Understanding basic statistical mechanics formulas

[phantomvommand]

Firstly, I would like to check my understanding of the first formula:
Using velocity distribution = f(v), speed distribution = fs(v):
fs(v) = f(vx)f(vy)f(vz)dxdydz, since dxdydz = 4pi*v^2*dv, fs(v) = 4piv^2f(v)

The second formula is the confusing one:
What does it mean? What is the significance/meaning of the "angle"?

 

[kuruman]

Velocity is a vector, which means it has magnitude and direction. As stated, (2.33) gives the fraction of molecules that are within a speed interval and moving at an angle between $\theta$ and $\theta+d\theta$ measured from some axis. Equation (2.33) does not take direction into account and gives you the distribution in all directions.

 

[vanhees71]

If you have the distribution $f(\vec{v})$ for the velocity, then you get the distribution for any function of $u=F(\vec{v})$ by
$$p(u)=\int_{\mathbb{R}^3} \mathrm{d}^3 v f(\vec{v}) \delta[u-F(\vec{v})].$$
For the Maxwell-Boltzmann distribution it's easy to calculate, because in this case
$$f(\vec{v})=N \exp[-m v^2/(2 k T)].$$
Then just introduce spherical coordinates $v,\vartheta,\varphi$ and $F(\vec{v})=v$ in the general formula
$$p(u)=\int_0^{\infty} \mathrm{d} v \int_0^{\pi} \mathrm{d} \vartheta v^2 \sin \vartheta N \exp[-m v^2/(2kT)] \delta(u-v)=4 \pi N u^2 \exp[-m u^2/(2kT)].$$

 

[aclaret]

let $g : R \times R \rightarrow R, (\theta, |v|) \mapsto g(\theta, |v|)$, where $v \in R^3$ and $|v| = \sqrt{\langle v,v \rangle}$. by spherical symmetrie, $f(v) = H(|v|)$ for some $H$$$\begin{align*}

\mathbb{P}(a < |v| < b \text{ and } \theta \leq c) &= \int_0^c \int_a^b g(\theta, |v|) d|v| d\theta \\

&\overset{!}{=} \int_0^{c} \int_{0}^{2\pi} \int_a^b f(v) |v|^2 \sin{\theta} \, d|v| d\phi d\theta \\

&= \int_0^c \int_a^b 2 \pi |v|^2 f(v) \sin{\theta} \, d|v| d\theta

\end{align*}$$follow that $$g(\theta, |v|) = 2\pi |v|^2 f(v) \sin{\theta} = \frac{1}{2} \left(4\pi |v|^2 f(v) \right) \sin{\theta} = \frac{1}{2} f_s(|v|) \sin{\theta}$$