2D Molecular Dynamics Simulation (Lennard–Jones Fluid)

This repository contains a 2D molecular dynamics (MD) simulation of interacting particles using a Lennard–Jones (12–6) potential with periodic boundary conditions.

The project includes:

• A C++ implementation used for numerical experiments and data generation
• A Python implementation that reproduces the same physics and provides a real-time animation with energy and temperature plots

The simulation models a classical Lennard–Jones fluid and demonstrates thermostatting, energy conservation, velocity distributions, and particle trajectories.

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Physical Model

Particles

• Identical particles with mass ( m = 1 )
• Motion restricted to two spatial dimensions
• Periodic boundary conditions in a square simulation box

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⚛️ Interaction Potential

The particles in this simulation interact via a shifted Lennard-Jones potential. To ensure numerical stability and prevent discontinuities, both the potential and its derivative are shifted to zero at the cutoff radius $r_c$.

The core potential is defined as:

$$ U(r) = 4 \left[ \left( \frac{1}{r} \right)^{12} - \left( \frac{1}{r} \right)^6 \right] $$

To guarantee the continuity of both energy and force, the implementation follows these conditions at the cutoff $r_c$:

• Zero Potential: $U(r_c) = 0$
• Zero Force: $\frac{dU}{dr} \Big|_{r=r_c} = 0$

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Numerical Method

Time Integration

• Velocity Verlet–type integration scheme
• Includes a Nosé–Hoover–style thermostat
• A thermostat variable dynamically adjusts the kinetic energy to control temperature

Thermostat Parameters

• Q: thermostat mass (controls coupling strength)
• T: target temperature (constant or time-dependent)

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Repository Structure

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C++ Implementation

Features

• Square-lattice initialization
• Maxwell–Boltzmann velocity distribution
• Removal of center-of-mass velocity
• Periodic boundary conditions (minimum image convention)
• Lennard–Jones forces with cutoff shifting
• Energy tracking (kinetic, potential, total)
• Velocity histograms
• Single-particle trajectories
• Temperature ramps
• Output to text files for analysis