Particle Statistics: Explaining Klimontovich's Formulas and Logic

[kaniello]

Hallo, I posted this in General Math, and I decided to post it here also because this room seems more appropriate. The formulas and part of the text are quoted from "Klimontovich - Statistical theory of non-equilibrium processes in a plasma":

Let $N_{a}(\textbf{x},t) =\Sigma_{i=1,N_{a}}\delta(\textbf{x}-\textbf{x}_{ai})$ be the phase density of particles of species $a$ and $f_{N}$ the distribution function of the coordinates and momenta of the all $N=\Sigma_{a} N_{a}$ particles of the system respectively.

The statistical average of $N_{a}$ is then

$\overline{N_{a}( \textbf{x},t )}$=$\int\sum_{i=1,N_{a}}\delta(\textbf{x}-\textbf{x}_{ai})f_{N} \prod_{a}d^{6}\textbf{x}_{a1}...d^{6}\textbf{x}_{a_{N_{a}}}$

and since all the particles of one kind are identical

=$N_{a} \int\delta(\textbf{x}-\textbf{x}_{a1})f_{N} \prod_{a}d^{6}\textbf{x}_{a1}...d^{6}\textbf{x}_{a_{N_{a}}}$

If we define

$f_{a}(\textbf{x}_{a1},t)=V \int f_{N}d^{6}\textbf{x}_{a2}...d^{6}\textbf{x}_{a_{N_{a}}} \prod_{b\neq a}d^{6}\textbf{x}_{b1}...d^{6}\textbf{x}_{b_{N_{b}}}$ where $V$ is the volume of the particle, then we can write

$\overline{N_{a}}( \textbf{x},t ) = n_{a} f_{a}(\textbf{x},t)$ where $n_{a}$ is the mean concentration of particles of the kind $a$

Up to here everything seems ok. He now tries to connect the mean values of the products of the phase densities $N_{a},N_{b}$ in the following way, where my problems come:

Splitting the double sum

$\Sigma_{i=1,N_{a}}\Sigma_{j=1,N_{b}} \delta(\textbf{x}-\textbf{x}_{ai}) \delta(\textbf{x}'-\textbf{x}_{bj})$

into the two parts (why?)

$\Sigma_{i=1,N_{a}}\Sigma_{j=1,N_{b}}\delta(\textbf{x}-\textbf{x}_{ai})\delta(\textbf{x}'-\textbf{x}_{bj})$

(for x_ai≠x_bj when a=b)

+

$\delta_{ab}\Sigma_{j=1,N_{a}} \delta(\textbf{x}-\textbf{x}_{ai}) \delta(\textbf{x}-\textbf{x}')$

we obtain, neglecting unity when compared with $N_{a}$ (when do we compare unity with $N_{a}$ ?)

$\overline{N_{a}( \textbf{x},t )N_{b}( \textbf{x}',t)}=n_{a}n_{b}f_{ab} ( \textbf{x},\textbf{x}',t)+\delta_{ab}n_{a}\delta( \textbf{x}-\textbf{x}')f_{a}(\textbf{x},t)$

where $f_{ab}(\textbf{x}_{1a},\textbf{x}_{1b},t)=V^{2} \int f_{N}d^{6}\textbf{x}_{a2}...d^{6}\textbf{x}_{a_{N_{a}}}d^{6}\textbf{x}_{b2}...d^{6}\textbf{x}_{b_{N_{b}}}\prod_{c \neq a,b}d^{6}\textbf{x}_{c1}...d^{6}\textbf{x}_{c_{N_{c}}}$

So, please, can anyone explain me the logic behind this?


Thank you very much in advance,
Kaniello

 

[Stephen Tashi]

It would be hard for a person only familiar with mathematics to interpret this excerpt from a book on physics. For example, I don't know the the $\delta$ is an indicator function, a Dirac $\delta$, or something else and I certainly don't know what $\delta_{ab}$ or x' represents. You'll probably get a better answer by posting in a section of the forum that deals with statistical physics.

If you want help from a mathematician, I suggest that you give a link that explains the physics that is going on. For example, one link I found about "phase density" is the PDF http://www.google.com/url?sa=t&rct=...sg=AFQjCNFYEck6SnQDMfBDreU8TobxNNzY4A&cad=rja