Reformulation of equilibrium distributions in few-particle systems

Abstract The derivation of equilibrium distributions via the most probable approximation constitutes a foundational component of statistical physics pedagogy. However, the conventional Stirling’s approximation employed in this framework introduces intrinsic deviations when applied to few-particle systems. Through systematic analysis of several typical ideal systems, including three-dimensional isotropic harmonic oscillator systems and three-dimensional infinite square wells, this work demonstrates that retaining higher-order terms in the Stirling series fails to converge toward true distributions. By implementing a highly accurate yet simple Stirling approximation formula, we derive a new equilibrium distribution expression for few-particle systems through the most probable approximation. Our theoretical results demonstrate perfect agreement with true distributions in prototypical systems.