A PROBLEM IN THERMODYNAMICS AND STATISTICAL MECHANICS: ANALYTICAL AND NUMERICAL STUDY OF AN EINSTEIN SOLID-Analytical Solution

IMAGE CREDIT/OBTAINED FROM:  https://mappingignorance.org/2015/12/17/einstein-and-quantum-solids/ Quite some time ago, I had posted a numerical study of an Einstein solid and I now present the analytical study of an Einstein solid. As this was one problem in one of my problem sets while studying thermodynamics and statistical mechanics, one may find this exact problem in the following text: … Continue reading A PROBLEM IN THERMODYNAMICS AND STATISTICAL MECHANICS: ANALYTICAL AND NUMERICAL STUDY OF AN EINSTEIN SOLID-Analytical Solution

Basics of Tensor Calculus & General Relativity|A Digression into Special Relativity

So far in this series I have given the definitions of vectors, scalars, tensors, and manifolds. As a result, much of this series has been mostly mathematics and not necessarily physics. To that end, the purpose of this post is to develop the salient points of special relativity. Namely, the intention of this post is … Continue reading Basics of Tensor Calculus & General Relativity|A Digression into Special Relativity

Basics of Tensor Calculus and General Relativity: An Introduction to Manifolds and Coordinates

SOURCE FOR CONTENT: General Relativity: An Introduction for Physicists, Hobson, M.P., Efsttathiou, G., and Lasenby, A.N., 2006. Cambridge University Press.  D-Dimensional Hypersphere and Gamma Function: Introduction to Thermal Physics, Schroeder D.V. 2000. Addison-Wesley-Longmann.  IMAGE CREDIT: NASA/JPL.  The intended purpose of the post is to introduce the concept of manifolds in the context of physics (mathematicians beware!). Furthermore, … Continue reading Basics of Tensor Calculus and General Relativity: An Introduction to Manifolds and Coordinates

Derivation of the Euler-Lagrange Equation for a Function of Several Dependent Variables

IMAGE CREDIT: NASA/JPL SOURCE FOR CONTENT: Classical Dynamics of Particles and Systems. Thornton and Marion. 5th Edition.    Consider a functional $latex \displaystyle \phi = \phi(y_{\mu},y_{\mu}^{\prime}; x), (1)$ where $latex \mu = 1,2,...,n$. By the method used in a previous section of the aforementioned text, we may write $latex \displaystyle y_{\mu}(\alpha, x) = y_{\mu}(0,x) +\alpha \eta_{\mu}(x). (2)$ … Continue reading Derivation of the Euler-Lagrange Equation for a Function of Several Dependent Variables

Astrophysics Series: Derivation of the Total Energy of a Binary Orbit

SOURCE FOR CONTENT: An Introduction to Modern Astrophysics, Carroll & Ostlie, Cambridge University Press. Ch.2 Celestial Mechanics Here is my solution to one of the problems in the aforementioned text. I derive the total energy of a binary system making use of center-of-mass coordinates. In order to conceptualize it I have used the binary Alpha Centauri … Continue reading Astrophysics Series: Derivation of the Total Energy of a Binary Orbit

A Problem in Thermodynamics and Statistical Mechanics: Analytical and Numerical Study of an Einstein Solid

Every physics major at some point in their undergraduate career takes a course in thermodynamics and statistical mechanics. One of my problem sets included a problem that considers an Einstein solid with 50 oscillators and 100 units of energy and then increases the number of oscillators to 5000. I will be presenting my solution to … Continue reading A Problem in Thermodynamics and Statistical Mechanics: Analytical and Numerical Study of an Einstein Solid

A “Proof” of the Sturm-Liouville Theorem/Problem

IMAGE CREDIT: NASA/JPL: This shows Jupiter's Great Red Spot; a storm that has been occurring for over 300 years now. Quite recently, however, observations show that the Spot appears to be shrinking in size.  About a week ago, I was looking through my notebooks and came across an unfinished problem posed by one of my … Continue reading A “Proof” of the Sturm-Liouville Theorem/Problem

A Narrow, Technical Problem in Partial Differential Equations

While I was in school, one of my professors set this problem to me and my classmates and challenged us to solve it over the next few days. I found the challenge intriguing and it fascinated me, so I thought it was worth sharing. The problem was this: Show that $latex \displaystyle v(x,t) = \int_{-\infty}^{\infty} … Continue reading A Narrow, Technical Problem in Partial Differential Equations