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}<br /> \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}<br /> \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}}}<br /> \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}&#039;-\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}&#039;-\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}&#039;)

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}&#039;,t)}=n_{a}n_{b}f_{ab} ( \textbf{x},\textbf{x}&#039;,t)+\delta_{ab}n_{a}\delta( \textbf{x}-\textbf{x}&#039;)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