Langevin Dynamics: From Basics to Brownian Bees
Author: Paul Severino Abstract This article reviews the basics of Langevin dynamics in order to gain a deeper understanding into non-equilibrium statistical physics. Beginning with a description of the Langevin equation, the importance of noise in the equilibrium limit is stressed. The quantities of interest are then calculated for simple Langevin dynamics, making reference to…
An Introduction to Kosterlitz-Thouless Theory and Superfluid Helium
Author: Paul Severino Abstract This paper introduces Kosterlitz-Thouless theory and its application to condensed matter physics, specifically superfluid He-4. First, the expectations for 2D systems – specifically from simple Bose-Einstein condensation and the Mermin-Wagner theorem – are discussed. Then the Berezinskii-Kosterlitz-Thouless (BKT) transition and its properties are introduced in the XY model where they were…
Prime Numbers in Fortune Cookies: An Introduction
Author: Paul Severino Abstract This paper studies the appearance of prime numbers in idealized fortune cookie lucky numbers. First, different ways to see prime numbers in fortune cookies are discussed. Then analytical and computational tools are used to find good approximations for the probability of finding each of these primes. The methodology and results from…
Schramm-Loewner Evolution and Random Curves in Statistical Physics
Author: Paul Severino This article was inspired by my recent interest in statistical physics, a new professor in the department of mathematics at the University of Chicago, and two questions on the 2021 International Theoretical Physics Olympiad. Abstract This article hopes to motivate the study of Schramm-Loewner evolution (SLE
Brief Overview of Complex Analysis
Author: Tabin Dharanikota I wanted to share a document that gives a short summary of complex analysis. The goal of this document is to give some context to some important results in complex analysis, and perhaps be a little referral sheet if you need to quickly remind one of a result or jog one’s memory of…
Methods of 2D Turbulence: Cascade Theory and Statistical Physics Approaches
Author: Paul Severino Abstract This paper reviews theories of 2D turbulence, referencing results and theories in 3D where relevant. Beginning with a review of 2D Euler flow, a proof of Kelvin’s circulation theorem is presented. From this, the quadratic invariant of enstrophy is found. Based on this enstrophy, the double cascade theory of 2D turbulence…
The Cosmological Constant
Author: Paul Severino The accelerated expansion of the universe is one of the most exciting and philosophically influential results of modern observation cosmology. Our knowledge of the history of the universe and its eventual fate rely heavily on understanding the so-called dark energy responsible. But a lot of times this topic is covered very generally…
Radon Transform and Central Slice Theorem in Computed Tomography
Author: Paul Severino Imaging a Cursed Box Imagine you are given a mysterious box, and you are dying to know what is inside. It could very well be empty, but it could contain enough cash to pay off your student debt. But you do not know where the box came from, and you are scared…
Billiard Motions on Simple, Flat Tables
Author: Paul Severino Billiard dynamics are extremely important to physics; complex, many-body systems can be directly modeled by the fundamental motion of billiard balls on a table. One might think that mathematicians and physicists have everything about billiard motion figured out by now, but this assumption is sorely wrong! The mathematics behind even the motion…
A Stochastic View of Quantum Mechanics
Author: Shiv I wrote a paper for my advanced quantum mechanics class on a somewhat unconventional topic: the relationship between quantum mechanics and stochastic processes. A stochastic process refers to a system that evolves over time under some probabilistic rule. Quantum mechanics is an inherently probabilistic theory. One can solve Schrodinger’s equation to for the…
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