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| -rw-r--r-- | stokes.tex | 39 |
1 files changed, 30 insertions, 9 deletions
@@ -442,15 +442,6 @@ imaginary energy join. \section{Numerics} To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically. -\begin{equation} - \mathcal L(z(t), z'(t)) - = 1-\frac{\operatorname{Re}(\dot z^*_iz'_i)}{|\dot z||z'|} -\end{equation} -\begin{equation} - \mathcal C[\mathcal L]=\int dt\,\mathcal L(z(t), z'(t)) -\end{equation} -$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ whose tangent is everywhere parallel to the direction of the dynamics. Therefore, finding functions which reach this minimum will find time-reparameterized Stokes lines. For the sake of numeric stability, we look for functions whose tangent has constant norm. - \begin{acknowledgments} @@ -517,4 +508,34 @@ Putting these pieces together, we find \frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i} \end{equation} +\section{Numerics} + +To study Stokes lines numerically, we approximated them by parametric curves. +If $z_0$ and $z_1$ are two stationary points of the action with +$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we +take the curve +\begin{equation} + z(t) + =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_it^i +\end{equation} +where the $g$s are undetermined complex vectors. These are fixed by minimizing +a cost function, which has a global minimum only for Stokes lines. Defining +\begin{equation} + \mathcal L(t) + = 1-\frac{\operatorname{Re}[\dot z^*(z(t))\cdot z'(t)]}{|\dot z(z(t))||z'(t)|} +\end{equation} +this cost is given by +\begin{equation} + \mathcal C=\int_0^1 dt\,\mathcal L(t) +\end{equation} +$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ +whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics. +Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes +lines. + +We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to +the parameter vectors $g$. Stokes lines are found or not between points by +using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$, +and approximating the cost integral by a finite sum. + \end{document} |