Appendix F Vlasov-Poisson Equations

The number of particles in the six-dimensional phase space element $dVdP$ at time $t$ can be expressed as

$N(t)={\displaystyle \int f(r_i,p_i,t)𝑑V𝑑P},$

(F.1)

where $f(r_i,p_i,t)$ is the so called distribution function of particles in phase space. The particle density can be expressed in terms of the distribution function as

$\rho (r_i,t)=mn(r_i,t)=m{\displaystyle \int f(r_i,p_i,t)𝑑P},$

(F.2)

where $n(r_i,t)$ is the number density of all particles and $m$ is the particle mass.

Without collisions, the distribution function satisfies the equation

${\displaystyle \frac{d}{dt}}f(r_i,p_i,t)=0.$

(F.3)

Writing the time derivative explicitely we get the collisionless Boltzmann equation

${\displaystyle \frac{\partial }{\partial t}}f+{\displaystyle \frac{\partial r_i}{\partial t}}{\displaystyle \frac{\partial f}{\partial r_i}}+{\displaystyle \frac{\partial p_i}{\partial t}}{\displaystyle \frac{\partial p_i}{\partial f}}=0.$

(F.4)

With the velocity $v_i=\frac{\partial r_i}{\partial t}$ and the gravitational force $F_i=\frac{\partial p_i}{\partial t}=m{g}_i$, this equation can be expressed as

${\displaystyle \frac{\partial }{\partial t}}f+v_i{\displaystyle \frac{\partial f}{\partial x_i}}+m{g}_i{\displaystyle \frac{\partial p_i}{\partial f}}=0.$

(F.5)

Together with the Poisson equation of gravity

${\displaystyle \frac{\partial }{\partial r_i}}{g}_i=4\pi Gm{\displaystyle \int f(r_i,p_i,t)𝑑P}$

(F.6)

equations F.5 and F.6 are often called the Vlasov-Poisson system of equations (Peebles, 1980). This system is used in astrophysics to describe the evolution of collisionless matter interacting only by gravity.

However in cosmological N-Body simulations it is not the Vlasov-Poisson system of equations that is solved. In fact, one assumes that the solution of the trajectories of $N$ particles determined by Newtons laws

	${\displaystyle \frac{\partial p_{i,j}}{\partial t}}$	$=m_j{g}_{i,j},$		(F.7)
	$g_{i,j}$	$=G{\displaystyle \sum _{l=1}^N}{m}_l{\displaystyle \frac{r_{i,j}-r_{i,l}}{{(r_{i,j}-r_{i,l})}^3}},$		(F.8)

where $m_j,m_l$ and $r_{i,j},r_{i,l}$ are the position of the $j$th and $l$th particle respectively, can be interpreted as Monte-Carlo-Approximation of the Vlasov-Poisson system (Steinmetz, 1999). So every particle in a cosmological N-Body simulation can in fact represent a huge number of particles, which is a major conceptual difference to N-body simulations used to model planetary systems or stars in star clusters, where each particle intends to mimic an actual physical body.