Analytical solution of space probability distributions of particles in a one-dimensional ring

Spatial probability distributions of a few particles having nonidentical masses in one-dimensional space with both periodic and fixed boundary conditions are analytically computed in statistical equilibrium states, and explicit solutions of the probability distributions are obtained. Some nontrivial interesting features of the probability distributions are predicted and fully confirmed by numerical simulations. The realization of a microcanonical equilibrium state of the system is justified by the agreement between the theoretical predictions and numerical observations.