Applied Physical Simulation

Computational models of gravitation, fluids, electromagnetism, quantum mechanics, and coupled systems.

General idea of the topic

Applied physical simulation translates Newtonian mechanics and thermodynamics into complex many-body systems: real gases, liquids, galaxy mergers, and fluid dynamics. In conservative systems, standard integrators fail due to secular energy drift; modern computational physics relies on symplectic integrators (Verlet, Velocity-Verlet, Leapfrog) that strictly preserve phase-space volume (Liouville's theorem).

This node covers intermolecular force calculations (Lennard-Jones potential), periodic boundary conditions, temperature/pressure thermostats (Berendsen, Nosé-Hoover), and Smoothed Particle Hydrodynamics (SPH).

What you will truly understand

You will learn how microscopic liquid structure is quantitatively extracted via the radial distribution function $g(r)$ computed from simulated particle trajectories.

You will determine macroscopic thermodynamic pressure via the virial theorem and self-diffusion coefficients $D$ via mean squared displacement (MSD) and Einstein's relation.

Recommended learning route

1. Symplectic integrators: Verlet and Velocity-Verlet formulations, energy drift elimination, and phase-space volume conservation.
2. Interatomic potentials and truncation: 6-12 Lennard-Jones potential, cutoff radius $r_c$, and long-range tail corrections.
3. Thermostats and ensembles: velocity rescaling, Nosé-Hoover dynamics, and NVT/NPT ensemble sampling.
4. Macroscopic property extraction: radial distribution function $g(r)$, virial pressure, and molecular diffusion rates.

Physical criterion you must acquire

Distinguishing between exact instantaneous energy conservation and bounded oscillations around a mean value guaranteed by symplectic geometry.

Connection with real problems

• Biophysics: membrane protein stability and ion transport in cellular channels.
• Nanotechnology: thermal transport and mechanical strength in carbon nanotubes and graphene.
• Astrodynamics: spacecraft orbital planning and N-body stellar cluster evolution.

Final goal of the topic

Construct, run, and analyze complete molecular dynamics simulations to extract macroscopic thermodynamic properties from micro-mechanical interactions.