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P131-Computational Physics
Adrian E. Feiguin
How to read these notes
Ordinary differential equations: a review
Euler's method
Exercise 1.1: Newton's law of cooling
Higher order: Taylor's series
Multistep or Predictor-Corrector methods
Runge-Kutta methods
2nd order Runge-Kutta
4th order Runge-Kutta
Equations of motion (2nd order ODEs)
Exercise 1.2: One dimensional motion
Exercise 1.3: Air resistance
Exercise 1.5: Gravitational force
Exercise 1.6: Harmonic oscillator
Two dimensional trajectories
Exercise 1.7: Trajectory of a shot
Stability
Motion in a central potential
Kepler's problem
Circular motion
Elliptical orbits
Astronomical units
Exercise 2.1: Simulation of the orbit
A mini solar system
Exercise 2.2: A three body problem
Scattering by a central potential
Exercise 2.3
Trajectories in phase space and integrability
Oscillatory Motion
The harmonic oscillator
Exercise 3.1: Energy conservation
Exercise 3.2: Large oscillations
Exercise 3.3: Damped oscillations
Exercise 3.4: Linear response to external forces
Exercise 3.5: Resonance
Chaotic structure in phase space
Molecular Dynamics
The intermolecular potential
Tricks of the trade
Units
Boundary conditions
Starting configuration
Adjusting density and temperature
MD simulation of Hard Spheres
Exercise 4.1: MD for Hard Spheres
MD simulation: Continuous potentials
Exercise 4.2: Approach to equilibrium I
Exercise 4.3: Approach to equilibrium II
Exercise 4.4: Distribution of speeds
Evaluation of observables
Exercise 4.5: Equation of state of a non-ideal gas
Exercise 4.6: Ground state energy
Exercise 4.7: The solid state and melting
Simple transport properties
How much complicated can it get?
Waves!
Coupled oscillators
Exercise 5.1
Exercise 5.2: two coupled oscillators
Exercise 5.3: response to an external force
Exercise 5.4: superposition of motion
Exercise 5.5: three coupled oscillators
Exercise 5.6:
coupled oscillators
Exercise 5.7: propagation speed in a linear chain
Fourier analysis
Exercise 5.8: Fourier analysis
Waves on a string
Numerical solution: finite differences
Exercise 5.9: Finite differences for the wave equation
Exercise 5.10: String with friction
Heat Flow
Finite differences solution
Exercise 6.1: Finite differences program
Exercise 6.2: Two bars in contact
Electrostatic potentials
Units
The finite differences algorithm
Exercise 7.1: Verification
Exercise 7.2: Numerical solution inside a rectangular region
Exercise 7.3: Capacitance of concentric squares
Exercise 7.4: Poisson's equation
Time dependent Schrödinger equation
The time-evolution operator
Exercise 8.1: Single-slit diffraction
Random sequences
Pseudo-random number generators
Testing for randomness and uniformity
Moments
Autocorrelation
Visual test
Statistical errors
Non-uniform random distributions
Exponential distribution
von Neumann rejection
Random walk methods: the Metropolis algorithm
Exercise 9.1: The Gaussian distribution
Monte Carlo integration
Simple Monte Carlo integration
Monte Carlo error analysis
Exercise 10.1: One dimensional integration
Exercise 10.2: Importance of randomness
Variance reduction
Importance Sampling
Exercise 10.3: Importance sampling
Exercise 10.4: The Metropolis algorithm
Random walks
A one-dimensional random walk
Exact enumeration
Monte Carlo
Exercise 11.1: Random walks in 1d
Exercise 11.2: Random walk with steps of variable length
The continuum limit
Two and three dimensional random walks
Exercise 11.3: A random walk in two dimensions
Exercise 11.4: Individual particle diffusion in a lattice gas
Restricted random walks
Self-avoiding walks
Monte Carlo simulation of the SAW in 2d
The traveling salesman
The Microcanonical Ensemble
Enumeration
Monte Carlo simulation
One-dimensional Classical Ideal Gas
Exercise 12.1: MC simulation of the 1D ideal gas
Temperature and the Canonical Ensemble
Exercise 12.2: The Boltzmann distribution
The Ising model
Boundary conditions
Physical quantities
Monte Carlo simulation of the Ising model
Exercise 12.3: MC simulation of the Ising model in 1D
The Canonical Ensemble
The Metropolis algorithm
Exercise 13.1: Classical gas in 1D
Exercise 13.2: One-dimensional Ising model
Simulation of the 2D Ising model
The heat capacity
The magnetic susceptibility
Metropolis algorithm
Boundary conditions
Initial conditions and equilibration
Tricks
Exercise 13.3: Equilibration of the 2D Ising model
Measuring observables
Exercise 13.4: The correlation time
Exercise 13.5: Comparison with exact results
The Ising phase transition
Exercise 13.6: Qualitative behavior of the 2D Ising model
Exercise 13.7: Critical slowing down
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Adrian E. Feiguin 2004-06-01