Termodinâmica estatística e redes neurais aplicadas ao estudo de fluidos atômicos - Felipe Silva Carvalho (2022)
The study of classical fluids can be carried out by different techniques, as for example: molecular dynamics, classical density functional theory or integral equations. One of the most successful integral equation was obtained in 1914 by L. S. Ornstein e F. Zernike and published in the paper "Accidental deviations of density and opalescence at the critical point of a single substance". This equation connects the total and direct correlation functions, h(r) and C(r). However, solving this equation requires the knowledge of an additional relation between these functions, which is called a closure relation. In this work it will be given attention to this problem. The Percus-Yevick (PY) and Hypernetted-Chain (HNC) closures will be presented, since these are very important and wildely applied in sciences. Then, two general closure relations will be proposed, for which PY and HNC are special cases. The hard sphere and Lennard-Jones fluids will be discussed, developing new closure relations that improves results for the radial distribution function, g(r), for hard spheres and to acquire thermodynamic consistent results using the Lennard-Jones potential. Also, an indirect method for solving the Ornstein-Zernike equation will be given in which the Hopfield Neural Network (HNN) is used along with experimental results for neutron scattering process. This method is further generalised using the Gâteaux derivative definition. Them, a recent proposal for neural networs, called Physics Informed Neural Networks, will be used to carry out both direct (solution of Ornstein-Zernike equation) and inverse (acquisition of g(r) from S(q)) problems discussed previously in this work. Although the theoretical background is given in several textbooks, all definitions and deductions necessaries in this work will be presented through the chapters so the reader will not need to search for these concepts in an external material.