Copasetic Flow

This installment of “It’s Obvious. Not!” looks at: Book: “Statistical Mechanics” Edition: second Authors: R.K. Pathria Publisher: Elsevier Butterworth Heinemann Page: 14 Section 1.3 follows the derivation laid out in 1.2 , but with a variable volume to arrive at: As in 1.2, a parameter is defined that must be equalized: The above subscript notation indicates that N and E are held constant and that V is equal to its equilibrium value. The basic formula of thermodynamics is then stated as: Immediately after which, it is stated that: But how? First, remembering that N and E are constant, rewrite the basic formula above as: Now, using the expression from section 1.2 relating the micro and macro states: We can write: However, so,

While doing my statistical mechanics homework today I arrived at a sum that looked like Sums like this come up frequently when you're working with random walks, or flipping coins, or counting states of quantum mechanical systems where only two energies are allowed, or in any other number of contrived situations. It's almost a good looking sum because everything to the right of the factor of m looks like the sum for a binomial expansion : which simply evaluates to: It turns out that there's an easy way to get the factor of m out of the sum and get on with your life! First notice that: so that the sum can be re-written as: So, we rather handily got rid of the factor of m. The extra factor of p can be taken outside of the sum since it has nothing to do with the summation index m. Furthermore, the order of summation and differentiation can be interchanged to arrive at: Now the sum actually is a binomial expansion and after simplifying and performing the derivative that was i...

This installment of “It’s Obvious. Not!” looks at: Book: “Statistical Mechanics” Edition: second Authors: R.K. Pathria Publisher: Elsevier Butterworth Heinemann Page: 12 The idea in Pathria is to start with a function for the number of microstates as a function of energy and then maximize it to study the implications of a system in equilibrium, (the maximum number of microstates). Pathria skips a few steps in the differentiation and simplification. There shown below to help me and others along. Have fun! Starting with Maximize with respect to . Keep in mind that is a function of : Use the chain rule of differentiation to expand the second partial derivative: The last derivative term simplifies to -1: So, to maximize we have: Now, consider the differentiation of a function . The chain rule gives: Applying this to our result above, we get:

I just saw a link to this article come across the HAMRADIOHELPGROUP . NASA will be collecting data from two observer satellites on either side of the sun and would like the help of ham radio operators to collect the data on 10m. Anyone with a 10m dish antenna is welcome. "St. Cyr notes that experienced ham radio operators can participate in this historic mission by helping NASA capture STEREO's images. The busy Deep Space Network downloads data from STEREO only three hours a day. That's plenty of time to capture all of the previous day's data, but NASA would like to monitor the transmissions around the clock. "So we're putting together a 'mini-Deep Space Network' to stay in constant contact with STEREO," says Bill Thompson, director of the STEREO Science Center at Goddard. The two spacecraft beam their data back to Earth via an X-band radio beacon. Anyone with a 10-meter dish antenna and a suitable receiver can pick up the signals. The data rat...

Photo Credit I'm exploring piezoelectric cyrstals, (crystals that produce a small electric potential when mechanically strained). I stumbled across a great concise history [pdf]. The outline is shown below. I also found an interesting summary of another amateur's experiments

I got a quick lesson on the importance of graphing my work this afternoon. After diligently calculating the expectation position of a particle as predicted by the Schrodinger equation: I came up with the value 1/2λ. After a little bit of thought, it occurred to me that the Schrodinger equation was symmetric about 0 and that if I was guessing the expectation, or average, value for the position of a particle predicted by the equation, I'd guess zero. So, I graphed the Schrodinger equation for the potential. Just for good measure, I went ahead and graphed the integrand of the expectation integral as well. Sure enough, the Schrodinger equation is symmetric about 0 and the areas under the expectation integrand curve to the left and right of (x=0) cancel out. Upon checking my work I realized that I was missing a factor of x in my original integration. I multiplied the x back in and everything worked out just like the graph said it should! Not bad for taking a 20 year break between ...

This installment of “It’s Obvious. Not!” looks at: Periodical: “Physical Review” Volume: 21 Page: 483, 486 Title: "A Quantum Theory of the Scattering of X-Rays by Light Elements" Author: Arthur H. Compton Excerpt from page 486: In the above excerpt, Compton discusses how to calculate the momentum of an electron that caused x-ray or gamma scattering. The momentum added to the electron is the momentum of the incident photon minus the momentum of the scattered photon. Problem: The angle of scattering, (theta), appears to be mislabeled in the above figure vs. the usage of the angle in formula 1. In formula 1, Compton calculates the magnitude of the electron momentum as the vector difference of the momentum of the incident and scattered photons. To subtract two vectors, you place their tails together, the resulting vector that points from the head of the second to the head of the first is the difference vector as shown below and described in this Wikipedia article . To get t...