🔴🔵 ENTROPY – Interactive Entropy Simulation

An educational 3D (and AR-capable) simulation demonstrating the irreversible increase of entropy in a closed system, inspired by the classic thought experiment of mixing two types of gas particles.

Red and blue particles bounce elastically inside a cubic box with no external forces. Over time, they mix irreversibly — entropy rises from near zero (ordered/separated state) to maximum (fully mixed).

The simulation calculates and displays Shannon entropy in real-time based on spatial distribution, making the Second Law of Thermodynamics visually intuitive.

entropy.mp4

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🚀 Features

• Realistic Physics: Elastic collisions between particles and walls (conservation of momentum and energy).
• Entropy Calculation: Space divided into 3D grid cells; Shannon entropy computed from red/blue ratios per cell.
• Real-Time HUD: Current entropy value displayed (desktop) or floating panel (AR).
• AR Support: View the simulation in augmented reality on supported devices.
• Adjustable Parameters: Particle count, speed, box size, grid resolution (configurable in code).
• No External Forces: Purely statistical mechanics — demonstrates spontaneous entropy increase.

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📂 Project Structure

eslint.config.js # ESLint configuration for code quality and React hooks
index.html # HTML entry point with meta, Launchar SDK script, and root div
package.json # Dependencies (R3F, Rapier, XR, stdlib gammaln) and scripts
package-lock.json # Locked dependency versions

main.jsx # React entry: renders with StrictMode
index.css # Global styles for layout and dark theme
App.jsx # Main app: XR store, Canvas setup, Physics, ParticleContainer, EntropyGrid, AR button

EntropyGrid.jsx # Grid visualization and entropy calculation (uses gammaln for ln!)
ParticleContainer.jsx # Particle spawning, RigidBody setup, position polling for entropy

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🧠 Theoretical Background

This simulation visualizes one of the most profound concepts in physics: the Second Law of Thermodynamics.

1. Macrostate vs. Microstate

• Initially, all red particles on one side, blue on the other → highly ordered macrostate.
• Many possible microstates (particle positions/velocities) correspond to separated state → low multiplicity.
• Fully mixed state has vastly more microstates → high multiplicity.

2. Entropy Definition (Boltzmann)

$$ S = k \cdot \ln W $$

• $k$: Boltzmann constant
• $W$: number of microstates corresponding to the macrostate

Higher $W$ → higher entropy. The system naturally evolves toward states with maximum $W$.

3. Shannon Entropy (Used in Simulation)

Since exact $W$ is computationally infeasible, we approximate using spatial Shannon entropy: $$H = - \sum_{i} p_i \log_2 p_i$$

• Grid divides space into cells.
• $p_i$: fraction of red (or blue) particles in cell $i$.
• Averaged over all cells → global entropy measure.

$H = 0$ when particles are perfectly separated (pure cells).
$H = 1$ (maximum) when perfectly mixed (50/50 in every cell).

4. Irreversibility

Even though individual collisions are time-reversible, the overwhelming statistical preference for high-entropy states makes return to ordered state practically impossible.

This is why "unmixing" gases spontaneously never happens — even though it's not forbidden by Newton's laws.

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📦 Installation & Running

Prerequisites

• Node.js (v16+ recommended)

Steps

cd code/entropy
npm install
npm run dev