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arXiv:1111.0626

Quantum Diffusion-Limited Aggregation

David Johnson, Ordóñez, Gonzalo

Published Nov 2, 2011• 1 citation
arXivHTMLPDF

Abstract

Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.

DOI
10.48550/arxiv.1111.0626

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Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.
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 ratc 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.
Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.
Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.
Time evolution of quantum particles like electrons is described by time-dependent Schrödinger equation (TDSE). The TDSE is regarded as the diffusion equation of electrons with imaginary diffusion coefficients. And the TDSE is solved by quantum walk (QW) which is regarded as a quantum version of a classical random walk. The diffusion equation is solved in discretized space/time as in the case of classical random walk with additional unitary transformation of internal degree of freedom typical for quantum particles. We call the QW for solution of the TDSE a Schrödinger walk (SW). For observation of one quantum particle evolution under a given potential in atto-second scale, we attempt a successive computation and visualization of the SW. Using Pure Data programming, we observe the correct behavior of a probability distribution under the given potential in real time for observers of atto-second scale.
The analogy between quantum 'paths' and random walks of a Brownian particle is extended to the relativistic regime. This yields a coherent picture of the fine structure of quantum paths: these can be viewed as analogies of the trajectory of a random walker. The fractal dimension of quantum paths is df=2 at scales large compared with the Compton wavelength Lambda but small compared with the de Broglie wavelength lambda and becomes df=1 at scales smaller than Lambda . The transition between the two regimes is exhibited explicitly and is analogous to the cross-over from ballistic to diffusive motion of Brownian particles. The analogue of the collisions in Brownian motion is electron-positron pair creations and annihilations, and the unit thermodynamic free energy kT corresponds to the rest mass mc2 of the particle.