# 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. Jan 6 '15 at 7:49
• I guess, there should be a method to it. Jan 6 '15 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 $V$. Jan 6 '15 at 9:21
• Essentially a duplicate of physics.stackexchange.com/q/143467/2451 Jan 6 '15 at 9:43

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