Implementing Quantum Random Walks in Two-Dimensions with Application to Diffusion-Limited Aggregation

Colin Sanberg, Butler University


This study simulates random movement and aggregation of particles in two-dimensional space based upon both quantum and classical mechanics. Using an original computer program to perform the calculations, the objective is to compare how quantum effects influence the random movement of a particle in comparison to the classical random movement. These effects are further studied by analyzing how the amassing of particles around a "seed" is affected by the differences in the random movement. Using the classical models that were generated as the basis of comparison, the initial results show that the quantum model aggregate grows at a slower rate than the classical case. Also, the quantum model grows in a more amorphous manner than the clear branching of the classical example, In an effort to more accurately simulate the behavior of the probability function as it encounters other particles, both the quantum and classical models were adjusted, This yielded a quantum aggregation that developed more similarly than the classical model The primary difference in the quantum model was a noticeable lack of symmetry as the particles amassed around the seeded particles. It is possible that this iteration of the quantum model develops more rapidly then the classical model, though more simulations are needed to further test this. The effect other particles have on the development of the probability function also needs to be further examined lo ensure that it is being modeled as accurately as possible.