# Anharmonic Oscillators and Equilibrium

## Group Members
* Ching-Yu Yao (B08202042), Department of Physics, National Taiwan University
* Kuei-Hung Tseng (B08202059), Department of Physics, National Taiwan University
* Jie-Ming Li (B08202054), Department of Physics, National Taiwan University
* Chi-Kai Yang (B08202033), Department of Physics, National Taiwan University

## Abstract
For a free system, we can always treat it as a number of independent simple harmonic oscillators. But if a system only consists of simple harmonic oscillators, there will be no interaction. Moreover, the system can't even reach equilibrium.
Therefore, the importance of the "anharmonic terms" arise. The anharmonic terms are also called the "interacting terms", which provide interactions between different oscillators (modes).
In this project, we want to use numerical simulation to discuss how the anharmonic terms affect the system.
Due to the incontinuity of numerical methods, it is a important task to reduce the deviations from the real system.
We expect to see the system reaches equilibrium after adding interacting terms. Futhermore, we want to discuss the relation between the coupling constant and the equilibrium state, and calculate some physical quantities of the system.

## Background and Objectives
Simple harmonic motion is one of the most basic and important model in physics. Professor Yu-Tin Huang once said,
> Physicists can only deal with simple harmonic oscillators and Gaussian intergrals. We can do nothing but these.

Which is quite true. The idea of Fourier transformation (or mode expansion) is basically considering that the system is consisting of oscillators.
For a free system, we can simply write down the Lagrangian
\begin{equation}
L(x,\dot{x})=\sum_{i,j}\frac{1}{2}m_{ij}\dot{x}_i\dot{x}_j-\frac{1}{2}k_{ij}x_ix_j,
\end{equation}
and we can treat this system as a combination of different modes by diagonizing the Lagrangian
\begin{equation}
L(y,\dot{y})=\sum_i\frac{1}{2}\dot{y}_i^2-\frac{1}{2}\omega_i^2y_i^2.
\end{equation}
Since the modes are independent, there will be no interaction between them. Moreover, this means that there will be no interaction between the (quasi-)particles after canonical quantization, and there will be no physics for a system of particles propagating in spacetime without having interactions.
Also, if we consider equilibrium as becoming Boltzman distribution, then the absence of interactions leads to impossibility of reaching equilibrium.

The key to the emergence of interactions is having "anharmonic terms" in the system.

In this project, we aim to show how the anharmonic terms provide interactions between different modes, and expect to see the system reaches equilibrium. Also, we want to discuss the relationship between the coupling constant and the behavior of how the system reaches its equilibrium, if it can reach.
Futhermore, we can calculate some physical quantities of the system that arise due to the apperence of anharmonic terms.

## Methods, Steps and Progress
### Methods and Steps
We are going to use programming language to simulate the anharmonic system.
The main tools we will be using are the programming skills related to the simulation and the animation on Python to visualize the obtained results. In addition, some of the familiar scientific packages like NumPy and SciPy will be used to simplify the calculation on differential equations.

### Timetable
| Week | Date | Task |
|:----:|:-----------:|:---------------------------------------------------------------------:|
| 1 | 11/15-11/21 | Preparing for Mechanics Midterm. |
| 2 | 11/22-11/28 | Programming for the SHO cases. |
| 3 | 11/29-12/5 | Observing and discussing the evolution of SHO systems. |
| 4 | 12/6-12/12 | Programming for the cases with potential containing anharmonic terms. |
| 5 | 12/13-12/19 | Continue programming for the cases with potential containing anharmonic terms. |
| 6 | 12/20-12/26 | Observing the evolution of systems with anharmonic term. |
| 7 | 12/27-1/2 | Discussing the overall result and the interpretation. |
| 8 | 1/3-1/9 | Preparing for poster presentation. |
| 9 | 1/10 | Poster presentation. |

### Responsibilities
* Kuei-Hung Tseng, coding, poster preparation.
* Jie-Ming Li, coding, preparing presentation.
* Chi-Kai Yang, coding, poster preparation.
* Ching-Yu Yao, physical interpretation of our works, presentation preparation.

## Expected Difficulties and Solutions
* Due to the incontinuity of the numerical simulation, the evolution of the system might die.
Solution: Try to optimize the numerical method.

* The result obtained by simulations may be far from our expectation and intuition, and may be hard to interpret.
Solution: Try to find the intepretation. (with team discussion, of course)

## Results and Evaluation
We expect to see that the anharmonic terms produce interactions between different modes, and the system reaches its equilibrium. We are going to determine whether the distribution of the system matches the Boltzmann's distribution as the criterion of whether the system reaches equilibrium.

After finishing the work, we can gain out understanding toward the anharmonic oscillators, and how important these anharmonic terms are for physics.

## References
1. Fermi, E.; Pasta, J.; Ulam, S. (1955). "Studies of Nonlinear Problems".