Which of these is the Boltzmann equation?

Question

I am slightly confused on terminology grounds which of the following two equations is the 'Boltzmann equation': $$\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\bra}[1]{\left<#1\right|} \newcommand{\braoket}[3]{\left<#1\right|#2\left|#3\right>} \p{f_\alpha}{t}+\vec v\cdot \nabla f_\alpha+\vec a \cdot \nabla_vf_\alpha=\l \f{\delta f_\alpha}{\delta t}\r_{col}\tag{1}$$ or $$ \p{f_\alpha}{t}+\vec v\cdot \nabla f_\alpha+\vec a \cdot \nabla_vf_\alpha=\sum_\beta\int_{\vec v_1} \int_\Omega(f'_\alpha f'_{\beta_1}-f_\alpha f_{\beta_1})d^3v_1 g \sigma(\Omega) d\Omega \tag{2}$$ where expression (2) is expression (1) with the Boltzmann collision integral subbed in.

My question is therefore, which of (1) and (2) is actually the Boltzmann equation. If (1) - is it still called the Boltzmann equation if we sub in say the Krook model for the collision operator.

Answer 1

The Boltzmann equation in its general shape is the first one. The substituted term on the right hand side of eq. 2 is just a model, as you rightly stated. And different models exist according to the collision physics you want to consider.

Look at it that way: $f$ is the probability density associated with the presence of a fluid particle in the volume $\mathrm{d}^3\vec{x}\, \mathrm{d}^3\vec{v}$ at a given time $t$ around the location $(\vec{x},\vec{v})$ of the phase space $\mathbb{R^3}\times \mathbb{R}^3$. If you consider a fluid particle $\mathrm{d}^3\vec{x}\, \mathrm{d}^3\vec{v}$ around $(\vec{x},\vec{v})$ at $t$, the state of that particle will move out of the volume $\mathrm{d}^3\vec{x}\, \mathrm{d}^3\vec{v}$ after a time $\Delta t$ because:

• Its velocity will move it out of the interval $\mathrm{d}^3\vec{x}$
• External forces will modify its acceleration, thus making it leave the interval $\mathrm{d}^3\vec{v}$
• It will collide with other particles.

That's exactly what's written in the first equation. Now, collision is very strenuously described, which is why you will find countless models to represent collision physics. The BGK model for instance is widely renowned for its simplicity (you replace the $\left(\frac{\delta f}{\delta t}\right)_{col}$ term by $\frac{f-f^{eq}}{\delta t}$ where $f^{eq}$ is the equilibrium density probability and $\delta t$ the relaxation time towards that equilibrium) and used to implement the lattice Boltzmann Method, but other declinations can also be used for that matter, they will give more or less adapted answers depending on the nature of the problem.

So yes, they're all Boltzmann equations because they describe the same phenomenon, only collision differs and you would thus talk about different models, not different equations.