Determinação da função energia potencial a partir do coeficiente de viscosidade quântico abordando a teoria de problemas inversos

Abstract

This thesis presents the problem of relating the quantum viscosity coefficient and potential energy function for the He-He system. First, in order to check a recent potential energy potential, the direct problem has been solved getting the viscosity coefficient from 1 to 100 K, range in which the quantum effects become important. The calculated values were found within the experimental uncertainty thus attesting the quality of the potential energy function for the description of the property. Then, with the main objective to get the potential energy function from the viscosity coefficient, first an intermediate inverse problem has been addressed, obtaining the potential energy function from the phase shift. Phase shift is found as a threshold value for distances at which the potential energy function is negligible, from Calogero equation. Calogero equation is a Ricatti differential equation which depends on the potential energy function in a nonlinear way. Accordingly, to apply the Functional Sensitivity Analysis Approach, a new method for get the sensitivity of the phase shift relative to potential energy function has been developed from equation Calogero’s. Finally, with the new methodology developed the potential energy function might be obtained from viscosity coefficient data within the quantum treatment. The procedure involved a linear step, to obtain the cross section through collision integrals, and a nonlinear step, obtaining the potential energy function from cross section. The problems are ill-posed and were solved with the regularization technique, Tikhonov regularization, which has been shown as an effective technique to stabilize the problem. The potential energy function recovered describes the viscosity coefficient with an average error of 1.6422 % that is less than the experimental error (5 %).