12 October, 2026
To do!
Unordered notes
So I am starting to further the the study of change of the topology and geometry of the attractive case of the model, I will describe this in ‘Renormalisation’
- I need to think about the time units
Renormalisation
So I am considering the case that the displayed, relative volume should be a constant value \(c\). I am looking for \(\epsilon\), to find how the width should be displayed.
\[\frac{\int_{-\epsilon}^{\epsilon} x(p)dp}{\int_{-\infty}^{\infty} x(p)dp}=c\]
The Lorenzian function \(x(p)\propto \tfrac{1}{2m_2E-p^2}\) is obtained by Looking at the boundary of the surface of section. If the boundary is considered, we obtain
\[x(p)= \frac{2m_2 \mathcal{C}}{L} \frac{1}{2m_2E-p^2}-\partial\mathcal{B}_{1\text{R}}\]
The integral of the lorenzian term is
\[\int_{-\infty}^{\infty}\frac{dp}{p^2+\kappa^2} = \frac{\pi}{\kappa}, \qquad \int_{-\varepsilon}^{\varepsilon}\frac{dp}{p^2+\kappa^2} = \frac{2}{\kappa}\arctan\frac{\varepsilon}{\kappa}\] \[\frac{2}{\pi}\arctan\frac{\varepsilon}{\kappa} = c \quad\Longrightarrow\quad \boxed{\varepsilon = \sqrt{-2mE}\;\tan\!\left(\frac{\pi c}{2}\right)}, \qquad 0<c<1.\]
As the boundary only funcitons as an offset, and only cases \(E<0\) shall be considered, this consideres the problem to be solved.
Lost Wonderers Thoughts:
When discussing with Awais, he suggested, attractive and repulsive should behave the same, with the difference of changing the order of the boxes?