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Uncertainty-Aware Liquid State Modeling from Experimental Scattering Measurements

Brennon L. Shanks

TL;DR

This work reframes liquid-state structure–property modeling as an uncertainty-aware inverse problem, leveraging Bayesian inference to connect experimental scattering data to interatomic forces. It introduces SOPR, a structure-inversion method that combines Henderson's inverse theorem with Gaussian process priors to derive transferable pair potentials from neutron scattering data, validated by RDFs and vapor–liquid equilibria in noble gases. To tackle computational bottlenecks, it introduces Local Gaussian Process surrogates, enabling rapid Bayesian inference on complex observables like RDFs with uncertainty quantification and parameter sensitivity analysis. A systematic study of experimental noise shows that state-of-the-art neutron instruments can recover detailed force-field parameters, highlighting the potential to use scattering data to constrain interatomic forces and inform thermodynamics for liquids, with broad implications for force-field design and many-body physics. The results collectively argue for a Bayesian, uncertainty-aware, structure-based approach as a foundational framework for linking quantum-scale interactions to macroscopic liquid behavior.

Abstract

This dissertation is founded on the central notion that structural correlations in dense fluids, such as dense gases, liquids, and glasses, are directly related to fundamental interatomic forces. This relationship was identified early in the development of statistical theories of fluids through the mathematical formulations of Gibbs in the 1910s. However, it took nearly 80 years before practical implementations of structure-based theories became widely used for interpreting and understanding the atomic structures of fluids from experimental X-ray and neutron scattering data. The breakthrough in successfully applying structure-potential relations is largely attributed to the advancements in molecular mechanics simulations and the enhancement of computational resources. Despite advancements in understanding the relationship between structure and interatomic forces, a significant gap remains. Current techniques for interpreting experimental scattering measurements are widely used, yet there is little evidence that they yield physically accurate predictions for interatomic forces. In fact, it is generally assumed that these methods produce interatomic forces that poorly model the atomistic and thermodynamic behavior of fluids, rendering them unreliable and non-transferable. This thesis aims to address these limitations by refining the statistical theory, computational methods, and philosophical approach to structure-based analyses, thereby developing more robust and accurate techniques for characterizing structure-potential relationships.

Uncertainty-Aware Liquid State Modeling from Experimental Scattering Measurements

TL;DR

This work reframes liquid-state structure–property modeling as an uncertainty-aware inverse problem, leveraging Bayesian inference to connect experimental scattering data to interatomic forces. It introduces SOPR, a structure-inversion method that combines Henderson's inverse theorem with Gaussian process priors to derive transferable pair potentials from neutron scattering data, validated by RDFs and vapor–liquid equilibria in noble gases. To tackle computational bottlenecks, it introduces Local Gaussian Process surrogates, enabling rapid Bayesian inference on complex observables like RDFs with uncertainty quantification and parameter sensitivity analysis. A systematic study of experimental noise shows that state-of-the-art neutron instruments can recover detailed force-field parameters, highlighting the potential to use scattering data to constrain interatomic forces and inform thermodynamics for liquids, with broad implications for force-field design and many-body physics. The results collectively argue for a Bayesian, uncertainty-aware, structure-based approach as a foundational framework for linking quantum-scale interactions to macroscopic liquid behavior.

Abstract

This dissertation is founded on the central notion that structural correlations in dense fluids, such as dense gases, liquids, and glasses, are directly related to fundamental interatomic forces. This relationship was identified early in the development of statistical theories of fluids through the mathematical formulations of Gibbs in the 1910s. However, it took nearly 80 years before practical implementations of structure-based theories became widely used for interpreting and understanding the atomic structures of fluids from experimental X-ray and neutron scattering data. The breakthrough in successfully applying structure-potential relations is largely attributed to the advancements in molecular mechanics simulations and the enhancement of computational resources. Despite advancements in understanding the relationship between structure and interatomic forces, a significant gap remains. Current techniques for interpreting experimental scattering measurements are widely used, yet there is little evidence that they yield physically accurate predictions for interatomic forces. In fact, it is generally assumed that these methods produce interatomic forces that poorly model the atomistic and thermodynamic behavior of fluids, rendering them unreliable and non-transferable. This thesis aims to address these limitations by refining the statistical theory, computational methods, and philosophical approach to structure-based analyses, thereby developing more robust and accurate techniques for characterizing structure-potential relationships.
Paper Structure (133 sections, 23 theorems, 477 equations, 43 figures, 15 tables)

This paper contains 133 sections, 23 theorems, 477 equations, 43 figures, 15 tables.

Table of Contents

  1. Introduction
  2. The Origins of Liquid Structure Analysis
  3. Measuring Liquid Structure with Experimental Neutron Scattering
  4. A Comprehensive Review of Structure-Potential Analysis
  5. Modern Scattering Analysis: A New Perspective
  6. Outline and Scope of the Thesis
  7. Learning Interaction Potentials from Scattering Data
  8. Abstract
  9. Introduction
  10. Results and Discussion
  11. Theory and Computational Methods
  12. Supporting Information
  13. Grand Canonical Monte Carlo Simulations
  14. Tabulated SOPR Results
  15. Convergence Stability
  16. ...and 118 more sections

Key Result

Lemma 1.1

Let $\rho_1$ and $\rho_2$ be positive, trace-class, and linear density operators on a Hilbert space, $H$, such that $Tr(\rho_i) = 1$. Then,

Figures (43)

  • Figure 1: The statistical mechanical version of a ball and stick model.
  • Figure 2: The radial distribution function keeps track of the particle density of a system as a function of radius away from a reference atom. The dashed radial shell in the particle picture (left) is represented as a radial interval in the radial distribution function (right).
  • Figure 3: Incident scattering vector $\mathbf{k_i}$ scattered through solid angle $d \Omega$.
  • Figure 4: Site-site radial distributions for amorphous $GeSe_2$ determined from different EPSR runs differ by $\sim$16%. Figure reproduced from Alan Soper, On the uniqueness of structure extracted from diffraction experiments on liquids and glasses, Journal of Physics : Condensed Matter, Volume 19, Issue 41, Page 12, 01/01/1989. © IOP Publishing. Reproduced with permission. All rights reserved.
  • Figure 5: Partial radial distribution functions for HH, OH, OO atom pairs with uncertainty estimates from 6 EPSR runs (vertical bars). Figure reproduced from Alan Soper, The Radial Distribution Functions of Water as Derived from Radiation Total Scattering Experiments: Is There Anything We Can Say for Sure?, ISRN Physical Chemistry, Volume 2013, 11/28/2013.
  • ...and 38 more figures

Theorems & Definitions (41)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 1.1
  • proof
  • Corollary B.3.1
  • Corollary B.3.2
  • Corollary B.3.3
  • Corollary B.3.4
  • ...and 31 more