The particle size distribution (PSD) is one of fundamental physical properties of soil. Compared with the conventional graphic parameters, the continuous function description has obvious advantages for characterizing the mathematical relationship of the PSD. In this study, unlike previous phenomenological and statistical studies, continuous functions were derived by directly developing physical and mathematical models of the key processes responsible for soil formation. The logarithmic gamma distribution was analytically derived as the solution of the fragmentation model, and a function with 5 parameters was derived as the unified distribution of real soil considering fragmentation and transport processes. Then, the proposed continuous functions were verified through a Lunar soil PSD database and the UNSODA2.0 database. The proposed functions can be used to replace the graphic parameters in soil mechanics and engineering practice to take advantage of its clear physical background and strict verification based on a large amount of data.