Introduction of probability density alternation method for inverse analyses of integral equations in surface science

¹ Graduate School of Agriculture, Meijo University - Nagoya 468-8502, Japan
² Faculty of Agriculture, Meijo University - Nagoya 468-8502, Japan

Received: 11 January 2026
Accepted: 12 May 2026

Abstract

Integral equations frequently arise in surface science, and in some cases, they must be treated as inverse problems. In our previous work on optical tweezers, atomic force microscopy, and surface force measurement apparatus, we performed inverse calculations to obtain the pressure between parallel plates from measured interaction forces. These inverse analyses were used to reconstruct solvation structures near solid surfaces and density distribution profiles of colloidal particles. In the course of these studies, we developed a method that enables inverse analyses through a unified and systematic procedure, hereafter referred to as the Probability Density Alternation (PDA) method. The central idea of this method is to reformulate a given integral equation in terms of probability density functions. In this letter, we demonstrate the validity of the PDA method both analytically and numerically. While the PDA method is less advantageous for single integral equations, it becomes a convenient and powerful approach for inverse analyses involving double or higher-order integral equations.