Tetra Quark

Don’t drink and derive

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A refresher on statistical mechanics

This article provides a comprehensive refresher on fundamental concepts in statistical mechanics, drawing inspiration from Leonard Susskind’s lectures. Beginning with probability theory and Shannon’s information-theoretic definition of entropy, we establish the mathematical foundations that bridge information theory and thermodynamics. We explore the derivation of entropy formulas using both Shannon’s axioms and combinatorial approaches with Stirling’s approximation. The article presents the zeroth, first, and second laws of thermodynamics, with particular emphasis on the relationship between entropy, energy flow, and temperature in interacting systems. Using calculus of variations and Lagrange multipliers, we demonstrate how entropy maximization principles lead to the uniform and Boltzmann distributions. Throughout, we supplement theoretical discussions with visual representations and detailed derivations to provide intuitive understanding of these abstract concepts. This refresher serves as an accessible entry point for readers seeking to revisit or develop a deeper understanding of statistical mechanics and its connections to information theory.

Biot-Savart Law

This article presents a rigorous derivation of the Biot-Savart law from Maxwell’s equations. Starting from the fundamental observation that magnetic fields are divergence-free, we develop the vector potential formulation and use Green’s function methods to solve the resulting differential equation. The derivation demonstrates how the familiar Biot-Savart law emerges naturally from these first principles, providing both the general form for continuous current distributions and the specialized case for line currents. This treatment emphasizes the deep connection between the absence of magnetic monopoles and the mathematical structure of magnetic fields.

Inductance of a Wire Pair with Neumann’s Method

This article presents a detailed derivation of the mutual inductance between two parallel wire segments using Neumann’s method. Starting from the fundamental electromagnetic energy expression involving the vector potential and current density, we evaluate the mutual inductance through direct integration. The analysis assumes thin wire approximation and provides the final result in terms of the wire length and separation distance. This approach offers an alternative perspective to the more commonly used flux-based calculations, while arriving at the same well-known logarithmic dependence on the geometric parameters.

Inductance of a Wire Pair

This article examines the mutual inductance between parallel wire segments, a fundamental configuration in electrical circuits and transmission lines. Building upon our previous analysis of single-wire self-inductance, we derive the magnetic coupling between current-carrying conductors using the Biot-Savart law. We address the mathematical challenges of finite-length conductors and present a complete solution that includes both the self-inductance of each wire and their mutual coupling.

Self-Inductance of a Wire

This article explores the calculation of self-inductance in a wire segment using the Biot-Savart law and energy methods. We derive expressions for both the external and internal contributions to the self-inductance. The external component is calculated by integrating the magnetic flux over a finite region, addressing the inherent challenges of infinite wire assumptions. The internal contribution is determined through energy considerations of the magnetic field within the wire.

Magnetic Dipole

This blog post provides a comprehensive treatment of magnetic dipoles, starting from first principles. We begin by deriving the vector potential for an arbitrary current distribution and apply it to the specific case of a circular current loop. The exact solution is expressed in terms of complete elliptic integrals, and we provide explicit forms for the magnetic field in both spherical and cylindrical coordinates. We then develop the magnetic dipole approximation, showing how it emerges naturally as the leading term in a multipole expansion. Finally, we extend our analysis to continuous distributions of magnetic moments, introducing the concept of bound currents and demonstrating how they provide an elegant framework for describing magnetized materials. Throughout, we emphasize the mathematical techniques and physical insights that connect these various aspects of magnetic dipole physics.

Red balls, Green Balls

This work delves into a probability puzzle involving an urn with an unknown number of red and green balls. Initially, the number of red balls is randomly selected. After observing a red ball on the first draw, the likelihood of drawing another red ball is calculated using Bayesian probability, which adjusts the probability distribution over possible urn configurations. The puzzle is further explored by modifying the setup, introducing a scenario where the number of red balls is determined by repeated coin flips, resulting in a binomial distribution. Theoretical results suggest a 2/3 probability of drawing a red ball again in the original setup, and a 1/2 probability when using the binomial model. These outcomes are validated through simulations, highlighting the nuanced effects of initial conditions on probability outcomes and underscoring the importance of rigorous reasoning in probabilistic analysis.

Fabry-Perot Cavity

This article explores the physics and applications of Fabry-Perot cavities, with a particular focus on their role in LIGO (Laser Interferometer Gravitational-Wave Observatory). We derive the fundamental equations governing cavity transmission and reflection, introduce the concept of finesse, and analyze different coupling regimes. The formation of interference patterns and bullseye fringes is explained through rigorous mathematical treatment. Building upon previous posts about LIGO’s electronics, this article complements the series by delving into the optical principles that make precision gravitational wave detection possible.

Renewal Processes

This article explores renewal processes and their relationship to Poisson processes in probability theory. We examine the fundamental concepts of interarrival times, counting processes, and their distributions, with particular emphasis on exponential and gamma distributions. The discussion includes rigorous derivations of key probability distributions, including the distribution of arrival times and the number of events. Special attention is given to clearing up common misconceptions about Poisson processes and their relationship with exponential distributions. The article provides both intuitive explanations and mathematical proofs, making it accessible to readers with basic probability theory knowledge while maintaining mathematical rigor.

Musings on the Gamma Distribution

In this post we demonstrate the derivation of the Gamma distribution through two different approaches. First, we show how the sum of \(n\) independent exponentially distributed random variables can be derived iteratively using convolution integrals. Starting with the case of \(n=2\), we explicitly calculate the probability density function and cumulative distribution function, then extend the result to \(n=3\) and generalize to arbitrary \(n\). Second, we present an elegant alternative derivation using Laplace transforms, leveraging their properties to convert convolutions into multiplications in \(s\)-space. Both methods arrive at the same result, showing that the sum of \(n\) independent exponentially distributed random variables is Gamma distributed.

Musings on the Exponential Distribution

This article presents a rigorous mathematical proof demonstrating why exponential distributions naturally emerge in memoryless physical processes. We show that the exponential distribution is uniquely characterized by its memoryless property through two distinct approaches. The first proof utilizes the survival function and its homomorphism properties, extending from natural numbers to rational and real domains. The second, more physics-oriented derivation employs differential equations to arrive at the same conclusion. We conclude by connecting these results to the hazard function formalism, providing an intuitive framework for understanding why exponential distributions are fundamental to describing microscopic systems such as radioactive decay.

Dirac delta with a Function inside

Dirac delta function appears frequently in physics. In certain cases, it takes a function as an argument. Such cases require care, and that is what we will take a quick look at in this post.

\(\text{Integral of the month: } \int dr \cos r^2\)

Fresnel integrals are a pair of integrals that are used to calculate the diffraction patterns of light waves. This post explores the mathematics of Fresnel integrals, a fundamental wave phenomenon that limits the resolution of optical instruments like telescopes. We first derive the results in the large \(r\) limit using the residue theorem. Then we will see how to evaluate the integrals numerically.

Diffraction

This post explores the physics and mathematics of diffraction, a fundamental wave phenomenon that limits the resolution of optical instruments like telescopes. Starting from Maxwell’s equations, we derive the wave equation in free space and develop the mathematical framework using the Huygens-Fresnel principle. We examine how light waves interfere through apertures, leading to characteristic diffraction patterns. The analysis includes interactive visualizations that demonstrate how parameters like aperture size, wavelength, and distance affect the resulting intensity distributions. Special attention is given to Fresnel integrals, which are essential for calculating these diffraction patterns.

Coil calculations

This article provides a comprehensive analysis of electromagnetic coil calculations, from basic principles to advanced mathematical treatments. We begin with simplified models to build intuition about magnetic fields in solenoids, then progress to more sophisticated analyses using elliptic integrals. The work covers magnetic field calculations both on and off axis, vector potential derivations, and practical considerations like resistance calculations and wire selection. Special attention is given to the effects of coil geometry, number of turns, and material properties on the magnetic field distribution and electrical characteristics. The analysis includes both analytical solutions and practical engineering considerations, making it valuable for both theoretical understanding and practical applications.

Separation of variables in spherical coordinates

This article presents a comprehensive derivation of the separation of variables technique applied to partial differential equations in spherical coordinates. We examine the process of decomposing the Laplace equation into its radial and angular components, leading to solutions involving spherical harmonics. The discussion includes a detailed analysis of the radial dependence, angular components, and their relationship through Sturm-Liouville theory. This mathematical treatment is fundamental to various physics applications, including quantum mechanics, electromagnetism, and gravitational field theory.

An introduction to conformal maps

This blog introduces conformal maps, which are complex functions that locally preserve angles between curves. We begin with the fundamentals of complex derivatives and the Cauchy-Riemann equations, demonstrating how these conditions lead to the Laplace equation. We explore the geometric interpretation of conformal mappings and their crucial property of preserving angles between intersecting curves. The article then shows how conformal maps can transform harmonic functions while preserving their harmonic properties, making them particularly useful for solving problems in electrostatics and fluid dynamics. We conclude with a practical example, using conformal mapping to solve for the electric field of an infinite line charge, demonstrating how these mathematical tools can simplify complex physical problems.

Radial Green’s Function in Cylindrical Coordinates

This document discusses the Radial Green’s Function in cylindrical coordinates, a fundamental concept in mathematical physics and engineering. The Green’s function serves as a crucial tool for solving differential equations, particularly in systems exhibiting cylindrical symmetry. We begin by deriving the Laplace operator in cylindrical coordinates, which is essential for understanding the behavior of physical systems. The focus is on defining the Green’s function as the solution to the corresponding differential equation, specifically in scenarios devoid of angular and axial dependencies, simplifying our analysis to radial dependence. The document explores the implications of the derived Green’s function in solving boundary value problems and its applications in various fields, including electrostatics, heat conduction, and fluid dynamics. Special attention is given to the careful handling of singularities and the application of Gauss’s theorem to validate the results.

Abrikosov-Nielsen-Olesen flux tubes

This article explores the physics of Abrikosov-Nielsen-Olesen (ANO) flux tubes, which are topological defects arising from spontaneously broken local symmetries in superconducting materials. We begin by examining how topological defects emerge from both global and local symmetry breaking, using examples from magnetic materials and superconductors. The mathematical framework of spontaneously broken U(1) gauge symmetry is presented, leading to the formation of vortices and flux tubes. We derive the Bogomol’nyi equations that describe these configurations and discuss their physical properties, including mass, central charge, and BPS saturation. The article provides insights into how these fundamental concepts connect quantum field theory with condensed matter physics.

Difference of two linearly distributed random numbers

We explore the statistics of the absolute difference between two random variables, each of which is linearly distributed within a specified range. We derive the probability density function of the absolute difference, \(S=|R_1-R_2|\). We employ various methods for simulating random numbers that adhere to the derived distribution, including inverse transform sampling and geometric approaches. The results are validated through graphical comparisons of the theoretical and simulated distributions.

Curl your Poynting vector

This blog post is a quantitative analysis of the concepts discussed in Veritasium’s “The Big Misconception About Electricity” video, and explores the dynamics of electrical energy propagation in transmission lines, with a particular focus on understanding how energy and signals travel from a source to a load. While the video suggests that energy flows through the electromagnetic fields surrounding the conductors via the Poynting vector, we provide a detailed mathematical treatment showing how the traditional circuit theory approach remains valid and complete for understanding energy transfer. We examine both ideal and lossy transmission lines, analyzing how voltage and current waves propagate when subjected to different load conditions including open circuits, short circuits, and reactive loads. Through mathematical analysis of wave equations and reflection coefficients, we demonstrate that the Poynting vector is not necessary for understanding energy transfer in circuits, and that the traditional circuit theory approach is sufficient for describing the energy flow in both DC and AC circuits. We also build an interactive plot to visualize voltage and current waves propagating in a transmission line.

The math of low pass filtered PWM

This article explores the mathematical principles behind low-pass RC filters, with a focus on their response to Pulse Width Modulation (PWM) signals. Through interactive visualizations, we demonstrate how these fundamental circuits process digital signals, showing the relationship between the time constant (RC), duty cycle, and the resulting output voltage. Readers can experiment with different parameters to understand how the filter smooths out PWM signals into analog voltages, making this complex topic both accessible and practical. This understanding is essential for anyone working with digital-to-analog conversion, motor control, or signal processing applications.

Interactive Timeline of JPL Missions

This interactive visualization presents a comprehensive timeline of Jet Propulsion Laboratory (JPL) missions throughout history. The timeline features an intuitive interface that allows users to explore missions by their operational status, mission type, and destination. Users can filter and analyze JPL’s diverse portfolio of space exploration initiatives, from planetary missions to Earth observation satellites. This interactive tool serves as both an educational resource and a historical record of JPL’s contributions to space exploration and scientific discovery.

Eigenvectors of \(\hat{n}\cdot\vec{\sigma}\)

In this blog post, we explore an alternative method for finding the eigenvectors of the operator \(\hat{n}\cdot\vec{\sigma}\), where \(\vec{\sigma}\) represents the Pauli matrices and \(\hat{n}\) is a unit vector. Rather than using conventional eigenvalue methods, we demonstrate how to obtain the eigenvectors through a series of rotations in spin space. This approach not only yields the correct results but also provides deeper insights into why the Pauli matrices transform as vector quantities under rotations.

Quantum scattering in one dimension

In this blog post, we explore quantum scattering in one-dimensional systems, focusing on the case of a rectangular potential barrier. We start by examining the Schrödinger equation in one dimension and then move towards a more efficient solution for the scattering problem. Rather than following the conventional approach of solving from left to right and imposing continuity conditions at the boundaries, we introduce a faster method by assigning coefficients directly from the transmitted wave and building in the boundary conditions. We cover both scenarios where the particle’s energy is below and above the potential barrier, and we detail how to calculate the transmission and reflection coefficients in each case. To simplify the calculations, we introduce dimensionless parameters, which allow us to rewrite the transmission coefficient in a more intuitive form. This also helps us identify resonance wavelengths, which occur when integer multiples of half-wavelengths fit within the potential barrier.