The discussion focuses on the derivation of the canonical ensemble expression for the variance of energy, specifically the expression for <(Delta E)^3>. The participants confirm that the grand-canonical partition function, represented as $$Z(\beta,\alpha,V)=V \int_{\mathbb{R}^3} \exp(-\beta p^2/(2m)+\alpha)$$, is applicable for ideal gases. The evaluation of the average energy and its variance is achieved through the relations ##U=\langle E \rangle=-\partial_{\beta} Z/Z## and ##\langle \Delta E^2 \rangle=\partial_{\beta}^2 Z/Z- \langle E \rangle^2##. The discussion highlights the importance of using explicit example systems to derive expressions in statistical mechanics.
PREREQUISITESStudents and researchers in physics, particularly those specializing in statistical mechanics, thermodynamics, and anyone interested in the mathematical foundations of energy distributions in ideal gases.
For any canonical ensemble.vanhees71 said:I guess, that's for an ideal gas? If so then just get the grand-canonical partition function
$$Z(\beta,\alpha,V)=V \int_{\mathbb{R}^3} \exp(-\beta p^2/(2m)+\alpha)$$
and evaluate ##U=\langle E \rangle=-\partial_{\beta} Z/Z##, ##\langle \Delta E^2 \rangle=\partial_{\beta}^2 Z/Z- \langle E \rangle^2##,...