I Phase space density function and Probability density function

[shahbaznihal]

I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun.

While talking about the speed distribution, it then moves to take the full phase space density and attempts to write everything in those terms. The relation it uses looks like this, $$\text{Phase space density} = \tilde{f}(v)=n_v f(v)$$ Here, $f(v)$ is the speed distribution function and $n_v$ is said to be the phase space number density.

From what I understand about these terms, and what I have been able to gather: $f(v)dv$ is the probability that a certain particle, out of $N$, has a speed between $v$ and $v +dv$. Moreover, $\tilde{f}(v)d^Nv$ is the probability that the state of the system, the speed of all $N$ particles, lies in a cube of $d^Nv$. What is referred here as the "phase space number density $n_v$" is still unclear to me and I am not able to find it's definition.

From what I understand, if I try to write the phase space density in terms of the speed distribution I would write it something like this: $f(v_1)dv$ is the probability of finding a particle in the speed range $v_1$ and $v_1 +dv$ and same goes for $f(v_2)dv$ and so forth for all $N$ particles. The probability of finding one particle in speed range $v_1$ and $v_1 +dv$ and then particle 2 in speed range $v_2$ and $v_2 +dv$ and then so forth should be then $f(v_1)f(v_2)f(v_3)...f(v_N)d^Nv$ which should be just $\tilde{f}(v)d^Nv$. But I can't understand how it is reduced to the relationship given in the literature I am following i.e. $\tilde{f}(v)=n_v f(v)$, if I am doing it right.

So am I understanding the terms "speed distribution" and "phase space density" correctly? And if yes then is my calculation correct? And finally, what is the relationship $\tilde{f}(v)=n_v f(v)$ and what is phase space number density $n_v$?

 

[vanhees71]

The only thing I wonder about is that it's assumed to factorize. First of all, and that's important, the phase-space distribution function is a function of time, position, and momenta. I also guess you need the relativistic version of all that. Let's discuss the classical limit first. Then you have a phase-space distribution function $f(x,\vec{p})$, where $x=(x^{\mu})$ is the space-time four-vector and $p=(p^{\mu})=(p^0,\vec{p})$ the "on-shell" energy-momentum fourvector of particles with mass $m$, $p_{\mu} p^{\mu}=m^2$ (working in natural units, where $\hbar=c=1$).

The phase-space distribution function is defined to be a Lorentz-scalar quantity. This uses the fact that $\mathrm{d}^3 x \mathrm{d}^3 p$ is a Lorentz invariant for particles on the mass shell. The particle density has to be defined as time component of a four-vector current. Since $f(x,\vec{p})$ is a scalar, it's given by the fact that $\mathrm{d}^3 p/E_p$ with $E_p=\sqrt{\vec{p}^2+m^2}$ is an invariant too. Thus one defines the number-density current
$$N^{\mu}(x)=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{p^{\mu}}{E_{p}} f(x,\vec{p}).$$
The $1/(2 \pi)^3$ is conventional and comes from the natural unit of the single-particle phase-space cell $h^3=(2 \pi \hbar)^3$ which reduces to $(2 \pi)^3$ in our natural units.

As an example take the Boltzmann-Jüttner distribution describing an ideal gass in local thermal equilibrium
$$f(x,\vec{p})=g \exp[-p_{\mu} u^{\mu}(x)/T+\mu(x)/T],$$
where $\mu(x)$ is a local chemical potential of some conserved charge like baryon number (which can be extended by more chemical potentials if you have more than one conserved charge).

For more details on how to define the phase-space distribution function as a scalar quantity and about relativistic transport theory, see

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

 

[shahbaznihal]

Thanks for your comments.

Actually, at the moment, it is all classical. The orbits of WIMPs within the solar system follows classical physics principles, or assumed to do so for most part. I am confused about the following parts:

1) What is the phase space number density? How do you define it?
2) Is my conversation from speed distribution function to phase space distribution function correct?