We develop a nonperturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation ...functions and dimensional regularization. Therefore, the nonperturbative spectral renormalization setup here respects all symmetries of the theories at hand. In particular, this includes space-time symmetries, as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalization can be applied within general functional approaches such as the functional renormalization group, Dyson-Schwinger equations, and two- or n -particle irreducible approaches. As an application, we compute the full, nonperturbative, spectral function of the scalar field in the ϕ4-theory in 2 + 1 dimensions from spectral Dyson-Schwinger equations. We also compute the s -channel spectral function of the full ϕ4-vertex in this theory.
Correlation functions are becoming one of the major tools for quantification of structural information that is usually represented as 2D or 3D images. In this paper we introduce ▪ open-source package ...developed in Julia and capable of computing all classical correlation functions based on imaging input data. Images include both binary and multi-phase representations. Our code is capable of evaluating two-point probability S2, phase cross-correlation ρij, cluster C2, lineal-path L2, surface-surface Fss, surface-void Fsv, pore-size P and chord-length p distribution functions on both CPU and GPU architectures. Where possible, we presented two types of computations: full correlation map (correlations of each point with other points on the image, that also allows obtaining ensemble averaged CF) and directional correlation functions (currently in major orthogonal and diagonal directions). Such an implementation allowed for the first time to assemble a completely free solution to evaluate correlation functions under any operating system with well documented application programming interface (API). Our package includes automatic tests against analytical solutions that are described in the paper. We measured execution times for all CPU and GPU implementations and as a rule of thumb full correlation maps on GPU are faster than other methods. However, full maps require more RAM and, thus, are limited to available RAM resources. On the other hand, directional CFs are memory efficient and can be evaluated for huge datasets – this way they are the first candidates for structural data compression of feature extraction. The package itself is available through Julia package ecosystem and on GitHub, the latter source also contains documentation and additional helpful resources such as tutorials. We believe that a single powerful computational tool such as ▪ presented in this paper will significantly facilitate the usage of correlation functions in numerous areas of structural description and research of porous materials, as well as in machine learning applications. We also present some examples as applied to ceramic, soil composite and oil-bearing rock samples based on their 3D X-ray tomography and 2D scanning electron microscope images. Finally, we conclude our paper with discussion of possible ways to further improve presented computational framework.
Program Title: CorrelationFunctions.jl
CPC Library link to program files:https://doi.org/10.17632/6gb9gfm3dw.1
Developer's repository link:https://github.com/fatimp/CorrelationFunctions.jl
Licensing provisions: MIT
Programming language: Julia
Supplementary material: Numerous Jupiter notebooks with examples are available on the GitHub page
Nature of problem: Correlation functions are invaluable universal statistical descriptors of structures used in numerous scientific fields such as astronomy, material science, rock and soil physics, hydrology and biology, to name just a handful of examples. While computational approaches are available in the literature for some functions, they are fragmented and are usually implemented in proprietary interpreted languages for CPU architecture alone.
Solution method: We contribute an open source and cross-platform solution with well documented API for computation of all classical correlation functions from both 2D and 3D images on CPU and GPU architectures. The package computes correlation functions using two approaches: computation of correlation maps and computation along predefined directions. These two approaches can be thought of as an execution time - memory trade-off, but the choice may also depend on the application. The computations are based on a) fast Fourier transform with preprocessing steps such as cluster labeling or edge detection, and b) linear scan approach to evaluate correlation functions along predefined directions. Where justified, the algorithms can be executed on both CPU and GPU which results in high execution speed on modern hardware.
Theory of Diffusive Fluctuations Chen-Lin, Xinyi; Delacrétaz, Luca V; Hartnoll, Sean A
Physical review letters,
2019-Mar-08, Volume:
122, Issue:
9
Journal Article
Peer reviewed
Open access
The recently developed effective field theory of fluctuations around thermal equilibrium is used to compute late-time correlation functions of conserved densities. Specializing to systems with a ...single conservation law, we find that the diffusive pole is shifted in the presence of nonlinear hydrodynamic self-interactions, and that the density-density Green's function acquires a branch point halfway to the diffusive pole, at frequency ω=-(i/2)Dk^{2}. We discuss the relevance of diffusive fluctuations for strongly correlated transport in condensed matter and cold atomic systems.
The subject of the manuscript is the algorithms for radar imaging. This research develops imaging methods and algorithms for wideband and ultrawideband active aperture synthesis systems with antenna ...arrays and stochastic probing signals. The use of antenna arrays makes it possible to obtain radar images without the need to move radar or antenna system in space. The use of wideband and ultra-wideband stochastic probing signals is justified by their narrow autocorrelation functions. This increased the resolution of the obtained images. The main idea of the proposed algorithms is to filter the original wideband signal into several narrowband processes. Furthermore, only the central frequencies of each narrowband signal were processed. This approach allows us to use the classical widespread methods of aperture synthesis for the case of a wideband signal. Usually, they are applicable only for narrowband signals that satisfy the condition of a quasi-monochromatic approximation. This significantly reduces the overall computational complexity of the imaging algorithm, which simplifies its further practical implementation on the existing radioelement base. Because of the simulation, a primary radar image has been obtained and the overall performance of the proposed approach to processing wideband signals has been confirmed. An increase in the quality of the obtained image when using a multiple of frequency ranges is shown. An experimental study of the effect of processing a wideband signal only at its centre frequency instead of the entire frequency band is conducted. During the experiment, the correlation functions of the signals received by two spaced receivers were obtained. As a result, the Van Cittert-Zernike theorem has been experimentally confirmed. It allows signal processing only at its centre frequency instead of the entire frequency band. Simultaneously, the prospect of expanding the bandwidth of the probing signal is indicated. It, in the presence of a wideband element base and devices for high-speed signal processing, will further increase the imaging resolution of a radar system.
Vibrational spectroscopy is an essential tool in chemical analyses, biological assays, and studies of functional materials. Over the past decade, various coherent nonlinear vibrational spectroscopic ...techniques have been developed and enabled researchers to study time-correlations of the fluctuating frequencies that are directly related to solute–solvent dynamics, dynamical changes in molecular conformations and local electrostatic environments, chemical and biochemical reactions, protein structural dynamics and functions, characteristic processes of functional materials, and so on. In order to gain incisive and quantitative information on the local electrostatic environment, molecular conformation, protein structure and interprotein contacts, ligand binding kinetics, and electric and optical properties of functional materials, a variety of vibrational probes have been developed and site-specifically incorporated into molecular, biological, and material systems for time-resolved vibrational spectroscopic investigation. However, still, an all-encompassing theory that describes the vibrational solvatochromism, electrochromism, and dynamic fluctuation of vibrational frequencies has not been completely established mainly due to the intrinsic complexity of intermolecular interactions in condensed phases. In particular, the amount of data obtained from the linear and nonlinear vibrational spectroscopic experiments has been rapidly increasing, but the lack of a quantitative method to interpret these measurements has been one major obstacle in broadening the applications of these methods. Among various theoretical models, one of the most successful approaches is a semiempirical model generally referred to as the vibrational spectroscopic map that is based on a rigorous theory of intermolecular interactions. Recently, genetic algorithm, neural network, and machine learning approaches have been applied to the development of vibrational solvatochromism theory. In this review, we provide comprehensive descriptions of the theoretical foundation and various examples showing its extraordinary successes in the interpretations of experimental observations. In addition, a brief introduction to a newly created repository Web site (http://frequencymap.org) for vibrational spectroscopic maps is presented. We anticipate that a combination of the vibrational frequency map approach and state-of-the-art multidimensional vibrational spectroscopy will be one of the most fruitful ways to study the structure and dynamics of chemical, biological, and functional molecular systems in the future.
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