Particle distribution in the longitudinal beam dynamics in accelerator

[M.A.M.Abed]

In longitudinal beam dynamics, particles exhibits a synchrotron motion. The motion has an amplitude (action) dependent synchrotron frequency. The motion is defined in terms of rf phase devotion deviation $\phi$ as:
$H=\dot \phi^2/2 - \Omega_s^2 \cos(\phi)$
I am trying to write the particle distribution in terms of Hamiltonian $f_H(H)$ and action $f_I(I)$, I know that the distribution has Gaussian distribution In terms of action $f_I(I)=1/\sigma_I^2 \exp(-H/\sigma_I^2)$, but I do not know what is $\sigma_I$ means and how to find it. I have $\sigma_\phi$ and $\sigma_{\dot phi} = h \eta w \sigma_{\delta_p}$.

where $\Omega_s$ is the nominal synchrotron frequency, h is the rf harmonic of accelerating voltage, $\eta$ is the slip factor, w is the revolution frequency, $\sigma_{\Delta_p}$ is the standard devotion deviation of the momentum deviation.

One more question, in the machine parameters its written that the $\sigma_{\delta_p}$ is not given, instead the max $\Delta_p/p$ is given, is it ok to assume that $\sigma_{\delta_p}=(\Delta_p/p)_{max}/4$

 

[Mark44]

@M.A.M.Abed I have attempted to revise your LaTeX into a more readable form, but I'm not certain that my changes are true to what you wrote. Please let me know if you have any questions about what I did.

 

[Vanadium 50]

I am still not sure what is going on with your question, but a Hamiltonian describes the behavior of one particle and you seem to be asking about distributions. Normally, one considers a statistical/phase space approach to understand distributions,

Note that the time it takes to fill phase space can be long compared to the time the beam is in the accelerator, but it doesn't have to be. So you will need to take care that your calculational assumptions are accurate.