I understand that my title may be a little confusing, but I have been reading a paper on phase space descriptions of many particle systems. The paper focuses on the longitudinal dimensions ($z, \delta$) and describes what happens to the distribution while undergoing first order (linear transformations) transformations.
The initial distributions are gaussian in both the $z$ and $\delta$ coordinate:
$$\rho_z (z) = \frac{N}{\sqrt{2\pi}\sigma_z} exp \left({-\frac{z^2}{2\sigma_z^2}}\right) \tag{1}$$
$$\rho_{\delta} (\delta) = \frac{N}{\sqrt{2\pi}\sigma_{\delta}} exp \left({-\frac{\delta^2}{2\sigma_{\delta}^2}}\right) \tag{2}$$
So, the overall distribution can be described by:
$$\rho (z,\delta) = \frac{N}{2\pi\sigma_z\sigma_{\delta}} exp \left({-\frac{\delta^2}{2\sigma_{\delta}^2}}-\frac{z^2}{2\sigma_z^2}\right) \tag{3}$$
Now, the particle beam goes through an energy chirper which imparts an energy kick that is linearly correlated with the $z$ coordinate: $uz$. Where u is the slope of the correlation i.e. particles with $z=0$ get no energy shift/kick.
So, because the energy chirper is correlated with the $z$ coordinate (which is a gaussian distribution), the distribution of the energy change due to the chirper will also be gaussian:
$$\rho_{chirper} (\delta) = \frac{N}{\sqrt{2\pi}u\sigma_{z}} exp \left({-\frac{\delta^2}{2\ u^2 \sigma_{z}^2}}\right) = \frac{N}{\sqrt{2\pi}\sigma_{\delta_{chirper}}} exp \left({-\frac{\delta^2}{2\sigma_{\delta_{chirper}}^2}}\right) \tag{4}$$
where the $\sigma_{\delta_{chirper}}=u\sigma_z$ was found by $\sqrt{<{u^2 z^2}>-<uz>^2}$ and $<uz>=0$. Now, my problem/confusion comes in trying to combine the effects of the chirper into the initial energy distribution.
The paper provides the answer/solution (with no explanation) as:
$$\rho (z,\delta) = \frac{N}{2\pi\sigma_z\sigma_{\delta}} exp \left({-\frac{(\delta+uz)^2}{2\sigma_{\delta}^2}}-\frac{z^2}{2\sigma_z^2}\right) \tag{5}$$
My confusion about this equation is:
- Shouldn't the $\sigma_{\delta}$ in eq. 5 used for the $\delta$ coordinate be the new $\sigma_{\delta-new}$ that includes both the initial distribution and energy chirper as so:
$$\sigma_{\delta-new}= \sqrt{<(\delta+uz)^2>-<\delta+uz>^2}=\sqrt{\sigma_{\delta}^2 + u^2 \sigma_{z}^2 }$$
given that the initial $\delta$-distribution is not correlated at all with the $z$-coordinate, $<z\delta>=0$.
- Is it true that the $\delta$ used in eq. 3 is not the same as the one in eq.5. The are related by: $\delta_3=\delta_5+uz$? My mathematical interpretation is $\delta_3$ is the overall energy coordinate, whereas $\delta_5$ is more of a intermediary value and $\delta_5 + uz$ is the true overall $\delta$ coordinate?
Hopefully, this makes some sense! I am trying to make sense of all this and would really appreciate any help from the experts! Thanks in advance!