Statistical Mechanics · Problem 3.18

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · canonical-ensemble-cubic-fluctuation

Problem

Show that for a system in the canonical ensemble

\langle(\Delta E)^3\rangle = k^2\left\{T^4\left(\frac{\partial C_V}{\partial T}\right)_V + 2T^3 C_V\right\}.

Verify that for an ideal gas

\left\langle\left(\frac{\Delta E}{U}\right)^2\right\rangle = \frac{2}{3N} \quad \text{and} \quad \left\langle\left(\frac{\Delta E}{U}\right)^3\right\rangle = \frac{8}{9N^2}.