Electromagnetism Basics in 1 or 2 Dimensions

This was a post that I had been thinking of doing for a while, but I couldn't get around to it until now. A lot of introductory electricity & magnetism problems constrain charges to only move in 1 or 2 dimensions, but in reality the constraint existed within a 3-dimensional space. I thought that would cover the bases for electrodynamics in 1 or 2 dimensions, but then I saw that in cylindrical coordinates, the order-0 multipole moment outside a line charge is $\phi \propto \ln(r)$ as opposed to $\phi \propto \frac{1}{r}$. That made me realize that there is in fact a distinction among 1 or 2 or 3 dimensions. In all of the following, I will make use of the conventions and relations \[ x^{\mu} = (ct, x, y, z) \\ \partial_{\mu} = \left(\frac{1}{c} \frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \\ \eta_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ F^{\mu \nu} = \begin{bmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & B_z & -B_y \\ -E_y & -B_z & 0 & B_x \\ -E_z & B_y & -B_x & 0 \end{bmatrix} \\ \partial_{\nu} F^{\mu \nu} = \frac{4\pi}{c} J^{\mu} \\ \epsilon_{\mu \nu \zeta \xi} \partial^{\nu} F^{\zeta \xi} = 0 \\ \mathbf{F} = q\left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right) \] in 3 dimensions, with Einstein summation and CGS implied (with more on that last point nearer to the end), with Latin indices representing only spatial components, and with Greek indices representing spacetime components. Also note that the fully antisymmetric tensor $\epsilon$ has $n$ Latin indices in $n$ spatial dimensions and $n+1$ Greek indices in $n+1$ spacetime dimensions; for example, in 2 spatial dimensions, the antisymmetric tensor over only space looks like $\epsilon_{ij}$, while over spacetime it looks like $\epsilon_{\mu \nu \xi}$, and I will frequently switch between the two as needed. Follow the jump to see what happens.


Gibbs Entropy and Two-Level Systems

Today, I was browsing through the MIT news page when I saw this article about how two mathematicians claim to have disproved the notion of negative temperature. My heart sank, because one of the coolest things I remembered learning in 8.044 — Statistical Physics I was the notion of negative temperature existing, being hotter than hot, and being experimentally realizable. I also became confused when the article referred to Gibbs entropy, because the definition I thought was being used for Gibbs entropy was \[ S = -\sum_j p_j \ln(p_j) \] which is exactly equivalent to the Boltzmann entropy \[ S = \ln(\Omega) \] where \[ p_j = \frac{1}{\Omega} \] in the microcanonical ensemble. I figured this would mean that the Gibbs entropy would exactly reproduce negative temperature results in systems with bounded energies such as two-level systems. I wasn't able to read the most recent paper as discussed in the news article, because it is behind a paywall, but I was able to read this article by the same authors, which appears to lay the foundational ideas behind the most recent paper. It seems like on my end, the misconception appears to hinge on what one would call the Gibbs entropy. The formula \[ S = \ln(\Phi) \] appears to be the correct one for the Gibbs entropy, where $\Phi$ is the total number of states with energy not greater than $E$ and $\Omega = \frac{d\Phi}{dE}$ is the number of states with energy exactly equal to $E$ quantum mechanically (or the number of states with energy within a sufficiently small neighborhood of $E$ in the classical limit). With this in mind, follow the jump to see how this might work for a two-level system and explore the other implications of this new definition of statistical entropy. (UPDATE: Note that in all of this, $k_B = 1$.)


Featured Comments: Week of 2013 December 15

There was one post this past week that got a few comments, so I'll repost all of those.

Review: Linux Mint 16 "Petra" Cinnamon + MATE

Reader Michael Freeman said, "Compiz doesn't work in Petra? It works fine in Maya LTS with MATE 1.6. What version of Compiz comes with Petra? Do you have all the needed packages installed?"
An anonymous commenter had this to say: "I must say that I am surprised by your assessment of Cinnamon 2.0. I have Petra installed and have been using it as my daily driver since the day it was released. I find it to be incredibly stable and not at all laggy, but rather snappy. Of course, everyone's hardware is different. Give it a go in a VM if you have the resources, you may find you like it. Cheers!"
Another anonymous reader followed up on the first comment: "Compiz worked fine for me on petra 32bit. First of all u have to install all the packages needed. Then you have to set an autostart entry like 'cinnamon --replace' and change the default desktop manager from 'marco' to 'compiz' using the dconf editor. If you're using live session without a persistance file to save settings, you have to install and run it manually every time you boot. check here - http://community.linuxmint.com/tutorial/view/1298".

Thanks to all those who commented on that post. I'm at home for break now, but I don't have any posts particularly planned. Anyway, if you like what I write, please continue subscribing and commenting!


Review: Linux Mint 16 "Petra" Cinnamon + MATE

Cinnamon: Main Menu
This is the second review that I'm doing at the moment. Linux Mint 16 "Petra" came out in MATE and Cinnamon guises recently, so as a fan of Linux Mint, I'll be reviewing those now. I tried each edition separately on a live USB made with UnetBootin. Follow the jump to see what each is like.


Done with 7th Semester!

It finally happened! The end of the semester rushed in and washed over just as quickly. My classes — 8.07 — Electromagnetism II, 8.09 — Classical Mechanics III, 8.333 — Statistical Mechanics I, and 14.12 — Economic Applications of Game Theory — were all together a bit more challenging than I anticipated. On top of that, I worked a lot on my UROP, and of course I had to submit graduate school applications by last weekend. Thankfully, my classes and graduate school applications are done. Now I can go home, relax, enjoy the company of family and friends...and probably work on my UROP. Of course, I'll be continuing my UROP over IAP, but I hope to be transitioning into a new project then, so I hope to get a fair amount of my current project done during the break. Happy holidays everyone!


Featured Comments: Week of 2013 December 8

There was one post that got a couple of comments, so I'll repost both of those.
Review: openSUSE 13.1 GNOME
Reader Kaf Shiel said, "Thanks for the excellent review. I tried the live usb openSUSE 13.1 RC with the Gnome desktop but it wouldn't boot to the desktop. Then I found out online that it was a bug affecting more users. After reading your review I'll give it another go. I have a friend that uses openSUSE and he swears by it so I'm curious, although I think that it's not for distro hoppers. It's more like a full install commitment. I use Ubuntu Studio (I'm an amateur musician) and Voyager (both have the Xubuntu engine) in both my workstation and netbook because I really love the Xfce desktop but I like to test other distros for fun (tested Tails 0.22 today). I love your blog and understand that your academic life won't give you enough time to write every week but when you post something new I always come here to read. Merry Christmas!"
Commenter alcalde had this to say: "I wish reviewers would stop reviewing the live CDs. That's Ubuntu thinking, and OpenSUSE is not Ubuntu. The Live CDs are not the default or preferred installation medium and are not tested to the same degree as the DVD installer, nor are they even close to being the most downloaded versions of OpenSUSE. They're presented for convenience only. When you use these live CDs you do not get the full OpenSUSE install experience. Unlike the CDs, the DVD offers many more options, including the ability to customize the install by choosing which packages to (not) install. Calc is included in a normal install of OpenSUSE; this is another artifact of trying to squeeze OpenSUSE onto a live disk. There are some other differences, including the DVD installing Flash during the update process, being able to update before being dropped into the desktop, etc. Why would one want to recommend GNOME for newbies? Far and away KDE is going to most resemble what they're used to already."

Thanks to both of those people for commenting on that post. I have another review written up that will be out this week. In addition, this week is my week of final exams, so I'll be writing a reflection on the semester at the end of that. Anyway, if you like what I write, please continue subscribing and commenting!


Review: openSUSE 13.1 GNOME

GNOME Activities Overview
I haven't been able to write up any reviews recently because of the confluence of classes, UROP, and graduate school applications. Now my classes are sort of wrapping up, in that my last problem sets were due yesterday (the publication date is after the date of writing), so I have a little time to relax and do some reviews. The first is openSUSE 13.1 GNOME. I've reviewed openSUSE before a number of times, so I won't try to introduce it again. I tried the live version of the GNOME edition on a live USB made with MultiSystem. Follow the jump to see what it's like.