THE OBJECTIVE is to learn and apply fundamental techniques used in (primarily classical) simulations in order to help understand and predict properties of microscopic systems in materials science, physics, chemistry, and biology.
- THE EMPHASIS will be on connections between the simulation results and real properties of materials (structural or thermodynamic quantities), as well as numerical algorithms and systematic and statistical error estimations
- FOR WHOM? This class is oriented for the first-year graduate or advanced undergraduate. It connects atomistics to observable, rather than investigates, e.g., cellular automata type approaches, and introduces all necessary concepts. A course project is required, rather than a final exam (see Teams and Projects in navigator bar).
- Methods and Applications:
- Molecular Dynamics: integration algorithms, static and dynamic correlations functions and their connection to order and transport
- Monte Carlo and Random Walks: variance reduction, Metropolis algorithms, Kinetic Monte Carlo, heat diffusion, Brownian motion, etc
- Phase Transitions: melting-freezing, calculating free energies
- Polymers: growth and equilibrium structure
- Quantum Simulation: zero temperature and finite temperature methods
- Optimization techniques such as simulated annealing
Researchers should cite this work as follows:
|Lecture Number/Topic||Online Lecture||Video||Lecture Notes||Supplemental Material||Suggested Exercises|
|Illinois PHYS 466, Lecture 1: Introduction||View Flash|
|Introduction to Simulation
Why do simulations?
Two Simulation Modes
Challenges of Simulation: Physical and mathematical underpinnings
|Illinois PHYS 466, Lecture 3: Basics of Statistical Mechanics||View Flash|
|Basics of Statistical Mechanics
Review of ensembles
Microcanonical, canonical, Maxwell-Boltzmann
Constant pressure, temperature, volume,…
|Illinois PHYS 466, Lecture 4: Molecular Dynamics||View Flash|
What to choose in an integrator
The Verlet algorithm
Boundary Conditions in Space and time
Reading assignment: Frenkel and Smit Chapter 4
|Illinois PHYS 466, Lecture 5: Interatomic Potentials||View Flash|
Before we can start a simulation, we need the model!
Interaction between atoms and molecules is determined by quantum mechanics
But we don’t know …
|Illinois PHYS 466, Lecture 6: Scalar Properties and Static Correlations||View Flash|
|Scalar Properties, Static Correlations and Order Parameters
What do we get out of a simulation?
Static properties: pressure, specific heat, etc.
Pair correlations in real space and …
|Illinois PHYS 466, Lecture 7: Dynamical Correlations & Transport Coefficients||View Flash|
|Dynamical correlations and transport coefficients
Dynamics is why we do molecular dynamics!
Diffusion constants, velocity-velocity auto correlation …
|Illinois PHYS 466, Lecture 8: Temperature and Pressure Controls||View Flash|
|Temperature and Pressure Controls
Constant Temperature MD
Brownian dynamics/Anderson thermostat
Nose-Hoover thermostat (FS 6.1.2)
|Illinois PHYS 466, Lecture 9: Probability tools & Random number generators||View Flash|
|Random Number Generation (RNG)
read “Numerical Recipes” on random numbers and chi-squared test
Today we discuss how to generate and test random numbers.
What is a random number?
A single …
|Illinois PHYS 466, Lecture 10: Sampling||View Flash|
|Fundamentals of Monte Carlo
What is Monte Carlo?
Named at Los Alamos in 1940’s after the casino.
Any method which uses (pseudo)random numbers> as an essential part of the algorithm.
|Illinois PHYS 466, Lecture 11: Importance Sampling||View Flash|
Today We will talk about the third option: Importance sampling and correlated sampling
Finding Optimal p*(x) for Sampling
Example of …
|Illinois PHYS 466, Lecture 12: Random Walks||View Flash|
Today we will discuss Markov chains (random walks), detailed balance and transition rules.
These methods were introduced by Metropolis et al. in 1953
who applied it to a hard …
|Illinois PHYS 466, Lecture 13: Brownian Dynamics||View Flash|
Let’s explore the connection between Brownian motion and Metropolis Monte Carlo. Why?
Connection with smart MC
Introduce the idea of kinetic Monte Carlo
Get rid of …
|Illinois PHYS 466, Lecture 14: Neighbor Tables, Long-Range Potentials, Ewald Sums||View Flash|
|Illinois PHYS 466, Lecture 15: Constraints||View Flash|
|Illinois PHYS 466, Lecture 16: Free Energies from Simulations||View Flash|
|Illinois PHYS 466, Lecture 17: Simulation of Polymers||View Flash|
|Illinois PHYS 466, Lecture 18: Kinetic Monte Carlo (KMC)||View Flash|
|Illinois PHYS 466, Lecture 19: The Ising Model||View Flash|