The Fokker-Planck equation

The state of a system of $N$ particles with velocities ${\bf v} =({\bf v}_i)$ and positions ${\bf x} =({\bf x}_i)$ ($i=1,\ldots N$) is represented by a point in $6N$-dimensional $\Gamma$-space of positional and velocity variables. The $N$-particle distribution $f^{(N)}({\bf x},{\bf v}, t)$ is defined as the probability to find the system in the volume element $d^3\!{\bf v}d^3\!{\bf x}$ around ${\bf x}$ and ${\bf v}$ at time $t$. $f^{(N)}$ is normalised as $1 = \int f^{(N)} d^3\!{\bf v}d^3\!{\bf x}$ The evolution of the system is a trajectory in $\Gamma$-space; the evolution of a continuous subspace of systems at $t=t_0$ (initial conditions) can be seen as a flow in $\Gamma$-space, which due to the absence of any dissipation is incompressible, and thus described by a continuity equation in $\Gamma$-space, which is Liouville's equation:

$$\frac{\partial f^{(N)}}{\partial t} + \sum_{i=1}^N \Bigl\{ \... ...v}_i} \Bigl[ f^{(N)} {d{\bf v}_i\over dt} \Bigr] \Bigr\} = 0 .$$ (1)

Using the definition $d{\bf x}_i/dt = {\bf v}_i$ and $\partial{\bf v}_i/\partial{\bf x_i} = 0$, since ${\bf v}_i$ and ${\bf x}_i$ are independent coordinates, and if the forces are conservative, $d{\bf v}_i/dt = -\partial\Phi_i/\partial{\bf x}_i$, where $\Phi_i$ is the potential at the position of particle $i$ due to the other particles, we can simplify last equation to

$$\frac{\partial{f^{(N)}}}{\partial{t}} + \sum_{i=1}^N \Bigl[ ... ... \frac{\partial{f^{(N)}}}{{\partial{\bf v}_i}} \Bigr] = 0 .$$ (2)

Here it has been utilised that $d{\bf v}_i/dt$ does not depend on ${\bf v}_i$ itself, since the potential only depends on the spatial coordinates.

Now we introduce the one-particle distribution function $f^{(1)}({\bf x},{\bf v}, t)$ as

$$f^{(1)}({\bf x},{\bf v}, t) = \int f^{(N)}({\bf x}_i,{\bf v}... ...x_2}\ldots d^3\!{\bf x_N}d^3\!{\bf v_2}\ldots d^3\!{\bf v_N} ,$$ (3)

the two-particle distribution function

$$f^{(2)}({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) = \int ... ...}_3\ldots d^3\!{\bf x}_Nd^3\!{\bf v}_3\ldots d^3\!{\bf v}_N \$$ (4)

and the two-particle correlation function $g$ by

$$g({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) = f^{(2)}({\b... ...f^{(1)}({\bf x}_1,{\bf v}_1,t) f^{(1)}({\bf x}_2,{\bf v}_2,t)$$ (5)

$g$ measures the excess probability of finding a particle at ${\bf x}_1$, ${\bf v}_1$ due to the presence of another particle at ${\bf x}_2$, ${\bf v}_2$. Since $f^{(1)}$ is normalised to unity, one has $\rho({\bf x}_2,t) = mN \int f^{(1)}({\bf x}_2,{\bf v}_2,t)d{\bf v}_2$, where $\rho$ is a mean mass density, and $m$ the individual stellar mass. Assuming that $f^{(N)}$ is symmetric with respect to exchange of particles (i.e. all particles are indistinguishable), and observing that $\Phi_1$ is also symmetric with respect to exchanges of the particles $2,\ldots, N$, one arrives at

$$\frac{\partial f^{(1)}}{\partial {t}} + {\bf v}_1 \cdot \fra... ... \frac{\partial {f^{(1)}}}{\partial {{\bf v}_1}} = \nonumber$$

$$-Gm(N\!-\!1) \int \frac{\partial}{\partial {\bf v}_1} \Bigl(... ... x}_1- {\bf x}_2\vert^{-1}\Bigr)d^3\!{\bf x_2}d^3\!{\bf v_2} .$$ (6)

Now one substitutes $f^{(1)}$ by the more common phase space density $f = f^{(1)}/N$ and drops for simplicity all subscripts ``1''. It follows

$$\frac{\partial{f}}{\partial{t}} + {\bf v} \cdot \frac{\parti... ...{{\bf v}}} = \bigg(\frac{\delta f}{\delta t} \bigg)_{\rm enc}$$ (7)

If the average particle distance $\bar{d}=1/n^{1/3}$ is bigger than the impact parameter $p_{90}$ related to a $90^o$ deflection, most of the scattering is due to small angle encounters, which change velocity and position of the particle only weakly. So, if we assume that all correlations stem from gravitational two-body scatterings of particles, not from higher order correlations, we have that the Fokker-Planck equation is

$$\bigg(\frac{\delta f}{\delta t} \bigg)_{\rm enc}= -\sum_{i... ...l( f({\bf x},{\bf v}) D(\Delta v_i)\Bigr) \Biggr] \nonumber$$

$$+ {1\over 2} \sum_{i,j=1}^3 \Biggl[ {\partial^2\over\part... ... f({\bf x},{\bf v}) D(\Delta v_i\Delta v_j)\Bigr) \nonumber$$

$$+ {\partial^2\over\partial x_i\partial v_j}\Bigl( f({\bf x... ...( f({\bf x},{\bf v}) D(\Delta v_i\Delta x_j)\Bigr) \Biggr] \$$ (8)

Here the convenient notation of diffusion coefficients has been introduced, which contain the integration over the velocity and position changes, as e.g.:

$$D(\Delta v_i) := \int \Delta v_i \Psi({\bf x},{\bf v},{ \D... ...x},{ \Delta \bf v}) d^3\!{\ \Delta \bf x}d^3\!{\Delta \bf v},$$ (9)

where $\Psi({\bf x},{\bf v}, {\Delta \bf x},{\Delta \bf v})$ $d^3\! {\Delta \bf x}d^3\!{\Delta \bf v}dt$ is defined as the probability for a star with position ${\bf x}$ and velocity ${\bf v}$ to be scattered into a new phase space volume element $d^3\!{\Delta \bf x}$ $d^3\!{\Delta \bf v}$ located around ${\bf x}+{\Delta \bf x}$, ${\bf v}+{\Delta \bf v}$ during the time interval $dt$.