Geometrical derivation of the Boltzmann factor

We show that the Boltzmann factor has a geometrical origin. Its derivation follows from the microcanonical picture. The Maxwell-Boltzmann distribution or the wealth distribution in human society are some direct applications of this new interpretation.

💡 Research Summary

The paper presents a purely geometric derivation of the Boltzmann factor, showing that the familiar exponential weighting of states emerges directly from the structure of phase‑space volumes in a microcanonical ensemble. The authors begin by recalling the standard statistical‑mechanical route: one starts from a closed system with fixed total energy E, assumes all microstates compatible with this constraint are equally probable, and introduces a Lagrange multiplier β to enforce the energy constraint when maximizing entropy. While this procedure yields the Boltzmann factor mathematically, it offers little physical intuition about why the exponential form appears.

To provide a more transparent picture, the authors consider a system that can be decomposed into N independent degrees of freedom (or particles). Each degree of freedom i carries a non‑negative amount of energy xi, and the conservation law reads Σi xi = E. Geometrically, the set of all admissible vectors (x1,…,xN) is an (N‑1)‑dimensional simplex embedded in the positive orthant of ℝN. The volume of this simplex is V(N‑1,E) = E^{N‑1}/(N‑1)!, a result that follows from elementary combinatorial geometry. Because the microcanonical hypothesis asserts equal probability for all points inside the simplex, the probability density for a particular component x1 is proportional to the volume of the remaining (N‑2)‑dimensional simplex defined by Σ_{i=2}^{N} xi = E – x1. This residual volume is V(N‑2,E – x1) = (E – x1)^{N‑2}/(N‑2)!.

The key step is to examine the large‑N limit, where the system contains many degrees of freedom. Expanding (E – x1)^{N‑2} using the approximation (1 – x1/E)^{N‑2} ≈ exp