**Physics 563: Phase Transitions and the Renormalization Group**

**Term essays 2010**

*These essays were written by students taking Physics 563 Phase Transitions and the Renormalization Group at the University of Illinois at Urbana-Champaign. The copyright of each essay is due to the author. *

*Please acknowledge the essay title, author, and this course in any citation to these articles. *

*The information, opinions and interpretations expressed are those of the authors, not necessarily those of the instructor. *

Author: Daniel Sussman

Title: Kinetic Ising models and the glass transition

Abstract

The Fredrickson-Andersen model, one of a class of Ising models that introduces constraints on the allowed dynamics of the system, was proposed in an attempt to capture some of the dynamic properties of glass formers while exhibiting trivial equilibrium behavior. After a brief introduction to some of the observed phenomena of glassy systems, the FA model and its scaling behavior is reviewed. In closing, two extensions and a few potential limitations of the FA model are discussed.

Author: Xianhao Xin

Title: Renormalization Group Approaches to the Kondo Effect

Abstract:

Kondo effect was first introduced by Knondo in 1964 for explaining the extraordinary resistivity of metals with magnetic impurities: The resistivity $\rho(T)$ increases as temperature $T$ goes to zero. This problem can be well understood by the method of the renormalization group. In this essay, the renormalization group is used to analyze the effective coupling of the impuity with the conduction electrons. One finds that the effectvie coupling diverges towards a fixed point for antiferromagnetic case, and tends to zero for ferromagnetic case. We will discuss the low temperature behavior of the antiferromagnetic Kondo model, which is well characterized by the Landau Fermi liquid approach. Some physical quantities such as specific heat capacity, impurity entropy and suscceptibility are obtained perturbatively in low temperature limit. The qualitative behaviors of those physical quantities for the whole temperature regime are also discussed in this essay.

Author: Tony Hegg

Title: Supercritical Fluid Near the Critical Point: The Piston Effect

Abstract

This essay describes thermalization at the critical point while under the effects of microgravity. In particular it covers the piston effect discovered in absence of Earth's gravity and the subsequent research this effect initiated.

Title: Neuronal Avalanches

Author: Ethan Brown

Abstract: It has recently been suggested that, in order to provide the richness of behavior which is seen empirically, the brain must operate in a critical regime between a phase of order and disorder [1]. In particular, such critical behavior is exhibited in what is termed neuronal avalanches. These cascades of synaptic activity have been seen both in vitro and in vivo [6]. Several models have been put forth to explain this apparently critical phenomena, ranging from simple dynamical models of neural networks to full fluctuation field theories, though the specific mechanism remains unclear. Despite the reproducibility of this behavior, its connection to high-order processes is yet unknown, though some have linked neuronal avalanches to large scale brain features such as information storage [2,4], synaptic stability [3], even epileptic shock [1]. In hopes of providing a larger context for this pervasive neural behavior, we examine the aforementioned experimental findings and compare various theoretical models which predict them.

Title: Two-Phase Diagram of Earthquake Distributions

Author: Anshul Kogar

Abstract:

This paper explains a model that shows that the conventional debate over whether earthquakes obey a Gutenberg-Richter Distribution or a Characteristic Distribution at a specific fault could be unfounded. The model, in the mean-field approximation, predicts a mode-switching phase and a Gutenberg-Richter phase that gives rise to the possibility of time-dependent seismic behavior. A qualitative phase diagram is obtained in parameter space and a second-order transition point is identified. Comparison with paleosiesmic data is made and discussed.

Title: Wetting and spreading phenomena

Author: Ponnuraj Krishnakumar

Abstract: Understanding wetting phenomena involves a careful and detailed study of the various surface tension forces, fluctuations and other contact line dynamics.Cahn's phenomenological theory combines the eff ects of short-range and some long-range forces to produce a phase diagram and the two critical exponents that characterize behavior near Tc. Experiments in the past two decades have been able to test and verify the phase diagram predicted in Cahn's theory. Further the critical exponents obtained in experiment closely agree with those obtained in theory. All of the theory and the results are presented here.

Title: Phase Transitions in Random Graphs- Outbreak of Epidemics to Network Robustness and fragility

Author: Mayukh Nilay Khan

ABSTRACT Inspired by empirical studies researchers have tried to model various systems like human populations, the World Wide Web or electric power grids by random graphs. Here we first examine different properties of random graphs(both undirected and directed) having arbitrary degree distributions using the generating functon formalism. We present some empirical data about the structure of random graphs in real life especially the WWW. Then we modify our results to include site and bond percolation to address questions about the onset of large scale connectivity and the formation of a giant component in random graphs. This point corresponds to a phase transition. We use this information to examine questions about the onset of epidemic or the robustness or fragility of a network to random or targeted deletion of nodes.

Title: Application of Statistical Physics to Terrorism

Name: Rodrigo Soto-Garrido

Abstract:

This paper reviews two models used to study terrorism from a statistical physics point of view. The first model describes terrorism using percolation theory based on individual passive supporters. The second one is an opinion dynamic model with memory that is used to understand the inter-event time (time between terrorism events). The first model provides us with some recommendations to reduce the terrorism threat or at least confine it to a specific region, while the second model offers a good agreement between the empirical data and their predictions.

Author: Olabode Sule

Title: Riemannian geometry of critical phenomena

ABSTRACT In this review, it is discussed how by incorporating the theory of fluctations into the basic axioms of thermodynamics, thermodynamic systems can be mapped into appropriate Riemannian geometries. For systems such as the ideal gas, ferromagnetic one-d Ising model and the Van der Waals Gas, it is found that the curvature of this manifold is related to the correlation volume of the system. The correlation volume is conventionally calculated from a statistical "microscopic" description of systems. For other systems such as the antiferromagnetic model and Kerr Newmann black holes the correlation volume is given a new interpretation. In the case of Kerr-Newmann black holes it is interpreted as the average number of correlated planck areas at the hole's surface.This idea may be formally extended to understand phase transitions in black holes thermodynamics where there is not a completely developed microscopic theory "quantum gravity".

Title: Renormalization Group Analysis of Complex Networks: Applications in Scientific Networks

Author: Po-Yao Chang

Abstract:

Complex networks have relevance to many real systems in biological and social fields. Applications of renormalization group (RG) theory technique give us the full pictures of the configurations in networks topologies (small-world or fractal types). Moreover using RG technique we can find the shortcuts structures hidden in the complex networks and even understand the growth of the networks. The mechanism of scientific credits diffusion and the ranking of scientists in academic fields is based on the structure of the publication networks and the citation system. Here I propose the RG technique to comprehend the scientific network structure and understand the evolution of scientific credits.

Author: Braden A. W. Brinkman

Title: Hysteresis and Phase Transitions when Grasping Objects of Different Sizes

Abstract

The state of many systems is observed to depend on its previous states, a phenomenon called hysteresis. Experiments on grasping behaviour in humans have found that if subjects are instructed to grasp objects which sequentially increase or decrease in size the size of the object at which the subject switches from one hand to two occurs at a different object size than when the subject switches from two hands to one- i.e., there exists a hysteresis effect in the behavioural transition. In this article we review experimental and theoretical work on these observations, and we briefly discuss how we might try to relate the observed behaviour to activity at the neural level.

Author: Qinglei Meng

Title: Metal Insulator Transition

Abstract Metal Insulator transition(MIT) is characterized by the conductivity which will be zero in the insulator phase. In this term paper, we focus on Mott insulator, and a simple theoritical way to describe this MIT is the Hubbard Model .Finally, we will look at the experiments of Mott insulator transition.

Author: Anuj Girdhar

Title: The Superconductor-to-Insulator Transition in Resistively-Shunted Josephson-Junction Arrays

Abstract This essay describes the, analytical theory, experimental observations and potential applications of superconductor-to-insulator quantum phase transitions in arrays of resistively-shunted Josephson junctions and superconducting islands dissipatively-coupled to an external source of single-particle excitations.

Title: Understanding glass-forming and "jamming" materials in the context of soft colloids

Author: Jian Yang

Abstract: Recent experiment and simulation research on the glass forming and “jamming” materials are reviewed from the perspective of using soft colloids to establish a general picture of molecular glasses, colloids and granular materials.

Author: Thomas F. Fehm

Title: Phase transitions in crowded behaviour

Abstract In this paper, I want to review theoretical models of two social phenomena: the dynamics of an applauding audience and the collective motion of people in a stadium forming a La Ola wave. In particular, I want to stress how phase transitions occur in these theoretical models and how these transitions emerge macroscopically.

Author: Ching-Kai Chiu

Title: Topological phase transition HgTe Quantum Wells

Abstract

Mercury telluride-cadmium telluride semiconductor quantum walls can change to Z2 topological insulator phase from conventional insulator phase when the thickness of the quantum well is varied to the critical thickness dc. In this report, I will introduce that Z2 topological insulator is protected by time reversal symmetry and discuss to topological quantum phase transition between conventional insulators and topological insulators.

Title: Nonlinear magnetization and the ''vortex liquid" state above Tc in cuprate superconductors

Author: Hryhoriy Polshyn

Abstract: This essay describes the observations of the magnetization M above the critical temperature Tc in cuprates. The nonlinear behavior of M suggests the existence of a "vortex liquid" region above Tc, which is a region of a strong superconducting fluctuations. This conclusion is consistent with the Nernst signal and other experimental data. At least in two families of cuprates, the low -H magnetization exhibits non-analytic divergent behavior.The possible explanations are based on the Kosterlitz-Thouless (KT) transition and incomprehensibility of vortex liquid are considered in this paper.

Author: Jonas J. Funke

Title: Forest-Fires and Phase Transitions

Abstract: A large group of natural hazards ( e.g. forest fires, earth quakes) are showing scaling law behavior even if they differ in size by magnitudes. As it turns out these class of systems can be described by Self-Organized-Criticality and it is indeed possible to relate them to the critical behavior of phase transitions. Using the example of forest-fires the question of how (well) the theoretical phase transition models fit the experimental results shall be studied.

Title: Geomagnetic reversal through critical models

Author: Ayah Massoud

Abstract: Geomagnetic polarity reversal intervals display power law distribution functions, which indicate a critical phenomenon as the mechanism of their source. The geodynamo is assumed to be a system of magnetic spins in a critical phase-transition state. Seki and Ito developed two models to simulate these reversals: the N-disc coupled model and the Coupled map lattice model both derived from the Rikitake Dynamo system and are both reviewed here. The simulations were done in two dimensions in a square lattice with periodic boundary conditions. The major difficulty is simulation times, which hinders a full three dimensional model.

Title: Phase transitions of traffic flow

Author: Agustinus Peter Sahanggamu

Abstract: This essay introduces a basic model for a traffic jam phase transition. Order parameter will be de fined, motivated from physical reasoning. Phase diagram of the system shows there is a critical behavior, from which we can deduce critical exponents similar to statistical mechanical exponents.

Title: Explosive phase transitions in percolation processes

Author: Maksim Sipos

Abstract: Percolation processes are well studied in physics. In theoretical physics, directed percolation (DP) is a representative of a well-known universality class of continuous phase transitions. DP has been used to model a variety of phenomena including turbulence, liquids percolating through porous media, epidemics and forest fires. In the Erdos-Renyi model, it is known that the order parameter (size of largest connected component) undergoes a continuous phase transition beyond a critical percolation treshold. However, recent numerical experiments indicate that an Erdos-Renyi model with a modified percolation rule can have an "explosive" phase transition in which the order parameter undergoes a discontinuous jump. Another model of a random network of integrate-and-fire neurons shows a similar transition.

Author: David Chen

Title: Continuum Model of Avalanches in Granular Media

Abstract

A continuum description of avalanches in granular systems is presented. The model is based on hydrodynamic equations coupled with an order parameter determined by a free-energy-like potential. The model successfully describes the transition between the static and the fluidized phases. This formalism is applied to avalanches on an inclined plane configuration. The theoretical predictions agree with the experimental results.

Author: Jun Li

Title: Phase transitions in crystalline materials

Abstract

We reviewed structural phase transitions in crystalline materials, especially recent simulations and experiments on the shock- induced transitions in iron. They both confirmed a phase transition wave in the picosecond to nanosecond time scale, inferred previously from wave profile analyses. The nonequilibrium molecular dynamics simulation suggests modification of pair potentials from temperature-induced to shock-induced transitions. The x-ray diffraction measurements are consistent with a compression and shuffle mechanism and in good agreement with simulation results.

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