Computational Physics Simulations ⚛️

A comprehensive collection of Python simulations and numerical solvers for modeling physical systems. This repository implements algorithms from scratch to solve problems in Fluid Dynamics, Stochastic Processes, Non-linear Dynamics, and Signal Processing.

Primary Focus: Scientific Computing, Numerical Analysis, and "First-Principles" Modeling.

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📂 Project Structure

The repository is organized into four core modules based on the physical domain and mathematical techniques used.

🔹 Module 1: Stochastic Systems

Simulating systems governed by randomness and probability.

• Monte Carlo Random Walks:
  • Simulates particle diffusion in 2D and 3D space.
  • calculates displacement statistics to model Brownian motion and diffusion processes.
• Ising Model & Phase Transitions:
  • (Optional/In-progress) Models magnetic dipole spins to study ferromagnetism and phase transitions using statistical mechanics.

🔹 Module 2: CFD & Differential Equations

Solvers for Partial Differential Equations (PDEs) in fluid mechanics and electromagnetism.

• Spectral Advection-Diffusion Solver:
  • Solves the time-evolution of transport phenomena using Fast Fourier Transforms (FFT) (Spectral Methods).
  • Demonstrates high-accuracy solutions for fluid flow compared to traditional grid methods.
• Poisson Equation Solver (1D):
  • Solves the electrostatic Poisson equation using Finite Difference Methods (FDM).
  • Implements boundary value problem solvers for potential fields.
• Wave Equation Solver:
  • Simulates wave propagation and interference patterns in time and space.

🔹 Module 3: Dynamic Systems & Chaos

Modeling time-dependent coupled systems and non-linear behavior.

• Lotka-Volterra Equations:
  • Numerical solution for Predator-Prey population dynamics using coupled ODEs.
• Chaos & Bifurcation:
  • Simulation of the Logistic Map to visualize the transition from stability to deterministic chaos (Period Doubling).
• Damped Harmonic Oscillator:
  • Solves the equation of motion for damped systems using Runge-Kutta (RK4) integration methods.

🔹 Module 4: Numerical Methods & Algorithms

Implementation of core mathematical algorithms from scratch.

• Fast Fourier Transform (FFT):
  • A custom recursive implementation of the Cooley-Tukey algorithm.
  • Application: Used for signal processing and spectral derivatives.
• Image Compression via SVD:
  • Application of Singular Value Decomposition (SVD) to compress images by retaining only principal singular values.
• Image Filtering (FFT):
  • 2D Fourier Transform for low-pass and high-pass filtering of images in the frequency domain.
• Linear Algebra Solvers:
  • Direct Methods: Gaussian Elimination and LU Decomposition (Doolittle’s Method).
  • Iterative Methods: Jacobi Method for solving large linear systems $Ax=b$.
  • Eigenvalue Solvers: QR Decomposition algorithm.

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🛠️ Tech Stack

• Language: Python 3.x
• Core Libraries: NumPy (Vectorized computation), Matplotlib (Visualization), SciPy (Scientific tools).
• Techniques: Monte Carlo, Finite Difference (FDM), Spectral Methods, Runge-Kutta, SVD.

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🚀 How to Run

1. Clone the repository:

   git clone https://github.com/VarunNair19/Computational-Physics-Simulations.git