problems

In this paper, we derive an a-posteriori error indicator for the Generalized Multiscale Finite Element Method (GMsFEM) framework. This error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high-contrast coefficients. The GMsFEM, which has recently been introduced in [12], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) the construction of the online space (the latter for parameter-dependent problems). In [12], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we study an adaptive enrichment procedure and derive an a-posteriori error indicator which gives an estimate of the local error over coarse grid regions. We consider two kinds of error indicators where one is based on the $L^2$-norm of the local residual and the other is based on the weighted $H^{-1}$-norm of the local residual where the weight is related to the coefficient of the elliptic equation. We show that the use of weighted $H^{-1}$-norm residual gives a more robust error indicator which works well for cases with high contrast media. The convergence analysis of the method is given. In our analysis, we do not consider the error due to the fine-grid discretization of local problems and only study the errors due to the enrichment. Numerical results are presented that demonstrate the robustness of the proposed error indicators.Comment: 26 page

A local scale-sensitive indicator of spatial autocorrelation for assessing high- and low-value clusters in multiscale datasets

Georeferenced user-generated datasets like those extracted from Twitter are increasingly gaining the interest of spatial analysts. Such datasets oftentimes reflect a wide array of real-world phenomena. However, each of these phenomena takes place at a certain spatial scale. Therefore, user-generated datasets are of multiscale nature. Such datasets cannot be properly dealt with using the most common analysis methods, because these are typically designed for single-scale datasets where all observations are expected to reflect one single phenomenon (e.g., crime incidents). In this paper, we focus on the popular local G statistics. We propose a modified scale-sensitive version of a local G statistic. Furthermore, our approach comprises an alternative neighbourhood definition that enables to extract certain scales of interest. We compared our method with the original one on a real-world Twitter dataset. Our experiments show that our approach is able to better detect spatial autocorrelation at specific scales, as opposed to the original method. Based on the findings of our research, we identified a number of scale-related issues that our approach is able to overcome. Thus, we demonstrate the multiscale suitability of the proposed solution

Global and local Complexity in weakly chaotic dynamical systems

In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We consider this indicator of local complexity of the dynamics and provide different examples of its behavior, showing how it can be useful to characterize various kind of weakly chaotic dynamics. We also provide criteria to find systems with non trivial orbit complexity (systems where the description of the whole orbit requires an infinite amount of information). We consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, taking into account systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive (roughly speaking: if the map can be defined up to any given accuracy using a finite amount of information) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that this notion comes out naturally in this kind of complexity questions.Comment: 23 page