Classical Statistical Mechanics with Nested Sampling

Baldock, Robert John Nicholas

doi:10.1007/978-3-319-66769-0

Cited by 5 publications

(9 citation statements)

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“…If the initial and boundary conditions are given in the form of probability, the solution must also appear in the form of probability, so the form of motion obtained is also in the form of probability. Therefore, the description of classical statistical mechanics can be obtained [16].…”

Section: Iintroductionmentioning

confidence: 99%

General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization

Huang¹,

Huang²,

Song³

2020

Preprint

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Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality (i.e., complementary principle), derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the square root of ( density function of ) classical statistical mechanics, which will make people renew thinking modern physics development. In addition, this paper discovers the reason why the time derivative of Schrȍdinger doesn’t takes the derivative of space coordinates. Therefore, this paper gives solution to the Crisis of the first quantization origin.

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Section: Iintroductionmentioning

confidence: 99%

General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization

Huang¹,

Huang²,

Song³

2020

Preprint

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“…Until very recently, there was no general computational method capable of performing such multi-dimensional integrals, but the discovery of Nested Sampling by Skilling,10) has opened up a whole new avenue of research, not only in Bayesian data analysis but in Statistical Physics and Material Science as well. 7,[24][25][26][27][28][29] Finally, in cases where no model is available, we must solve the so-called "non-parametric regression" problem, in which we learn functions from data. This is the standard problem in Machine Learning.…”

Section: Discussionmentioning

confidence: 99%

“…= P(Y | X, I) P(X | I) (5) As a corollary of the sum and product rules, one can easily derive the marginalization rule 1,2) P(X | I) = ∫ ∞ −∞ P(X,Y | I)dY (6) as well as Bayes' theorem P(X | Y, I) = P(Y | X, I) P(X | I) P(Y | I) (7) Marginalization will prove to be incredibly useful in cases where our model contains parameters that are unknown, or uninteresting to us (so-called nuisance parameters), but which are still required to evaluate the probabilities. In such cases, the answer is simply to integrate out these unwanted degrees of freedom.…”

Section: Submission Encouragement Award Articlementioning

confidence: 99%

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Select Applications of Bayesian Data Analysis and Machine Learning to Flow Problems

Seryo

Molina

Taniguchi

2021

J. Soc. Rheol., Jpn

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This review focuses on the use of Bayesian Data Analysis and Machine Learning Techniques to study and analyze flow problems typical to polymer melt systems. We present a brief summary of Bayesian probability theory, and show how it can be used to solve the parameter estimation and model selection problems, for cases when the model(s) are known. For the more complex non-parametric regression problem, in which the functional form of the model is not known, we show how Machine-Learning (through Gaussian Processes) can be used to learn arbitrary functions from data. In particular, we show examples for solving steady-state flow problem as well as learning the constitutive relations of polymer flows with memory.

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“…Additional volume-preserving simple shear (off-diagonal deformation gradient) and stretch (diagonal deformation gradient) moves are also proposed with a uniform distribution in strain. However, it can be shown that simulation cells that are anisotropic (long in some directions and short in the others) dominate the configuration space [46,47]. At early iterations and high energies, where the system is disordered and interatomic interactions are relatively unimportant, this does not significantly affect the sampled energies.…”

Section: Generating New Sample Configurationsmentioning

confidence: 99%

Nested sampling for materials

2021

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We review the materials science applications of the nested sampling (NS) method, which was originally conceived for calculating the evidence in Bayesian inference. We describe how NS can be adapted to sample the potential energy surface (PES) of atomistic systems, providing a straightforward approximation for the partition function and allowing the evaluation of thermodynamic variables at arbitrary temperatures. After an overview of the basic method, we describe a number of extensions, including using variable cells for constant pressure sampling, the semi-grand-canonical approach for multicomponent systems, parallelizing the algorithm, and visualizing the results. We cover the range of materials applications of NS from the past decade, from exploring the PES of Lennard–Jones clusters to that of multicomponent condensed phase systems. We highlight examples how the information gained via NS promotes the understanding of materials properties through a novel way of visualizing the PES, identifying thermodynamically relevant basins, and calculating the entire pressure–temperature(–composition) phase diagram. Graphic abstract

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Classical Statistical Mechanics with Nested Sampling

Cited by 5 publications

References 0 publications

General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization

General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization

Select Applications of Bayesian Data Analysis and Machine Learning to Flow Problems

Nested sampling for materials

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