# Is the Vlasov equation a good kinetic equation for solids or even liquids?

Suppose we know the intermolecular potential $$V(r-r')$$ acting between the positions $$r,r'$$. Then we have the kinetic equation for a one-particle Distribution function $$f(r,p,t)$$ given by:

$$\frac{\partial}{\partial t}f(r,p,t) = -\frac{p}{m} \frac{\partial}{\partial r}f(r,p,t) - \frac{\partial}{\partial r}\int d^3r' d^3p' f(r',p',t)V(r-r') \frac{\partial}{\partial p}f(r,p,t)$$.

This equation for a (collisionless) many-particle System with particle mass $$m$$ is called the Vlasov equation. It can be solved with the "deterministic particle trajectory" Ansatz:

$$f(r,p,t) = \prod_{i=1}^N \delta(r-r_i(t))\delta(p-p_i(t))$$.

Here, $$r_i(t),p_i(t)$$ is the time-dependent Position,momentum trajectory for the $$i$$-th particle in the System. It is easy to Show that These trajectories obeye the exact Newton's equation of Motion.

So now let the particle number $$N$$ be very huge; this is the case in dense liquids and solids. Then the 6-dimensional phase space generated by $$(r,p)$$ will have a huge amount of Peaks where the particles are localized. Like scattering many many marbles into space, nearly continuous structures in Phase space can be formed. These structures are similar to well-known statistical mechanics distributions like the Maxwell Distribution.

Question: Will Vlasov equation be a sufficient kinetic equation for some ideal solids or even liquids?

According to this paper

https://archive.org/details/arxiv-cond-mat9910070

liquids can be modelled with a pair correlation function (triplet correlations are neglected), where collective effects in the liquid are modelled in the kinetic equation for the 2-particle Distribution function.

Is it possible to assume pairwise collisions in liquids? Or are triplet or higher order correlations mandatory?

My idea:

According to the Ornstein-Zernike equation

$$h(r_1,r_2) = c(r_1,r_2) + n \int d^3r_3 c(r_1,r_3)c(r_3,r_2) + n^2 \int \int d^3r_3 d^3r_4 c(r_1,r_3)c(r_3,r_4)c(r_4,r_1) + \dots$$

for the total pair correlation function $$h(r_1,r_2)=g(r_1,r_2)-1$$ and a direct correlation function $$c(r_1,r_2)$$ (can be obtained by equation closure approximations), correlations are built up when other intervening particles in the interaction range between two particles are present. Typically, interparticle interactions have a few angstroms, but if the System is dense, particles can come in-between the interaction range. In case of a liquid, we have to take account triple correlation, since I guess the term $$\int d^3r_3 c(r_1,r_3)c(r_3,r_2)$$ is significant. Can I assume that the Expansion to higher powers in $$c$$ converges at some order (such that higher correlations can be neglected)?