## Retrospect on statistical mechanics Gibbs ensemble, or just say ensemble: do not focus on specific microscopic state, but a collection of system with: - identical composition - identical in macroscopic condition ($V, T, P$, etc.) - exists in different state A Gibbs ensemble is represented by a distribution of representative points in [[Potential, force, momentum, velocity#^phase-space|phase space]] $\Gamma$, and may be described as a ***distribution function*** $\rho(p,q,t)$. >[!note] >Here I think Gibbs ensemble is just a generalized concept, a "big" system called ensemble, and described as $\rho(p,q,t)$ with these generalized condition. >i.e., an ensemble being described by a distribution function. And one may define **ensemble average** as $\langle O\rangle = \frac{\int \mathrm{d}^{3N}p \mathrm{d}^{3N}q O(p,q)\rho(p,q,t)}{\int \mathrm{d}^{3N}p \mathrm{d}^{3N}q \rho(p,q,t)}$ ^23b0cd For dilute ideal gas, most probable velocity is $\bar{v} = \sqrt{\frac{2kT}{m}}$ and (corresponding) energy is $\frac{3NkT}{2}$. >[!note] >the velocity given above is at each direction, being the same as $\frac{1}{2}mv^2 = kT$, and the energy is just 3D case of this result.