Scaling and intermediate asymptotics for singular physical problems

Zhu, Minhui

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https://hdl.handle.net/2142/127428 Copy

Description

Title

Scaling and intermediate asymptotics for singular physical problems

Author(s)

Zhu, Minhui

Issue Date

2024-08-08

Director of Research (if dissertation) or Advisor (if thesis)

Goldenfeld, Nigel D.

Doctoral Committee Chair(s)

Cooper, S. Lance

Committee Member(s)

• Dahmen, Karin A.
• Leite Noronha, Jr, Jorge

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• nonequilibrium dynamics
• intermediate asymptotics
• singular perturbation
• pattern formation
• scaling laws
• superconductor
• turbulence

Abstract

This thesis focuses on modeling the singular nonequilibrium dynamics of complex systems, identifying hallmark behaviors such as scaling laws and crossovers and collaborating with experimentalists to ground the theoretical models with real data. We examine the dynamics of two seemingly disparate physical systems: cuprate superconductors and turbulence. Both systems are renowned for their fascinating emergent properties, yet scientific understanding has been hindered by their intricate microscopic complexities. The class of problems we encounter in this thesis are treated in perturbative fashion, but pose special difficulties on account of strong coupling effects. Thus, we also study advanced asymptotic methods necessary for addressing such singular behavior across a broader range of regimes beyond perturbative methods. The specific aspect of complex systems that we focus on is to understand the general principles governing the way that they relax back to an equilibrium state after a perturbation. In the first part of this thesis we focus on a pattern forming quantum system, namely a cuprate superconductor, and study its relaxation back to a state with periodic charge order after a laser pulse has photoexcited the system out of its equilibrium. In the second part of this thesis, we focus on a pattern forming classical system, namely the propagation and decay of turbulence after its creation through the collision of vortex rings in the center of a box of fluid. In both problems we find novel scaling behavior which we analyze using perturbation and renormalization group theory. Part I concerns the phase ordering dynamics of stripe charge orders in cuprate superconductors, which we have studied as part of a collaboration with experimentalists at the University of Illinois at Urbana-Champaign. Cuprates, a widely studied class of high-temperature superconductors, have neither their superconducting mechanism nor their normal state well understood due to their correlated nature and the interplay of superconductivity, charge orders and magnetic orders. Ultrafast X-ray scattering experiments done by our experimental collaborators reveal that collective charge order in doped LBCO undergoes relaxational dynamics after being excited out of an ordered phase, where dynamic charge order is propagated by Brownian-like diffusion. We model this process as a quench dynamics governed by the Swift-Hohenberg equation, a phenomenological model selected on the emergent symmetry of modulated charge orders, originally used to study pattern formation in fluid convection. By performing linear stability analysis, we find the initial dynamics are characterized by an exponential recovery process, with a relaxation rate indicating diffusive propagation of the excitation of the full equation. Notably, the pure dissipation energy, which is related to equilibrium transverse fluctuations by this narrative is close to the superconducting transition temperature $T_C$, suggesting that dynamic charge order might contribute to superconductivity, in contrast to the common belief that static charge orders compete with it. At later times, the charge order follows a dynamic scaling hypotheses, indicating a universal coarsening process dominated by transverse topological defects, as consistently described by the same model. This combined experimental and theoretical investigation reveals nonequilibrium dynamics in previously inaccessible time and energy regimes, introduces new tools into quantum systems and potentially provides support for a proposed pairing mechanism in cuprates. Part II reports on a collaboration with experimentalists at the University of Chicago on the decay and propagation dynamics of an isolated blob of turbulence, away from boundaries. A large part of our current understanding of turbulence is focused around the energy cascade and the quantification of intermittency and stochasticity in idealised systems that are homogeneous, isotropic and have periodic boundary conditions. However, the dynamics of emergent large-scale turbulent motions are not well understood, though they are crucial for applications ranging from engineering efficiency to disaster prevention. Experimentally, turbulence is challenging to control and measure due to its fluctuating nature, and hard to isolate its behavior because of complex boundary effects and emergent mean flows generated by the turbulence itself. Our experimental collaborators at the University of Chicago innovatively generate and control an isolated turbulence blob in the lab through colliding vortex loops, minimally coupled to mean flow. The dynamical evolution of this blob is measured with high spatial and temporal resolution using optical probing methods. Theoretically, we model an ideal scenario where an isolated turbulence blob expands and decays in free space. We generalize a phenomenological model based on turbulent energy diffusion and turbulence dissipation to allow time-dependence of the integral length scale, as observed experimentally. Through a combination of analytical and numerical approaches, we uncover the formation of a sharp front during the expansion, propagating non-diffusively in an approximately self-similar regime. This behavior is observed in experiments upon careful analysis of the data. In addition, we predict scaling and crossover behavior for the decay of the energy in time, which depends on both the growth of the integral length scale and the spatial distribution of the turbulence energy, although experimental observations are further complicated by boundary conditions in later times. By minimally incorporating boundary effects into our model through parameter tuning, our model shows remarkable agreement with the extended decay dynamics of a blob constrained by the experimental chamber. When studying the singular dynamics of complex systems, even minimal models often exhibit high nonlinearity and singularities, posing significant mathematical challenges. Exact solutions are typically unattainable. One common approach is perturbation theory, where we expand a system around a known solution, assuming a small deviation—such as the linear stability theory used in the cuprates project. However, in singular perturbation problems, a naive perturbation expansion can become divergent, as seen in the nonlinear diffusion absorption equation in the turbulence project. Such singularities can often be treated by perturbative renormalization group (RG) methods, but these methods might fail in the strong coupling regime. In Part III, we apply the self-consistent expansion (SCE) method to these singular dynamics problems, exemplified by the Barenblatt's equation for diffusion in a sponge --- a nonlinear diffusion process in the class of porous medium equations, but with anomalous scaling. We show that the SCE method can be combined with perturbative RG to provide improved analytical calculations of the anomalous dimensions even in the strongly-nonlinear regime. Furthermore, the calculations are surprisingly tractable. Additionally, we present a scheme to translate dynamics problems into field theory, and illustrate how to apply both RG and SCE in the context of similarity solutions. This comprehensive approach enhances our ability to tackle complex singular dynamics beyond the perturbative regime, providing more robust mathematical tools for their analysis, and laying the groundwork for the application of these techniques in singular perturbation problems more complicated than similarity solutions.

Graduation Semester

2024-12

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/127428

Copyright and License Information

Copyright 2024 Minhui Zhu

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