In the boltzmann equation,(adsbygoogle = window.adsbygoogle || []).push({});

[itex]{\bf{L}}\left[ f \right] = {\bf{C}}\left[ f \right][/itex], the right which is the collision term and in general it is[itex]{\bf{C}}[f]=\sum\limits_{p,p_1} {|Amplitiude|^2}{f(p)}[/itex]. and explicitly, the collision term for decaying process is

[itex]

\begin{eqnarray}

{\bf{C}}\left[ f \right] &=& \frac{1}{{E\left( p \right)}}\int {\frac{{{d^3}p}}{{{{\left( {2\pi } \right)}^3}2E\left( p \right)}}\frac{{{d^3}{p_2}}}{{{{\left( {2\pi } \right)}^3}2E\left( {{p_2}} \right)}}}\nonumber\\

&\times& {\left| {{\cal M}\left( {p \to {p_1} + {p_2}} \right)} \right|^2}f\left( p \right)\nonumber\\

&\times& {\left( {2\pi } \right)^4}{\delta ^4}(p - {p_1} + {p_2})

\end{eqnarray}

[/itex]

I want to know exactly how does the coefficient in the equation comes from,for it is very important for the result.

My questions are:

1) how does the [itex]E(p)[/itex]comes from

2) [itex]\begin{equation}

\int {\frac{{{d^4}p}}{{{{\left( {2\pi } \right)}^4}}}\left( {2\pi } \right)\delta ({E^2} - {m^2} - {p^2})} = \int {\frac{{{d^3}p}}{{{{\left( {2\pi } \right)}^3}}}\frac{{\delta (E - \sqrt {{m^2} + {p^2}} )}}{{2E}}}

\end{equation}[/itex],the[itex](2\pi)^4[/itex]in the left of the equation is the space element with[itex]\hbar=1[/itex],but how does the [itex]2\pi[/itex] comes from?

3) in the equation above, why cannot we write the [itex]\delta[/itex]function is the left with[itex]\delta (E - \sqrt {{m^2} + {p^2}} )[/itex] directly, if so, then [itex]2E[/itex]does not exist in denominator of the right equation.

4) why there is [itex](2\pi)^4[/itex]in the equation[itex]{\left( {2\pi } \right)^4}{\delta ^4}(p - {p_1} + {p_2})[/itex].

can anyone give give explanations or recommend some books to which I can refer?

**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!

# How does the coefficient in the Boltzmann equation comes from

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for does coefficient Boltzmann |
---|

I Does dark matter contribute to my body mass? |

I How does one actually compute a correlation function? |

B Does Hubble's Law imply that galaxies are accelerating away |

I Does space have its own density? |

B Why does the multiverse theory have only 4 levels? |

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