Hence, we expect that this new method could become a pivotal workhorse for numerically analytic continuation and be broadly useful in many applications. More specifically, Ma圎nt applies inference techniques rooted in Shannon information theory, Bayesian probability, and the principle of maximum entropy. As a byproduct, this method could derive a fitting formula for the Matsubara data, which provides a compact approximation to the many-body Green's functions. The maximum entropy (Ma圎nt) model was used in this study to identify the current and potential distribution and habitat suitability of three pine species and B. In physics, maximum entropy thermodynamics (colloquially, Ma圎nt thermodynamics) views equilibrium thermodynamics and statistical mechanics as inference processes. The causality of spectral function is always satisfied even in the presence of sizable noises. More importantly, it exhibits excellent robustness with respect to noisy and incomplete input data. The sharp, smooth, and multi-peak features in both low-frequency and high-frequency regions of spectral functions could be accurately resolved, which overcomes one of the main limitations of the traditional maximum entropy method. The benchmark results demonstrate that this method is capable of reproducing most of the key characteristics in the spectral functions. The synthetic Green's functions, as well as realistic correlation functions from finite temperature quantum many-body calculations, are used as input. Then we employ this method to tackle the analytic continuation problems of matrix-valued Green's functions. The maximum entropy principle has been shown Cox 1982, Jaynes 2003 to be the unique consistent approach to constructing a discrete probability distribution. To demonstrate the usefulness and performance of the new method, we at first apply it to study the spectral functions of representative fermionic and bosonic correlators. In order to capture narrow peaks and sharp band edges in the spectral functions, a constrained sampling algorithm and a self-adaptive sampling algorithm are developed. This method is based on the pole representation of Matsubara Green's function and a stochastic sampling procedure is utilized to optimize the amplitudes and locations of poles. There are a few ways to measure entropy for multiple variables we'll use two, X and Y. In this paper, we propose a new analytic continuation method to extract real frequency spectral functions from imaginary frequency Green's functions of quantum many-body systems. According to the maximum entropy principle, the best guess is the one which maximises the information entropy under the given constraints. (8.1) Average information, surprise, or uncertainty are all somewhat parsimonious plain English analogies for entropy.
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