incremental integration of computational physics into traditional undergraduate courses kelly r....

Incremental Integration of Computational Physics into Traditional Undergraduate Courses Kelly R. Roos, Department of Physics, Bradley University Peoria, IL 61625, [email protected] 3D trajectory visualization - Projectile motion with Coriolis deflection Rotational reference frames - Verify Kepler’s laws - 2-body problem - 3-body problem Classical gravitation and central force motion more sophisticated programming - Double pendulum - Numerical integration of other dynamical systems Lagrangian dynamics - Higher order Runge-Kutta methods - introduction to Verlet and Gear algorithms - chaos identification - Poincaré sections - Lyapunov exponents - period doubling and transition to chaos - Simple pendulum - Damped driven pendulum Nonlinear oscillations - artifact identification - Basic Runge-Kutta method - increased programming skills - function plotting - phase space plots Simple harmonic oscillator Linear oscillations - Simulating a model - Euler method - programming and debugging fundamentals - necessity of error control Realistic projectile motion with air resistance Projectile Motion Computational Method and Skills Acquired Computational Assignment Course Topic Precis: In a department wherein the creation of specific computation physics courses has not been possible, I have devised a mode of computational physics instruction wherein I incrementally integrate computational physics instruction into the traditional format of two upper-level undergraduate course I have taught for many years. Summary of computational topics covered and skills acquired: Classical Mechanics Statistical Mechanics and Thermodynamics - kinetic MC method - box counting method for determining fractal dimension of DLA structures - application of periodic boundaries - finite simulation size effects - realistic model of non-equilibrium atomistic processes at surfaces - complicated programming - verification of analytic theory with simulation results - Diffusion-limited aggregation (DLA) - Kinetic MC simulation of initial stages of thin film growth - Numerical Integration of nonlinear stochastic microscopic rate equations Non-equilibrium physics - Monte Carlo (MC) method and the Metropolis algorithm - ferromagnetic phase transition in 2D system Ising Model Magnetism - Molecular Dynamics (MD) method - importance of material boundaries and appropriate boundary conditions - visualization of connection between macroscopic observables and microscopic states - utilization of Verlet and/or Gear algorithms - Hard sphere simulation of ideal gas - Non-ideal gas model with Lennard-Jones potentials Ideal and non-ideal Gases - application of random number generator - simulation of a stochastic model - stochastic vs. deterministic models - calculate probability distributions - importance of concept of ensemble - computational realization of large ensemble - Random occupation of a square lattice - 2D random walk Methods of statistical analysis random number sequence generation Linear Congruential Generator Mapping Computational Method and Skills Acquired Computational Assignment Course Topic Examples of student calculations from Statistical Mechanics: -400 -300 -200 -100 0 100 200 300 400 -400 -300 -200 -100 0 100 200 300 400 2D Random W alk T rajectories 100,000 steps 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 root-m ean-square- displacem ent x-direction y-direction num ber of steps 2D Random W alk 1000 S teps Im portance of a large ensem ble S e = ensem ble size S e = 1 S e = 10 S e = 1,000 S e = 100 S e = 10,000 -100 -5 0 0 50 100 0.000 0.005 0.010 0.015 0.020 Probability Px Py Gaussian Position 1000 steps S e = 10,000 2D Random Walk on a square lattice Equal probability in each of four directions Probability distribution Modeling Sub-Monatomic Layer Epitaxial Growth 0.00 0 .02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.0 5.0x10 -4 1.0x10 -3 1.5x10 -3 2.0x10 -3 2.5x10 -3 n 1 ,sin g le a to m s n 2, dim ers N ,sta b le isla n ds # p e r la ttice site co ve ra g e (fra ctio n o f o ccu p ie d la ttice site s) Stochastic rate equations Random Occupation of sites on a square lattice 0 2 4 6 8 10 12 14 50 100 0 2 4 6 8 10 12 14 20 40 60 80 100 20 40 60 80 100 Ave. site occupation = 4 40,000 depositions on a 100x100 lattice 0 10000 20000 30000 40000 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 10000 0 exp Poisson Distribution Effect of increasing temperature Kinetic Monte Carlo Simulations 100 x 100 lattice (periodic boundaries) Effect of increasing Edge diffusion Diffusion Limited Aggregation Hausdorff dimension N = r D D ≈ 1.6 High binding energy between atoms Low binding energy between atoms

incremental integration of computational physics into traditional undergraduate courses kelly r....

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