• Kinetic Monte Carlo codes (KMC) for two sorts of KMC
• Classical MC and MD for 2 or 3 dimensional liquids, plasmas and polymers (CLAMPS)
• Software on NCSA computers (we can help) find some.

A LIST OF IDEAS:

Environment Simulations: (Stochastic Evolution of River Valleys)

• For a area (2-D, i.e. x and y) of the earth, use MC to generate a terrain (height, z), as well as existing water levels, based on PRNG algorithm. Extending this, simulate the evolution of river basins based on algorithm to erode earth height and to distribute water, possibly including the oft found rain contribution. The algorithm could be physically based like Flux Equations, or non-physically based like cellular automata. Best of all, they also may be represented by faux color maps, since all this can be viewed topologically. (See overly fancy version of this LUCAS or example paper by Mitas et al. . But with your or simplier approach, many interesting things can happen.)

Dislocations:

• Using a empirical potential, such as the Double-Yukawa, write a, or modify an existing, MD code to study dislocation motion in a 2-D, finite simulation cell. (2-D is recommended for simplicity as well as the ability to more easily visualize.) How do you impose shear strain into boundary conditions? What additional requirements are needed for boundary conditions? For example, one dislocation in cell produces very large strain fields which cannot be handles well in finite cell. How can you arrange another dislocation so that strain fields are short-ranged and finite cell works reasonable well?

Kinetic Monte Carlo: (Diffusion, Concentration Gradients, Phase Transitions, Surface Chemistry...)

• Simulate via KMC a binary alloy system to determine the changes in the local environmental structure due to vacancy migration. Because the global alloy composition cannot change the simulation will be in canonical ensemble. There should be many interesting phase transitions depending on vacancy-element interactions and their sizes, which are related to the concentration fluctuations in non-equilibrium. Kinetics will depend on the effective chemical potentials, but the equilibrium thermodynamics will depend only on the effective pair-interactions, as discussed in class. (See, e.g., Athenes et al., Acta Mater. 44, 4739 (1996)). This could also be performed to study kinetics and equilibrium structure of concentration gradients in multi-layer alloy.
• Implement a KMC to determine to 2-D surface diffusion coefficient, which may be obtained via MC and Boltzmann-Matano analysis. It may be implemented as a study of time correlation of the concentration fluctuations. (See, e.g., independently invented by Reed and Ehrlich at UIUC, Surface Science 105, 603 (1981), Bortz et al., J. Comput. Phys. 17, 10 (1975), and, intro paper by Clark, Raff and Scott, Computers in Physics 10, 584 (1996).)
• Using KMC, study the evolution a concentration gradient around an anti-phase boundary defect in an binary alloy, which may be mapped to an effective 2-D problem if entire planes parallel to defect are considered identically occupied.
• Surface Chemistry: Produce a hybrid MC code and simulation consisting of kinetic plus equilibrium moves which applies to systems with N different processes whose rates vary by orders of magnitude, like chemical vapor deposition. See intro paper by Clark, Raff and Scott, Computers in Physics 10, 584 (1996).

Simulated Annealing: (Modeling the Stock Market or Finding Global Minima)

• Find a Needle in a Haystack: Choose a function which contains a large number of multi-minima, perhaps with even large deviations of minina depths, but with only one global minima. Use the technique of simulated annealing to produce an efficient code for finding the "ground state", i.e. global minina, and compare to more standard Conjugant Gradient or Newton-Rapheson Techniques. See Numerical Recipes for short explanation of these latter techiques, if you are unfamiliar with them. See also, Andricioaei and Straub, Computers in Physics 10, 449 (1996) for intro article on CONFORMATIONAL OPTIMIZATION, which is the fancy name for this approach, where they give some suggestions for study, including a function of the type used in simulations and conformation of small L-J clusters.
  • Note that multi-state spin models (Potts, etc.) always produce such multi-minima states. Such models are used on the New York Stock Exchange to model derivate buying.
  • The Travelling Saleman Problem is in fact a problem of this type (but you cannot choose this one). (E.g, Yoshiyuki et al., Computers in Physics 10, 525 (1996)

Random Number Generators:

• Test the ideas of quasi-random numbers for a integration of a function which is not feasible with a grid-based method and compare to Monte Carlo efficiencies. (see Computers in Physics Nov. 1989 )
• For various psuedo-random number generators, use both tests of their quality as far as correlation and randomness as well as their efficiencies within a particular simulation algorithm, such as 2-D Ising Model, to pick a Best Choice PRNG for your application. (For Portable PRNG, see Marsaglia and Zaman, Computers in Physics 8, 117 (1994); also, P. Coddington, Tests of random number generators using Ising model simulations , Int. J. of Mod. Phys. C, 7(3):295-303, 1996.)

Non-Equilibrium Phenomena: (Order-disorder, Spinodal Decompositions, Liquid-solid Trans., etc., using Langevin Simulations

• Study phase separation in a binary alloy (Cahn-Hillard type equations) using Langevin Dynamics. (See Ken Elder, Computers in Physics 7, 27 (1993) and note that CLAMPS can do some types of Langevin dynamics.)
• Active-walker models used to simulate growth phenomena in open systems in non-equilibrium conditions. Large-scale versions of this are the expansion or shrinkage of river basins (see Environmental Simulation Idea above), or smaller-scale versions are chemical and biological systems, like cell evolution, snowflake formation, and so on. Study a type active-walker model, but not just one out of sample papers, even your own. (See, e.g., Lam and Pochy, Computers in Physics 7, 534 (1993).) or about bacterial colonies by Ben-Jacob et al., Nature 368, 46 (1994).
• We're all getting older so, perform simulation of biological aging using MC algorithm. See, for example, D. Stauffer article, Computers in Physics 10, 341 (1996), where he has suggested several topics for futher study. (Some of these are nothing more than trivially altering code and running: Refrain from such inactivity. You certainly can come with more interesting extensions.)

Phase Transition:

• Simulate the phase coexistence for complex molecules by producing a configurational-biased, grand-canonical MC program for L-J chain polymers. CLAMPS may be modified to do this, or write your own specialized and short version. A good discussion and some code can be found by Frenkel and Smit, Computers in Physics 11, 247 (1997).
• Study the equilibrium thermodynamics (MC or MD) of simple binary gas based on empirical potentials (I would use Double-Yukawa type but it is not necessary) on the surface of a sphere so that there are no Periodic Poundary Conditions. What, if anything happens around the liquid-solid interface , for example?
• Implement so-called Cluster Algorithm for MC simulation, which flips multiple sites rather than single sites sequentially. Compare the MC efficiency from standard and cluster algorithm. See, e.g., Selke, Talapov, Scheir, "Cluster-flipping MC algorithm and correlation in a "good" RNG," JETP Lett. 58, 665 (1993). (For simplisicity, it may be best to due this for 2-D Ising Model. If so, then also compare what happens just below critical point for conifgurations using both algorithms.)
• Simulation of Polymer Chains on surface of Sphere by altering relevant parts of CLAMPS to change boundary conditions. (Check with Prof. Cepereley whether this is big job.) This acts like confined volume with artificial boundary conditions like bags or boxes and limits the complexity. Density of polymers can be controlled via sphere radius, for example. Does anything change by maintaining density while varying radius?
• Implement the Mori-Zwanzig-Daubechies (Wavelet) decomposition of Ising Model Monte Carlo Dynamics. (See, e.g., Phillies and Stott, Computers in Physics 9, 225 (1995).) But first answer, why would you want to do this?
• Implement MC of hard-sphere gas on a 2-D Sphere and study the entropy-driven phase transition. Determine the distribution of the velocities. Determine how PV=nRT is recovered, or not, from simulation. (see general discussion of problems in article by Gould, Tobochnik adn Colonna-Romana, Computers in Physics 11, 157 (1997)

Cellular Automata:

• While this is not really the basis of this course in Atomic Scale Simulations, it does have obvious connections and can be applied to numerous and widely varied systems: Image Processing, Urban Development, Ant Colonies, Reaction-Diffusion, Alloy Kinetics, ..... If yours is interesting and relevant, why not? (To get started, e.g., Cellular Automata)