Boltzmann’s Equation

The purpose of this post is to explain how Boltzmann’s equation in kinetic theory arises.

Problem #$1$: Write down Liouville’s equation from classical Hamiltonian mechanics governing the incompressible phase space flow (i.e. time evolution) of the joint probability density function $\rho(\textbf x_1,…,\textbf x_N,\textbf p_1,…,\textbf p_N,t)$.

Solution #$1$: Remembering the intuition that $\{\space\space,H\}$ implements an advective derivative on the joint phase space:

$$\frac{D\rho}{Dt}=0\Rightarrow\frac{\partial\rho}{\partial t}+\{\rho,H\}=0$$

The idea is that $\rho(\textbf x_1,…,\textbf x_N,\textbf p_1,…,\textbf p_N,t)d^3\textbf x_1…d^3\textbf x_Nd^3\textbf p_1…d^3\textbf p_N$ is the (purely due to classical ignorance) probability that the system of $N$ identical particles is, at some time $t\in\textbf R$, living within an infinitesimal volume $d^3\textbf x_1…d^3\textbf x_Nd^3\textbf p_1…d^3\textbf p_N$ centered around the microstate $(\textbf x_1,…,\textbf x_N,\textbf p_1,…,\textbf p_N)\in\textbf R^{6N}$ in phase space.

Problem #$2$: What is the marginal probability density $\rho_1(\textbf x,\textbf p,t)$ of the any given particle being in state $(\textbf x,\textbf p)\in\textbf R^6$ at time $t\in\textbf R$? How is $\rho_1$ related to the $1$-particle distribution function $n_1(\textbf x,\textbf p,t)$?

Solution #$2$: One simply takes the joint distribution and integrates away the $6(N-1)$ degrees of freedom of the other $N-1$ particles:

$$\rho_1(\textbf x,\textbf p,t)=\int d^3\textbf x_2…d^3\textbf x_Nd^3\textbf p_2…d^3\textbf p_N\rho(\textbf x,…\textbf x_N,\textbf p,…,\textbf p_N,t)$$

Moreover, because all $N$ particles are identical, the same marginal distribution $\rho_1$ applies equally well to all the other $N-1$ particles; thus $\rho_1(\textbf x,\textbf p,t)=\rho_2(\textbf x,\textbf p,t)=…=\rho_N(\textbf x,\textbf p,t)$. The one-particle distribution function $n_1(\textbf x,\textbf p,t)$ at $(\textbf x,\textbf p)\in\textbf R^6$ at time $t\in\textbf R$ is therefore:

$$n_1(\textbf x,\textbf p,t)=\sum_{i=1}^N\rho_i(\textbf x,\textbf p,t)=N\rho_1(\textbf x,\textbf p,t)$$

Problem #$3$: In terms of the $1$-particle distribution function $n_1(\textbf x,\textbf p,t)$, write down formulas for the number density $n(\textbf x,t)$ in configuration space. What about the momentum density and kinetic energy density (also in $\textbf x$-space)?

Solution #$3$: If one further marginalizes $n_1(\textbf x,\textbf p,t)$ over the momenta $\textbf p$, one obtains the number density of particles purely in configuration space:

$$n(\textbf x,t)=\int d^3\textbf p n_1(\textbf x,\textbf p,t)$$

Similarly, the momentum density in phase space is $\textbf pn_1(\textbf x,\textbf p,t)$ while the kinetic energy density in phase space is $\textbf |\textbf p|^2n_1(\textbf x,\textbf p,t)/2m$, and both of these can be similarly marginalized over $\textbf p$ to obtain corresponding density distributions in configuration space.

Problem #$4$: What is the analog of Liouville’s equation for the $1$-particle distribution function $n_1$?

Solution #$4$: For a generic Hamiltonian $H$:

$$\frac{\partial n_1}{\partial t}(\textbf x,\textbf p,t)=N\int d^3\textbf x_2…d^3\textbf x_Nd^3\textbf p_2…d^3\textbf p_N\{H(\textbf x,…\textbf x_N,\textbf p,…,\textbf p_N,t),\rho(\textbf x,…\textbf x_N,\textbf p,…,\textbf p_N,t)\}$$

Problem #$5$: To make further progress, it is necessary to actually specify some physics. The following typical dispersion relation for the Hamiltonian $H$ is chosen (here the identification $\textbf x_1:=\textbf x$ is being made):

$$H(\textbf x,…\textbf x_N,\textbf p,…,\textbf p_N,t)=\sum_{i=1}^N\left(\frac{|\textbf p_i|^2}{2m}+V_{\text{ext}}(\textbf x_i,t)\right)+\sum_{1\leq i<j\leq N}V_{\text{int}}(\textbf x_i-\textbf x_j)$$

Show that:

$$\frac{\partial n_1}{\partial t}(\textbf x,\textbf p,t)=\{H_1(\textbf x,\textbf p,t),n_1(\textbf x,\textbf p,t)\}+\left(\frac{\partial n_1}{\partial t}\right)_{\text{collision}}(\textbf x,\textbf p,t)$$

where $H_1(\textbf x,\textbf p,t):=|\textbf p|^2/2m+V_{\text{ext}}(\textbf x,t)$ is the $1$-particle Hamiltonian which features in the so-called streaming term $\{H_1,n_1\}$, and $\left(\partial n_1/\partial t\right)_{\text{coll}}$ is called the collision integral and is given by:

$$\left(\frac{\partial n_1}{\partial t}\right)_{\text{coll}}(\textbf x,\textbf p,t)=\int d^3\textbf x_2d^3\textbf p_2\frac{\partial V_{\text{int}}(\textbf x-\textbf x_2)}{\partial\textbf x}\cdot\frac{\partial n_2}{\partial\textbf p}$$

where the $2$-particle distribution function $n_2(\textbf x,\textbf x_2,\textbf p,\textbf p_2,t)$ is defined by:

$$n_2(\textbf x,\textbf x_2,\textbf p,\textbf p_2,t):=N(N-1)\int d^3\textbf x_3…d^3\textbf x_Nd^3\textbf p_3…d^3\textbf p_N\rho(\textbf x,…,\textbf x_N,\textbf p,…,\textbf p_N,t)$$

and has an interpretation entirely analogous to the one-particle distribution function $n_1(\textbf x,\textbf p,t)$.

Solution #$5$:

Problem #$6$: Explain why only the streaming term $\{H_1,n_1\}$ (and not the collision integral $(\partial n_1/\partial t)_{\text{coll}}$) matters for the time evolution of the real space number density $n(\textbf x,t)$. Is this also the case for the momentum density $\textbf pn_1$ or the kinetic energy density $|\textbf p|^2n_1/2m$?

Solution #$6$: Using $n=\int d^3\textbf p n_1$:

$$\frac{\partial n}{\partial t}=\int d^3\textbf p\{H_1,n_1\}+\int d^3\textbf p\left(\frac{\partial n_1}{\partial t}\right)_{\text{coll}}$$

But the integral over the collision integral vanishes for the same kind of reasons as in Solution #$5$:

(note that if instead one were interested in the time evolution of the momentum space number density $n(\textbf p,t):=\int d^3\textbf xn_1(\textbf x,\textbf p,t)$ then the collision integral does matter. Similar remarks apply to the real space momentum density $\int d^3\textbf p\textbf pn_1$ or the real space kinetic energy density $\int d^3\textbf p|\textbf p|^2n_1/2m$).

Problem #$7$: Explain how the methods of Solution #$5$ generalize to yield the BBGKY hierarchy of $k=1,2,…,N\sim 10^{23}$ coupled PDEs.

Solution #$7$: Using the same methods as Solution #$5$, one can check that:

$$\frac{\partial n_k}{\partial t}=\{H_k,n_k\}+\sum_{i=1}^{k}\int d^3\textbf x_{k+1}d^3\textbf p_{k+1}\frac{\partial V_{\text{int}}(\textbf x_i-\textbf x_{k+1})}{\partial\textbf x_i}\cdot\frac{\partial n_{k+1}}{\partial\textbf p_i}$$

with $k$-particle distribution function:

$$n_k(\textbf x_1,…\textbf x_k,\textbf p_1,…,\textbf p_k,t)=\frac{N!}{(N-k)!}\int d^3\textbf x_{k+1}…d^3\textbf x_Nd^3\textbf p_{k+1}…d^3\textbf p_N n(\textbf x_1,…,\textbf x_N,\textbf p_1,…,\textbf p_N,t)$$

and $k$-particle Hamiltonian $H_k$ including both $V_{\text{ext}}$ and interactions $V_{\text{int}}$ among the first $k$ particles but ignores interactions with the other $N-k$ particles:

$$H_k=\sum_{i=1}^{k}\left(\frac{|\textbf p_i|^2}{2m}+V_{\text{ext}}(\textbf x_i)\right)+\sum_{1\leq i<j\leq k}V_{\text{int}}(\textbf x_i-\textbf x_j)$$

The Boltzmann equation arises by truncating the BBGKY hierarchy. The idea is that there is a long time scale $\tau$ (interscattering time) and a short time scale $\tau_c\ll\tau$ (scattering time).