Introduction

MATLAB Functions used to plot for dynamics of the triangular billiard.

• standard map of Kicked system: Standard_Map_Plot.m
• Boundary map of triangular billiard: Standard_Map_tri_Plot.m
• Birkhoff coordinates calculator: tri_billiard2.m
• Density function: density.m
• transfer operator $T^n \rho$: billiard.m
• Boundary map of transfer operator: density_Plot.m

Usage:

Method 1 : use git to download: git clone https://github.com/Homingdung/2deg_billiard_matlab.git

Method 2: click Code, then Download zip

The poster is accessible, simply click the file 2021_poster.pdf to have a look, the corresponding file package is summer research 2021, thanks to the ''Henriques lab poster template''.

Background Knowledge

Kicked system

The model describes a particle constrained to move on a ring. The particle is kicked periodically. The model could be described by Hamiltonian: $$ H(p,\theta,t)=\frac{1}{2}p^2+\frac{K}{\pi^2}cos(\pi\theta)\sum_{-\infty}^{\infty}\delta(t-n) $$ Where $$ \begin{eqnarray} &&\theta_{n+1}=\theta_n+\pi p_{n}\ &&p_{n+1}=p_n+\frac{K}{\pi}sin(\theta_n+\pi p_n) \end{eqnarray} $$ both $\theta$ and $p$ will take modulo, $2\pi$ and $2$, respectively, which could be used to create a 2D standard map.

Poincaré map

In the theory of dynamical systems, a Poincaré map is an intersection of periodic orbits in state space with a certain lower-dimensional subspace, called Poincaré section.

Birkhoff coordinate

Birkhoff coordinate, also known as boundary coordinate, which can be used to define a natural Poincaré section, specifies the collision point, with which could be displayed on a phase-space with $(s,p)$, where $s$ is the position along the boundary, and $p$ is the projected momentum on the boundary. Based on Birkhoof coordinate, a list of points could be generated by Poincaré mapping and then get the boundary map which is a special type of Poincaré section by visualizing these points.

Bifurcation

Bifurcation of a system occurs when a small, smooth change in a parameter causes the system to suddenly behave in a qualitative or topological way. There are two principle types of bifurcation:

• Local bifurcations
• Global bifurcations

Transfer operator

The problem of propagation in cavities, which is widely seen in wireless communication, semiconductor manufacturing, etc., is most effectively solved by focusing on the boundary in the lower dimensions (e.g., the inner wall of an optical fibre) rather than in the higher dimensions of the interior. That is the reason why we create a mapping for the collision points rather than the specific orbits or the electrons moving in the cavity.

In a simple way, transfer operator could be seen as the behavior of a dynamical system on the mass density of initial conditions. Transfer operators methods have been successfully applied to Dynamical Energy Analysism, which has a wide range of industrial applications. In this research, the transfer operator arises from the equations of motion and can be expressed as an integral over initial conditions.

2DEG

Two-dimensional electron gas (2DEG) is a solid-state physics model. This project is to derive a formula for the 2DEG transport.