Applies to a macroscopic Nb. of atoms or molecules with randomly distributed velocities

Classical limit ⇒ it applies to high $T$ such that $k_bT\gg \Delta E$

Can treat the distribution of velocities and energies as continuous function

Probability distribution

💼 Case: Probability distribution with $p(\vec v)$ = probability density

$$ \begin{aligned} p(\vec v)\,\text d^3 v&=p(\vec v)\,\text d v_x\,\text dv_y\,\text dv_z \\ &\rightarrow \text{probability to find velocity vector in} \\ &\quad \;\text{ infinitesimal volume element in 3D } \vec v\text{-space} \end{aligned} $$

🔔 Properties:

• $p(\vec v) \ge 0$ for all $\vec v$

• Mean $\left < F \right >=\int\,\text d^3v \,p(\vec v) f(\vec v)$

• $\int p(\vec v) \, \text d^3 v=1$ (normalisation

• Variance $\sigma^2(F)=\left < F^2 \right > -\left < F \right >^2$

🗞️ Examples: Exponential distr. $p(x)=\lambda e^{-\lambda x} \; \lambda>0, \, x>0$ ; Gaussian $p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$

Gaussian integrals

$$ I_m=\int^\infin_{-\infin}x^m e^{-ax^2}\,\text dx \qquad (a<0) $$

• if $m$ is odd then $I_m=0$ so we will consider $m=2n$

$$ \begin{aligned} I_0&=\int^\infin_{-\infin}e^{-ax^2}\,\text dx \\ I_0^2&=\int^\infin_{-\infin}e^{-ax^2}\,\text dx \int^\infin_{-\infin}e^{-ay^2}\,\text dy=\int^\infin_{-\infin}\int^\infin_{-\infin}e^{-a(x^2+y^2)}\,\text dx\,\text dy \\ &\text{Use polar coordinates} \\ I_0^2&=\int^{2\pi}0\text d\theta \int^\infin{-\infin}e^{-ar^2}r\,\text dr \\ &\text{sub } u=ar^2 \text{ so } \text du=2ar\,\text dr \\ I_0^2&=2\pi\int^\infin_0 \frac{1}{2a}e^{-u}\,\text du = \frac{2\pi}{2a} \left [ -e^{-u}\right ]^\infin_0=\frac{\pi}{a}\\ \text{So: }I_0&=\sqrt{\frac{\pi}{a}}

\end{aligned} $$

• We can then get the general form for $I_{2n}$

$$ I_{2n}=(-1)^n\left ( \frac{\partial}{\partial a} \right )^nI_0 $$

🗞️ Examples:

$$ \begin{aligned} I_2&=\int^{\infin}{-\infin}x^2e^{-ax^2}\,\text dx=\frac 12 \sqrt{\frac \pi {a^3}} \\ I_4&=\int^{\infin}{-\infin}x^4e^{-ax^2}\,\text dx=\frac 34 \sqrt{\frac \pi {a^5}} \end{aligned} $$

Boltzmann factor and ideal gas

💼 Case: consider a gas of point mass ( no internal structure ) and non-interacting particles in a cubic box at temperature $T$

• Gas is isotropic ⇒ $p(\vec v)$ only depends on $v=|\vec v|$
• Velocity components are uncorrelated:

$$ p(\vec v)=f(v^2)=f_x(v^2_x)\,f_y(v_y^2)\,f_z(v_z^2) $$

where $f$ is a function

$$ p(\vec v)=f(v^2)=Ce^{-bv^2} $$

where $C$ and $b$ are positive constants

• Trying to find $C$ and $b$. Normalisation:

$$ \int p(v)\,\text d^3 v=1 \Rightarrow 4\pi C\underbrace{\int^\infin_0 v^2 e^{-bv^2}\,\text dv}_{\frac 12 I_2 }=1\Rightarrow C=\sqrt{\frac{b^3}{\pi^3}} $$