Fluctuation driven phenomena refer to a broad class of physical systems that are shaped and influenced by randomness. These fluctuations can manifest in various forms such as thermal noise, stochasticity, or even quantum fluctuations. The importance of understanding these phenomena lies in their ubiquity in natural systems, from the formation of patterns in biological systems, to the behavior of phase transitions and universality classes, to quantum computers. In this dissertation, we delve into the peculiar phenomena driven by fluctuations in the following scenarios:
We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a non-conserved order parameter reversibly coupled to the conserved total magnetization. We find that in equilibrium, the dynamics is well-separated from the statics and the static response functions are recovered in the limit ω → 0, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d = 2 + ε expansion about its lower critical dimension d<sub>lc</sub> = 2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d<sub>c</sub> = 4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε = 4 − d expansion near dc. This yields anomalous scaling features induced by the massless Goldstone modes, namely sub-diffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z<sub>Γ</sub>= d − 2 which may be observable in neutron scattering experiments.
We investigate the influence of spatial disorder on coined quantum walks. Coined quantum walks describe the time evolution of a quantum particle that is controlled by a quantum coin degree of freedom. We consider one-dimensional walks and use a two- level system as quantum coin. Each time step thus consists of the iterative application of a quantum coin toss and a conditional shift operator. Qualitative differences with classical random walks arise due to superpositioned states and entanglement between walker and coin. We consider spatially inhomogeneous coin tosses with every lattice site having a tossing amplitude. These amplitudes are noisy such that the walk is spatially disordered. We find that disorder deteriorates the ballistic transport properties of non-noisy quantum walks. This leads to an extremely slow spreading of the quantum walker and potentially induces localization behavior. We investigate this slow dynamics and compare the disordered quantum walk with the standard coined Hadamard walk.
Special focus is given to the influence of disorder on entanglement-related properties.
We apply a perturbative field-theoretical analysis to the symmetric Rock-Paper-Scissors (RPS) model and the symmetric May-Leonard (ML) model, in which three species compete cyclically. We demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation- induced renormalizations in the perturbative regime. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS model, whereas the spontaneous emergence of spatio-temporal structures features prominently in the ML model.
We delve into the action-to-absorbing phase transition in the Pair Contact Process with Diffusion (PCPD), which naturally generalizes the Directed Percolation (DP) reactions. We revisit the single-species PCPD model in the Doi-Peliti formalism and propose a possible perturbative solution for the model. In addition, we investigate the two-species effective model of PCPD and demonstrate its equivalence to the single- species PCPD at tree-level effective field theory. We also examine the fixed point of the model where all relevant parameters are set to zero. Our analysis reveals that the fixed-point theory is inconsistent with the PCPD critical condition. Thus, combining the effective field theory argument, this inconsistency suggests that the critical theory should already be completely encoded in the single-species model. / Doctor of Philosophy / Fluctuations are a ubiquitous aspect of the real world. For instance, even though a train schedule may be set, the train may arrive two minutes ahead of schedule or two hours late.
Similarly, if you were to flip a coin ten times, you would expect to get five heads and five tails based on simple probability, but in reality, you may not even come close to this result.
In classical situations, these fluctuations are a result of our lack of knowledge about the details of the system. However, in quantum mechanics, scientists have demonstrated that fluctuations are inherent to the system, even when every single detail of the system is known.
Therefore, understanding fluctuations is crucial to gaining insight into the fundamental laws of the universe.
In most cases, fluctuations are insignificant and the world can be accurately described by a set of deterministic equations. However, there are situations in which fluctuations play a significant role and can greatly deviate the system from the predictions of deterministic equations. In this dissertation, we study the following scenarios where fluctuations dominate and lead to peculiar phenomena:
Near continuous phase transitions, due to the divergence of the characteristic length, most systems become long-range correlated. This means that the changes at one point can affect another point very far away. We study the critical dynamics of two systems near their phase transitions: antiferromagnetic system in Chapter 2 and a simplified population dynamics model in Chapter 5. Through our analysis, we demonstrate how fluctuations significantly alter the behavior of these systems near their critical points.
In chapter 3, we examine the impact of spatial disorder on the quantum random walk, a quantum counterpart of the classical random walk or "drunkard's walk". Given that the quantum random walk has been shown to have universal quantum computing capabilities, this disorder can be considered as errors in the control of the system. We reveal how disorder effects drastically change the dynamics of the system.
The formation of patterns is typically studied in deterministic nonlinear systems. In Chapter 4, we analyze pattern formation in stochastic population dynamics models, and demonstrate emergent behavior that goes beyond what is seen in their deterministic counterparts.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/115048 |
| Date | 15 May 2023 |
| Creators | Yao, Hong |
| Contributors | Physics, Tauber, Uwe C., Barnes, Edwin Fleming, Pleimling, Michel Jean, Anderson, Lara Briana |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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