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"# Molecular Dynamics Simulation of Thermalization\n",
"\n",
"## Introduction\n",
"This code simulates the thermalization of a system of particles undergoing elastic collisions. It models a **2D gas** where particles are initially divided into two populations:\n",
"- **Hot particles** with a higher initial temperature\n",
"- **Cold particles** with a lower initial temperature\n",
"\n",
"As the simulation progresses, particles collide and exchange energy, leading to **thermal equilibration**. The key observables in this simulation include:\n",
"1. **Kinetic energy evolution of labeled particles**\n",
"2. **Histogram of kinetic energy distributions before and after thermalization** (now with a log-linear plot)\n",
"3. **Velocity distributions before and after thermalization**, compared to the **normalized 2D Maxwell-Boltzmann distribution**\n",
"4. **An animated visualization of particle dynamics** (optimized for inline rendering in Jupyter Book)\n",
"\n",
"## Physical Model\n",
"Each particle follows **Newtonian mechanics**, moving freely until it collides with a wall or another particle. \n",
"\n",
"### **Hard-Sphere Collision Model**\n",
"Particles are modeled as **hard spheres**, meaning their interactions follow **elastic collision laws**:\n",
"- Momentum and kinetic energy are conserved.\n",
"- Collisions with walls are perfectly elastic.\n",
"- **Particle-particle collisions are explicitly handled** using momentum and energy conservation equations.\n",
"\n",
"### **Initial Conditions**\n",
"- Particles are assigned **random positions** within a box.\n",
"- Half of the particles are initialized with **higher kinetic energy** (hot particles), while the other half have **lower kinetic energy** (cold particles).\n",
"- A small subset of particles is **labeled** for tracking.\n",
"\n",
"### **Thermalization Process**\n",
"As time progresses, the energy distribution evolves due to collisions, approaching a **Boltzmann distribution**. The final kinetic energy distribution should follow an **exponential distribution**, characteristic of a system in thermal equilibrium.\n",
"\n",
"## Implementation Details\n",
"- **Elastic collisions** are detected and handled via velocity updates.\n",
"- **Hard-sphere interactions** are included to ensure realistic thermalization.\n",
"- **Kinetic energy tracking** is performed for labeled hot and cold particles.\n",
"- **Animation subsampling** is implemented to reduce memory overhead.\n",
"- **Histograms** show the initial and final kinetic energy distributions, now also in a **log-linear plot**.\n",
"- **Velocity distributions are plotted and compared to the properly normalized 2D Maxwell-Boltzmann distribution**.\n",
"\n",
"## Visualization and Outputs\n",
"1. **Inline animation** of particle motion (compatible with Jupyter Book, avoids external windows).\n",
"2. **Time series plots** of kinetic energy for labeled particles (rendered inline in Jupyter Book).\n",
"3. **Histograms of kinetic energy distributions** before and after equilibration, including a **log-linear plot**.\n",
"4. **Velocity distributions before and after thermalization**, with an overlay of the **normalized 2D Maxwell-Boltzmann distribution**."
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