# Summation to Integration in Statistical Mechanics

In Statistical Mechanics, what is the procedure of replacing this summation by the integration given by $$\sum_{\vec k} \rightarrow \frac{V}{(2\pi)^3} \int_{0}^{\infty} 4\pi k^2 dk$$

• If $|\Delta \vec{k}| \ll |\vec{k}|$ holds for most of the states in the heat bath then we can approximate $\Delta \vec{k} \sim d\vec{k}$ and send the sum to an integral. There is no real procedure involved, you just replace the sum by an integral and send the summand to an integrand. Commented Jan 6, 2015 at 7:49
• I guess, there should be a method to it. Commented Jan 6, 2015 at 7:53
• @FenderLesPaul: I think maybe OP wants to understand where the $V$ comes from, etc. This is not completely trivial. Try explaining about the mode density in a box of volume $V$. Commented Jan 6, 2015 at 9:21
• Essentially a duplicate of physics.stackexchange.com/q/143467/2451 Commented Jan 6, 2015 at 9:43

\begin{align} \int\int\int\rho(k)dk_xdk_ydk_z &= \int^k_0\int^{2\pi}_{0}\int^{\pi}_{0}\rho(k)k^2\sin\theta d\theta d\phi dk\\ &= \frac{V}{(2\pi)^3}\int^k_04\pi k^2dk \end{align}

where $k$ is the "radius" of the spherical coordinate system.