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":(adsbygoogle = window.adsbygoogle || []).push({});

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

The statistical average of [itex]N_{a}[/itex] is then

[itex]\overline{N_{a}( \textbf{x},t )}[/itex]=[itex]\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}}}[/itex]

and since all the particles of one kind are identical

=[itex] 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}}}[/itex]

If we define

[itex]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}}}[/itex] where [itex]V[/itex] is the volume of the particle, then we can write

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

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

Splitting the double sum

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

into the two parts (why???)

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

(forx_{ai}≠x_{bj}when a=b)

+

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

we obtain, neglecting unity when compared with [itex]N_{a}[/itex] (when do we compare unity with [itex]N_{a}[/itex] ???)

[itex]\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) [/itex]

where [itex]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}}}[/itex]

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

Thank you very much in advance,

Kaniello

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Particle Statistics

**Physics Forums | Science Articles, Homework Help, Discussion**