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).