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PHY 289.04 Methods of Computational Science

0. Introduction -- How does computational science help us?
(3 lectures and 1 Lab)

0.1 What is computational science?
(Lecture#1 (HM): 1/18)

  • Scientific approach combining human intellect and computer technology

  • How do we achieve a productive interaction between human thinking and computers?

  • Keep in mind what computers can do and what they can't do.

  • You are the boss!

  • You must decide what you want a computer to do.

  • You must assess and interpret scientific significance of results you obtain.

  • Human scientific insight and imagination is the most important ingredient in computational science, just as in other scientific approaches (theoretical and experimental).

0.2 Impetus comes from the recent rapid progress in computer technology
(Lecture#1)

  • What can computers offer?

    • Large memory storage capacity

      Fast operation speeds

    • Sequential (or sometimes parallel) processing of sometimes repetitive well-defined step-by-step operations --> Algorithms --> Computer programs

  • What can't computers offer?

    • Numbers with an infinite # of digits

      • No rational numbers w/ an infinite # of digits

      • No exact irrational numbers

      • All the numbers on computers are either real numbers with finite # of digits or complex numbers whose real and imaginary parts are such real numbers.

      • Numbers w/ an infinite # of digits must be approximated w/ numbers with a finite # of digits. --> Round-off errors!

    • Operations involving an infinite # of steps

      • Continuous analysis (tools from calculus)

      • No differentiation and integration

      • Differentiation --> Difference or subtraction

      • Integration --> Summation

      • No limiting process --> Approximate with a finite # of approaching steps.

      • No infinite iterative procedures --> Truncate the procedures.

      • Symbolic computer mathematics (Mathematica, MACSYMA, and Reduce, etc.) can handle some of these operations now, though.

    • Human mental (conscious and subconscious) capacities

      • Scientific insights and intuition --> No moments of Eureka! or Aha!

      • Synthetic or global thinking

      • Making connections between seemingly different and remote notions and concepts.

      • Imaginative and creative thinking (i.e., coming up with new ideas)

0.3 Desire and needs for attacking previously intractable complex problems
(Lecture#2 (HM): 1/20)

    Complex problems:

  • With:

    • A few variables but with complex time evolution
      (e.g., chaotic dynamical systems)

    • Many (up to a few million?) variables and complex interactions or inter- relationships among these variables
      (e.g., complex mechanical systems such as automobiles and satellites; fluid motions w/ or w/o turbulence; chemical reactions w/ catalytic feedback; weather forecasting; microscopic motions of electrons, atoms, ions, and molecules in many-body systems such as gases, plasmas, liquids, glasses, solids, and macromolecules such as polymers, DNAs, and proteins, etc.)

  • Without:

    • Exact analytical solutions

    • Reliable, well-controlled, and systematic approximate analytical solutions

0.4 Goals and strategy in computational science (Lecture#2)

  • Numerical results:

    • Numerical calculus

      • Numerical integration and differentiation of functions (of a single or many variables) and data

      • Numerical solution of ordinary and partial differential equations

    • Numerical linear algebra

      • Numerical solution of a set of linear equations

      • Numerical diagonalization of symmetric or hermitian matrices

    • Numerical solution of a set of non-linear transcendental equations

    • Optimization problems
      (e.g., circuit designs for LSI (Large Scale Integrated circuit), VLSI, ULSI, etc: scheduling problems for air traffic, etc.)

    • Linear programming problems

  • Simulations:

    • Modeling of real systems

    • Based on detailed information (e.g., dynamics) at the level of constituent units (e.g., small parts of machines, atoms in a gas, etc.).

    • They allow for fine and close control of system parameters.

    • Answering what-if questions.

      • Setting system parameters to values unavailable or inaccessible in labs.

    • Visualization of system behaviors

    • Careful selection of quantities to be observed and analyzed

    • Comparison w/ analytical results for limiting cases.

  • Symbolic computer mathematics:

    • Symbolic manipulations of mathematical operations to get exact results

0.5 Programming versus canned (commercially available) software
(Lecture#2)

    You must make a sensible choice between using a canned software and writing your own program based on your goal and available resources including money, your time and efforts. Ask yourself: "Is it worth writing a program or spending money?"

  • Canned software:

    For well-established routines:

    • Numerical integration (Numerical Recipes, Mathematica, Matlab, etc.)

    • Numerical solution of a single equation or a set of several equations
      (Numerical Recipes, Mathematica, Matlab, etc.)

    • Numerical linear algebra

      • Numerical inversion of matrices

      • Numerical estimates of determinants

      • Diagonalization of symmetric and hermitian matrices and eigenvalue problems (LINPACK, EISPACK, IMSL)

    • Fourier analysis (Mathematica, Matlab, etc.)

    • Data analysis (Kaleidagraph)

    • Simple and quick graphing (Mathematica, Matlab, Kaleidagraph)

    Quick and dirty estimates

    Symbolic computer mathematics (Mathematica, MACSYMA, REDUCE)

    Even when you use a canned software, it's much safer to know a basic idea and well-known advantages and disadvantages of the algorithm used by the particular software so that you can make an intelligent judgment on the accuracy and reliability of the results the canned software produces.

  • Programming:

    No canned software is available, because:

    • Your problem calls for a special (not general) treatment.

    • Your problem requires a new algorithm recently developed in your field

    • You want to try a new algorithm you've just come up with.

    • You just happen not to have an access to an appropriate canned software.

    You need a precise result and you cannot trust canned routines not specifically designed for your problem.

    You know that your problem has delicate or special features for which canned software may not work.
    (e.g., a numerical integration of rapidly varying functions or functions with singularities.)

    You then tailor-make a program according to the particular nature of your problem.

    Sometimes your problem forces or prompts you to invent a new algorithm.

    • Development of new algorithms occupies a significant place in the discipline of computational science.

0.6 Programming language
(Lecture#3 (HM): 1/23)

  • We use FORTRAN in this course because

      (i) it is widely used in scientific and engineering communities

      (ii) it is fairly straightforward to translate a FORTRAN program into another in other languages such as PASCAL.

  • Become proficient in at least one programming language!

0.7 Structured programming (more on this in I) (Lecture#3)

  • Top-down hierarchical block structure of a program: a single task per block.

  • General recyclable subroutines.

  • Human errors and inflexible computers.

    • Don't take personally what a computer does.

  • Benefits:

    • To reduce bugs (human errors!) in programs.

    • To facilitate detection and correction of bugs.

0.8 How do we assess and trust computational results? (Lecture#3)

  • Collect as much background information related to your problem as possible.

    • Do some literature search and reading up on general background.

    • Try to find mathematically rigorous or exact results related to your problem.
      (e.g., existence and uniqueness of solutions for ordinary differential equations)

  • Try to get some analytical handle on your problem.

    • Derive analytical results for various limiting and analytically tractable cases.
      (e.g., linearize your problem)

    • Apply known approximate methods:

      • Expansion techniques such as perturbation theory

      • Variational methods

  • Use effective visualization of your results

    • Effective and meaningful graphs

    • You must know what you are trying to show with your computational results.

    • Animation

  • Scientific intuition and insights

  • Healthy skepticism

    • Try to estimate errors involved in your computation.

    In computational science, you must have not only an attitude of an experimentalist to carefully handle computational procedures and to analyze your results but also an attitude of a theorist to carefully handle analytical and numerical calculations and to draw a logical and intelligent theoretical conclusion.

Lab#1: 0.9 Introduction to our computer system and some software
(assigned on 1/23: due on 1/30)

  • How to use "NCSA telnet" to log on "entropy" (IBM RS6000).

  • How to create and edit a program using an editor "Jove".

  • How to execute a FORTRAN program.

  • How to transfer a file to a Macintosh using "ftp".

  • How to use Kaleidagraph on a Macintosh.

  • How to use Mathematica.

  • How to send and to receive e-mail using "Eudora".

I. Nonlinear dynamical systems in chemistry and physics: an introduction to chaotic motion
(20 lectures, 5 Labs, 1 hour exam)

How do we study dynamical systems?

    I.1 What are dynamical systems? (Lecture#4 (JS): 1/25)

  • Systems evolving (changing) in time:

    Physics: Newtonian dynamics

      (i) Second-order ordinary differential equations (ODEs)

      (ii) First-order ODEs

    Chemistry: chemical kinetics

      First-order ODEs

    Biology: population dynamics

      First-order ODEs

  • A dynamical system --> A set of ODEs!

I.2 Phase space (Lecture#4)

Solutions for ODEs?

    Phase space?

    • A phase space point --> a dynamical state

    A phase (space) trajectory

    • Watch how your system evolves in the phase space.

    • The existence and uniqueness of a solution for ODEs

      • There always exists a unique trajectory starting from an arbitrary phase space point.

      • The noncrossing property of trajectories

    Examples:

    Plane pendulum

    • The equation of motion

    • Angular velocity

    • Kinetic and potential energy

    Linear pendulum

    • The equation of motion and its general solution

    • Phase space trajectories

    Non-linear pendulum

    • Librational and rotational motion and separatrix in the phase space

    • In which direction does time flow along a trajectory?

    • Connect a motion along a phase trajectory to a motion in real space.

I.3 Plane pendulum and 1-D projectile motion (Lecture #5 (HM): 1/27)

    Newtonian dynamics in 1 dimension

    • Newton's 2nd law of motion

      • Newton's equation of motion (a 2nd-order ODE)

      • Hamilton's equations of motion (a pair of 1st-order ODEs)

    • Phase space or the x-v plane

      • Phase trajectory

    • Conservative forces

      • Force derived from a potential energy

      • Example: a gravitational force on a body at height x

    • Total energy conservation for a body subject to a conservative force

      • Constant-energy curves in the phase space

      • --> contours for the total energy landscape

    • Phase trajectories

      • Following constant-energy contours

      • No crossing

      • Which way does time run?

    Plane pendulum

    • Hamilton's equations of motion

    • Dimensionless equations of motion

    • The total energy in dimensionless form

    One-dimensional projectile motion

    • Hamilton's equations of motion

    • Dimensionless equations of motion

Lab#2: Plane pendulum and 1-D projectile motion by Mathematica
(assigned on 1/27: due on 2/6)

    Linear pendulum (analytical homework)

    • Phase trajectory

    • Total energy conservation

    Plane pendulum (Mathematica)

    • Numerical solution: librational and rotational motion

    • Phase trajectories in the phase space

    • Energy contours in the phase space

    • The energy landscape

    1-D projectile motion

    • Analytical solution

    • Total energy and the energy landscape

    • Phase trajectories in the phase space

I.4 How to numerically solve a set of first-order ODEs?
(Lecture#6 (HM):1/30)

    Basic strategy:

    • Discretization of time --> discretization of time derivatives

    Types of errors

    • Round-off errors

      • Due to the finite # of digits used to represent a number on computers

    • Truncation errors

      • Due to a particular algorithm for approximating a mathematical expression

      • Due to discretization of time derivatives

    Main criteria for accuracy of numerical solution

    • Local truncation error: errors created in each time step

    • Global truncation error: errors accumulated after many time steps

    Criteria for an optimal algorithm

    • Small well-controlled local truncation error

    • Small well-controlled global truncation error

        --> Convergence of numerical solutions

    • Stability of numerical solutions

      • No divergence of a numerical solution away from the exact solution as time goes to infinity.

    • Fast execution time

    • Simplicity of an algorithm

      • Easy debugging

      • Easy to modify

    • Portability: recyclable subroutines

    You must weigh your priorities, available resources (money, time, and efforts), and future plans (are you going to need similar algorithms later in solving different but similar problems?). --> An optimization problem!

I.5 Types of ODE algorithms (Lecture#7 (JS): 2/1)

    Derivatives in calculus and on computers

    • Taylor series and discretization error

    Dynamical systems: a set of 1st order ODEs

    Numerical solution of 1st order ODEs

    • Strategy: discretize time

    Simplest approximation for dx/dt: the Euler method

    • Local truncation error

    • Global truncation error

    • Example: a plane pendulum

    Numerical algorithms for ODEs

      I. Single step methods

        A. The Euler method

        B. The Runge-Kutta methods

          1. Second-order Runge Kutta method

          2. Fourth-order Runge Kutta method

      II. Linear Multistep Methods

      Explicit and implicit methods

        A. The predictor-corrector method

        Combining an explicit and implicit methods.

        B. The Burlisch-Stoer method

        Extrapolation with respect to time step size

I.6 The Euler method (Lecture#8 (JS): 2/3)

    Programming the Euler method

    • Implementation as a subroutine

    Solving the linear pendulum using the Euler method

    Solving the nonlinear pendulum using the Euler method

    More programming tips

    • The global truncation error for the Euler method

I.7 More programming tips (Lecture#8)

    Lab#3: Programming the Euler method (assigned on 2/3: due on 2/10)

    • Creating a main program to simulate pendulums

    • Creating a subroutine for the Euler method

    Solving the linear pendulum

    • Compare numerical results with analytical solutions

      • Time series starting w/ various initial conditions

      • Phase space plots for various initial conditions

      • Absolute error and time steps

    Solving the non-linear pendulum

    • Compare numerical results with the solutions by Mathematica in Lab#2

      • Time series starting w/ various initial conditions

      • Phase space plots for various initial conditions and total energies

    • We need a better algorithm!

I.8 The Runge-Kutta (R-K) methods

    The second-order R-K method (Lecture#9 (JS): 2/6)

    • Basic idea: controlling local discretization (truncation) errors

    • Derivation

    • Comparison with the Euler method

    • Local and global truncation error

    The fourth-order R-K method (Lecture#10 (JS): 2/8)

    • Systematic derivation of local truncation error

      • Double Taylor series expansion

      • Local truncation error in the second-order R-K method

    • Algorithm and local truncation error in the fourth-order R-K method

    Adaptive step size control (Lecture#10)

    • To get the most out of the fourth-order R-K method.

I.9 Dimensional analysis and scaling of dynamical variables
(Lecture#11 (HM): 2/10)

    Procedures

    • Dynamical variables in dimensionless forms

    • Time evolution equations in dimensionless form

    Benefits

    • Simplified equations

    • Numerical solution ~ O(1)

    • Scaling of dynamical variables

    • Scaling forms for solution

I.10 Conservative and dissipative systems (Lecture#11)

    Conservative systems?

    • Force derived from a potential energy function

    • Conservation of the total energy

    Dissipative systems?

    • Force that cannot be derived from a potential energy function

    • No conservation of the total energy

    How do we identify them?

    • Physical criterion:

      • Preserved phase space volume --> a conservative system

      • Contracting phase space volume --> a dissipative system

    • Mathematical criterion:

      • Rewrite the first-order ODEs as first-order autonomous ODEs.

    Examples:

    • Linear pendulums

    • Damped linear pendulums

    • Damped driven plane pendulums

I.11 Damped pendulums and Fixed points in the phase space
(Lecture#12 (JS): 2/13)

    Damped pendulums

    • Newton's equation of motion

    • Dimensionless variables and dimensionless Hamilton's equations

    Fixed points in the phase space

    • Stationary points

    • Examples:

      (a) Undamped nonlinear pendulums

        Elliptic points

        Hyperbolic points

      (b) Damped pendulums

        Stable focus

        Hyperbolic points

Lab#4: The fourth-order Runge-Kutta method: application to the pendulum
(assigned on 2/13: due on 2/22)

    Homework on conservative and dissipative systems

    • The damped pendulum

    • The Henon and Heiles model

    The Runge-Kutta method with adaptive step size control

    • Modifying the main program for pendulums to include the adaptive step size control

    • Creating a subroutine for the fourth-order R-K method

    • Adding adaptive step size control

    Simulating the plane pendulum without adaptive step size control

    • Compare the phase space plot with that obtained by the Euler method.

    • Compare the energy conservation in the R-K method with that in the Euler method.

    Simulating the plane pendulum with adaptive step size control

    • Phase space plots

    Simulating the damped pendulum

    • Modifying the program

    • Comparing damped and undamped pendulum trajectories

    • Exploring the phase space of the damped pendulum

I.13 Damped driven pendulums (Lecture#13 (JS): 2/15)

    Newton's equation of motion with a driving force

    Extended phase space and equations of motion in autonomous form

    Different types of motion as the driving amplitude is increased

    • Limit cycles

    • Rotational motion

    • Rotational chaotic motion

    • Focal points (point attractors)

    Fixed points related to extrema in the potential energy function

    • Elliptic points with minima

    • Hyperbolic points with maxima

I.14 Limit cycles and chaotic orbits: Poincare surface of section
(Lecture#14 (JS): 2/17)

    Limit cycle

    • Stable and unstable limit cycles

    Poincare surface of section

    • Limit cycles and chaotic attractors

    Chaotic orbits

    • Sensitive dependence on initial conditions

I.15 Chaotic trajectories (Lecture#15 (JS): 2/20)

    Review: conservative and dissipative systems

    • Nonlinear pendulums and damped pendulums

    Properties of chaotic trajectories

    • Sensitive dependence on initial conditions

    • Chaotic attractor in the Poincare surface of section

      • Self-similar structures --> fractals

I.16 Attractors (Lecture#16 (JS): 2/22)

    Point attractors

    Stable Limit cycles

    Chaotic (strange) attractors

I.17 Fractals (Lecture#16)

    Fractal dimension

    Cantor set

    Koch snowflakes

I.18 What is chaos? (Lecture#16)

  • Aperiodic long-term behaviors

  • Deterministic systems

  • Sensitive dependence on initial conditions

I.19 Review for Hour exam#1 (Lecture#16)

Hour exam#1 on Part I.1 - I.19 (2/24)

I.20 Review on Hour exam#1 (Lecture#17 (JS): 2/27)

I.21 Period doubling route to chaos (Lecture#17)

  • Period doubling as the driving amplitude in the damped driven pendulum is increased

Lab#5: The damped driven pendulum and the stroboscopic surface of section method
(assigned on 2/27: due on 3/8)

    Programming the stroboscopic method for the damped driven pendulum

    Testing the stroboscopic method for the damped driven pendulum

    • Limit cycles

    • Chaotic attractor

    Exploring the dynamics of the damped driven pendulum

    • Limit cycles to chaotic attractors

    • Period doubling

I.21 Application 1: chemical kinetics (Lecture#17)

    Kinetics

    • How do concentrations of chemical species change in time?

    General chemical reaction

    • Reaction rate

    Rate law for a chemical reaction

I.22 Application 1: chemical kinetics (continued) (Lecture#18 (JS): 3/1)

    Comments on the rate law

    Elementary reactions

    • Unimolecular reactions

    • Bimolecular collisions

    Mechanisms for chemical reactions

    • Consecutive reactions

      • Fixed points (chemical equilibrium)

I.23 Application 1: chemical kinetics (continued) (Lecture#19 (JS): 3/3)

    Autocatalysis

    • Mechanism of cubic autocatalysis

    Cubic autocatalator

    • Mechanism

    • Dimensionless formalism

    • Fixed points and limit cycles (chemical oscillations)

I.24 Application 1: chemical kinetics (continued) (Lecture#20 (JS): 3/6)

    Autocatalator (continued)

    • Comments

    • Non-isothermal autocatalator

      • T-dependence of rate constants

    Oregonator model

I.25 Application 1: chemical kinetics (continued) (Lecture#21 (JS): 3/8)

  • A class demonstration of oscillating chemical reactions.

I.26 Application 2: molecular vibrations (Lecture#21)

    Triatomic (ABA) molecules

    • Bond displacement coordinates

    Potential energy

    • Morse potential

    Hamilton's equations of motion

    • Hamiltonian

    Equations of motion for an ABA triatomic molecule

I.27 Application 2: molecular vibrations (continued)
(Lecture#22 (JS): 3/10)

    Dimensionless variables and equations of motion

    • Dimensionless energy

    Initial conditions

    Coordinate space projection

    Example: H2O

    • Local modes

    • Normal modes

    • Bond energies

    Laser control of chemical reactions

SPRING BREAK (3/11 - 3/19)

I.28 REVIEW ON PART I (Lecture#23 (JS): 3/20)

    Dynamical systems with a few degrees of freedom in PART I

      A. Dissipative systems

        (a) Damped pendulums

        (b) Damped driven pendulums

        (c) Nonlinear chemical kinetic equations: the C-feedback system

      B. Conservative systems

        (a) Linear and non-linear pendulums

        (b) Molecular vibrations in an ABA triatomic molecule

    Main computational methods in PART I

    • The Euler method

    • The fourth-order Runge-Kutta method

    • Adaptive step size control

    • The stroboscopic Poincare surface of section method

Lab#6: Chemical kinetics and molecular vibrations
(assigned on 3/20: due on 3/29)

    PART A. Chemical kinetics

      C-feedback system or a 3-variable autocatalator

      • Rate equations

      • Dimensionless variables

      The adaptive step size R-K program for the C-feedback system

      Numerical investigations of the C-feedback system

      • Limit cycles

      • Period doubling

      • Chaotic motion and a strange attractor

    PART B. Molecular vibrations

      The fixed step size R-K program for an ABA triatomic molecule

      Investigation of the vibrational dynamics of water

      • Local and normal modes

      Investigation of the vibrational dynamics of different molecules

      • Local modes to chaotic motion

      • Stable normal modes

II. Molecular dynamics (MD) simulations of two-dimensional argon cluster
(12 lectures, 3 labs, and 1 hour exam)

    How do we study the thermodynamics of many-body dynamical systems?

II.1 Molecular dynamics (MD) simulations
(Lecture#24 (HM): 3/22)

    Simulating time evolution of many-body systems

    • A large number of interacting particles

    Examples

    • Condensed matter (solid and liquids)

    • Macromolecules

    • Clusters

    • Galaxies, etc.

    Few-body vs. many-body

    Few: behaviors of each individual degree of freedom

    • Many: overall, collective, macroscopic behaviors

    • Thermodynamics with average quantities

    Microscopic atomic motion

    • Length scale ~1A = 10-10 m

    • Time scale ~ 10-12 sec

    Macroscopic thermodynamics

    • Length scale ~ 1 m

    • Time scale ~ 1 sec

    Computer memory and speed

    • N < 106

    • Time step < 106 --> 10-8 sec in real time if each time step = 10-14 sec

    • A MD simulation is still looking at short length and time scale phenomena.

    Time evolution by solving equations of motion

    • Classical MD -- Newton's equations of motion

      • Inert gases, ionic and covalent systems, etc.

      • Electrons are localized around nuclei.

    • Quantum MD -- Schroedinger equations

      • Metals / semiconductors

      • Ions (classically) and electrons (quantum mechanically)

      • Light atoms: H, He.

      • Quarks and gluons, etc.

II.2 MD simulations on a two-dimensional Ar clusters
(Lecture#25 (HM): 3/24)

    Newton's equations of motion

    Verlet's algorithm

    • Discretizing Newton's equation of motion

      • No force --> inertial motion

      • Force and inertia --> accelerated motion

    • Velocity

    Thermodynamics

    • Total energy --> internal energy

    • Total kinetic energy --> temperature

    • Internal energy vs. temperature --> an equation of state

    Atomic structures

    • Low T: solid with an ordered lattice structure

    • High T: liquid with disordered structure

    • Pair distribution functions

    Atomic motion

    • Low T: thermal vibrations

    • High T: migration (diffusion)

    • Atomic trajectories

    • Mean square displacements

II.3 Quantities to be computed and analyzed (Lecture#26 (HM): 3/27)

    Quantities computed during a simulation

    • Atomic positions --> interatomic forces and the total potential energy

    • Velocities --> the total kinetic energy --> temperature

    • The total energy: almost constant

    Data analysis

    • Time averaged total energy and temperature --> the equation of state

    • Atomic trajectories

    • Pair distribution functions

    • Mean square displacements

II.4 Derivation of Verlet's algorithm (Lecture#26)

    Derivation by Taylor series

    Local and global truncation error

    • Better than Euler but worse than 4-th order R-K

    Advantage

    • Fewer evaluations of forces: better than 4-th order R-K

II.5 Interatomic pair potential (Lecture#27 (HM): 3/29)

    The total potential energy as a sum of pair potentials

    The Lennard-Jones 6-12 potential for argon

    • Short-range repulsive force: Pauli's exclusion principle

    • Long-range attractive force: dipole-dipole interactions

II.6 Dimensionless variables (Lecture#27)

  • Atomic positions

  • Pair potential

II.7 Interatomic forces (Lecture#27)

    Hamilton's equations and interatomic forces

    • 2-body forces due to the pair potential

    • Dimensionless forces

    • Net forces: Newton's 3rd law of motion

Lab#7: Molecular dynamics (MD) simulations on a 2-dimensional Ar cluster
(assigned on 3/29: due on 4/7)

    Initial data

    • Create the initial data program

    MD program

    • Create the MD program

    • A test run

II.8 How to control truncation error (Lecture#28 (HM): 3/31)

    Remedy

    • Displacement vectors

    • Revised Verlet's algorithm

II.9 Dimensionless variables (Lecture#28)

  • Dimensionless atomic positions

  • The dimensionless Lennard-Jones potential

  • Dimensionless 2-body forces

II.10 Interatomic forces revisited (Lecture#28 & Lecture#29 (HM): 4/3)

    Physical meaning of 2-body forces

    • Short-range repulsive force

    • Long-range attractive force

II.11 Verlet's algorithm revisited (Lecture#29 & Lecture#30 (HM):4/5)

    Dimensionless time evolution equations

    • Updating the displacement vectors

    • Atomic positions from the displacement vectors

    Dimensionless variables

    • Velocity vectors from the displacement vectors

    • The dimensionless total kinetic energy

    • The dimensionless instantaneous kinetic temperature

    • The dimensionless total energy

    Time step

    • Shorter than a typical vibrational period

    Initial conditions

    • Initial positions and displacements

    • Vanishing initial displacement vectors

    • Vanishing total linear and angular momenta

II.12 MD for molecular systems (Lecture 31 (JS): 4/7)

    Intermolecular and intramolecular potential energy functions

    • Bond stretching and bending potentials

    • Torsional potentials

    • van der Waals interactions

    • Electrostatic terms

    • Hydrogen bonds

    Applications

    • Conformational energy differences

    • Solvent effects

    • Macromolecular simulations

Lab#8: Data analysis -- <E> vs. <T> and atomic trajectories
(assigned on 4/7: due on 4/14)

    Production runs

    • At 8 different values of the total energy

    The total energy as a function of time steps (Kaleidagraph)

    Time average program

    • Time averages of the total energy and temperature

    The equation of state (Kaleidagraph)

    • <E> vs. <T>

    Atomic trajectory program (NCAR graphics)

II.13 Thermodynamic quantities (Lecture#32 (HM): 4/10)

    Total energy: almost constant

    Kinetic and potential energies

    • Approach toward thermal equilibrium: transient and equilibrium time intervals

    Macroscopic quantities as time averages

    The equipartition theorem

    • Time averaged temperature and time averaged kinetic energy

    How to control the total energy

II.14 Atomic structure and motion (Lecture#33 (HM): 4/12)

    Pair distribution functions

    • Definition

    • Atomic structures at low and high temperatures

    Mean square displacements

    • Definition

    • Atomic motion at low and high temperatures

Lab#9: Data analysis -- atomic structure and motion
(assigned on 4/14: due on 4/21)

    Pair distribution function program

    Mean square displacement program

    Animation of atomic motion (Mathematica)

II.15 Review for Hour exam#2 (Lecture#33 (HM & JS))

Hour exam#2 on Part I.21-I.27 and Part II (4/14)

II.16 Review on Hour exam#2 (Lecture#34 (HM & JS): 4/17)

II.17 REVIEW ON PART II (Lecture#35 (HM): 4/19)

    Various types of MD

      A. Microcanonical MD

      Closed system with constant total energy

      Types of boundary conditions (BC)

        (i) Free BC

        (ii) Box BC

        (iii) Periodic BC

      B. Canonical MD with fixed volume

        System in contact with a heat bath at constant temperature

      C. Canonical MD with constant pressure

III. Monte Carlo (MC) simulations of phase transitions
(8 lectures and 1 lab)

    How do we study phase transitions in many-body systems?

III.1 Stochastic simulations (Lecture#35)

    Deterministic simulations

    • Few-body: numerical solution of ODEs with the Runge-Kutta method

    • Many-body: MD with Verlet's algorithm

    Stochastic simulations

    • Updating a system's state with probability

    • Example: a random walk by tossing a coin

III.2 Two-dimensional Ising model (Lecture#35)

    Two-dimensional Ising model on a square lattice

    • A spin on a lattice site

    • 2 states for a spin

    • A state for the entire system: a spin configuration

    • Total energy for each spin configuration

III.3 Ferromagnetic interactions (Lecture#36 (HM): 4/21)

    No kinetic energy in the total energy

    Spins as magnetic moments

    • Favoring a parallel spin pair

    • Magnetization

    • Doubly degenerate states at T = 0

III.4 Temperature control (Lecture#36)

  • System in contact with a heat reservoir at constant temperature

III.5 Time evolution of the system (Lecture#36)

    Time averages

III.6 The Boltzmann factor (Lecture#36)

    Probability for the system to be in a state with energy E

    • States with lower energy are favored.

    • At T = 0: the lowest energy state has probability one

    • As T is increased, more and more high E states become probable.

III.7 Replacing time averages with expectation values
(Lecture#37 (HM): 4/24)

    The partition function

III.8 Phase transitions and critical phenomena in the Ising model
(Lecture#37)

    What do we already know about the 2-D Ising model analytically?

    Spontaneous magnetization as a function of temperature

    • The low temperature ordered phase

    • The high temperature disordered phase

    Total energy as a function of temperature

    Divergence in the specific heat at the critical temperature

    Evolution of spin domain patterns and critical fluctuations

III.9 Monte Carlo simulations (Lecture#38 (HM): 4/26)

    Goal:

    • Create a sequence of states whose histogram is proportional to the Boltzmann factor.

    Markov process

    • A single step process

    • The accessibility condition

    • The detailed balance

III.10 The heat bath algorithm (Lecture#38 & Lecture#39 (HM): 4/28)

    Procedure

    Flow chart

    Dimensionless scheme

Lab#10: Monte Carlo (MC) simulations on a 2-dimensional Ising model
(assigned on 4/26: due on 5/5)

    The MC program

    • The heat bath algorithm

    Production runs

    • At 6 different values of temperature

    The total energy per spin as a function of MC sweeps

    The magnetization per spin as a function of MC sweeps

    The time average program

    • Times averages for the total energy per spin, magnetization per spin, and the heat capacity per spin.

    The average total energy per spin, the average magnetization per spin, and the average heat capacity per spin as functions of temperature.

III.11 Quantities to be calculated (Lecture#40 & 41 (HM): 5/1 & 5/3)

    How do we calculate the following quantities in MC simulations

    • The total energy per spin as a function of temperature

    • The spontaneous magnetization per spin as a function of temperature

    • The specific heat as a function of temperature

    • The spin domain patterns as a function of temperature

    Specific heat and fluctuations in the total energy

    • The specific heat as a measure for energy fluctuations

    The spin domain patterns and fluctuations in magnetization

III.12 The heat bath algorithm revisited (Lecture#42 (HM): 5/5)

  • The accessibility condition

  • The detailed balance

Review for the Final exam (Lecture#42 (JS & HM): 5/5)

Final exam on Part I, II, and III (5/10)


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