Indirect Solution of Ornstein-Zernike Equation Using the Hopfield Neural Network Method
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ATOMIC PHYSICS
Indirect Solution of Ornstein-Zernike Equation Using the Hopfield Neural Network Method F. S. Carvalho1
· J. P. Braga1
Received: 21 May 2020 © Sociedade Brasileira de F´ısica 2020
Abstract Microscopic information, such as the pair distribution and direct correlation functions, can be obtained from experimental data. From these correlation functions, thermodynamical quantities and the potential interaction function can be recovered. Derivations of Ornstein-Zernike equation and Hopfield Neural Network method are given first, as a theoretical background to follow the present work. From these two frameworks, structural information, such as the radial distribution (g(r)) and direct correlation (C(r)) functions, were retrieved from neutron scattering experimental data. The problem was solved considering simple initial conditions, which does not require any previous information about the system, making it clear the robustness of the Hopfield Neural Network method. The pair interaction potential was estimated in the Percus-Yevick (PY) and hypernetted chain (HNC) approximations and a poor agreement, compared with the Lennard-Jones 6-12 potential, was observed for both cases, suggesting the necessity of a more accurate closure relation to describe the system. In this sense, the Hopfield Neural Network together with experimental information provides an alternative approach to solve the Ornstein-Zernike equations, avoiding the limitations imposed by the closure relation. Keywords Ornstein-Zernike equation · Radial distribution function · Direct correlation function · Hopfield Neural Network
1 Introduction Correlation functions are important for thermodynamics and structural studies in liquid state theory [1]. These functions are related to the structure factor (S(q)), which can be measured experimentally. Studies based on theoretical models and molecular dynamics (MD) simulations to fit experimental data can be found in literature, as those presented by L. Verlet [2], in which a hard sphere model is used to fit experimental data of S(q) for noble gases, and Martin et al. [3], in which MD data is used to propose an “experimental” bridge function for molten lithium. Machine learning techniques have been applied to describe the potential energy function for a more realistic simulation [4], from which the correlation functions may be accurately calculated. J. P. Braga
[email protected] F. S. Carvalho felipe.s.carvalho [email protected] 1
Departmento de Qu´ımica - ICEx, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil
Prediction of an experimental result from a theoretical model is known as a direct problem, whereas retrieving information from experimental data is considered an inverse one [5]. In this last case, if one of the three conditions is not satisfied [6]: (a) for each f ∈ Rn there exists a g ∈ Rm , such that Kf = g, (b) the solution, f, is unique in Rn , and (c) the dependence of f on g is continuous, the problem is to be classified as ill-posed. Some methodologies, such as th
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