Saturday, January 30, 2010

Compactified Dementia - The hierarchy problem of cranks

Over the last couple of days I have come across several blogs (and tweet's) relating to cranks (crackpots) and how they choose to interact with scientists, the general public, and themselves.
I have had a special place in my heart for crackpots (cranks, kooks, nutters) for the last 15 years since running into them abruptly on Usenet (now google groups). I have framed my very first snail mail crackpot letter I received shortly after publishing a paper in Nature. The rest I have consigned to a cardboard box.

There are many useful web sites and resources out there regarding cranks, and I very highly recommend reading Underwood Dudley's fantastic books on cranks.

 


I have battle cranks for many years, but some comments on a blog about how cranks never seem to communicate with other cranks reminded me of a Usenet post I made several years ago. At the time I was doing research on short range tests of gravity, which would be able to put constraints on the size of compactified extra dimensions according to some string theory models by Arkani-Hamed et. al. I decided to write the following post. Enjoy. =)



Compactified Dementia - The hierarchy problem of cranks




The hierarchy problem of cranks is an interesting topic. Why are their arguments so weak compared to the average person? One answer might be that they reside in their own dimensions, compactified and placed out of the way somewhere.
Many people think that cranks (crackpots) live in their own universe, a universe that has nothing to do with our own. However, the Yankovich theory states that universes cannot interact directly. Since it is empirically obvious that cranks do in fact interact with our universe, this would mean that cranks in fact reside somewhere in our universe. This would mean that crackpots (cranks) live in their own compactified dementias, with bits of their crankiness "leaking" across to our regular Space-time.
There are some interesting properties associated with those compactified dementias.
1. There are absolute reference frames in Dementia-Space. This allows such things as faster than light travel, and the Ether.
2. Any GPS device entering such a Dementia is rendered useless.
3. Computer Keyboards have broken shift keys.
4. While communication from Dementia-Space is easy, communication into Dementia-Space is very difficult. In fact, one could postulate that there is no information transfer into Dementia-Space. Further tests are needed.
5. Residents of Compactified Dementias are very xenophobic. They think that occupants of regular space-time are in a vast conspiracy to "Hide the truth" and "Steal tax payers money".
6. Mathematics in Dementia-Space do not allow proper multiplication of negative numbers, but seem ideally suited to solve Fermat's last theory.
7. Time does not exist.
8. Clocks are broken.
9. The second law of thermodynamics does not hold in Dementia-Space.
10. Intelligence in Dementia-Space is dictated by the NIT. [1]
11. Diversified population makeup. [2]
This is but a short list. At this time, very few studies have been made into the fundamental physics of Compactified Dementias. At this time, most study has been simply to categorize (and catalog) the crank species that are emitted. For example, the list compiled by OM [3], while interesting, does not in fact explain why the interaction length of a crank is so small. Usually, crank interaction is limited to such an extent that the general population is rarely aware of cranks. Cranks have little effect on the physical world, much less any socio-political clout.
There is a new parameter space to explore when it comes to Compactified Dementias.

[1] M. Crank, Negative Intelligence, sci.physics, March 3, 2003 http://www.google.com/groups?&selm=oNwda.123485%24L1.17357%40sccrnsc02
[2] www.crank.net
[3] http://www.google.com/groups?as_q=crank%20list&as_uauthors=old%20man

Friday, January 29, 2010

Meniscus Fermi


As you have probably guessed from a previous post, I find that dimensional analysis is a powerful tool for back of the envelope type problems.

Therefore I present this problem on surface tension as an exercise in dimensional analysis. Of course, most dimensional analysis solutions, as with most solutions to Fermi problems, are order of magnitude. So after having some fun with the following problem, see if you can refine your model a bit.

---------- Start Fermi Problem -----------

Add water to a clean glass, and you will notice that the water likes to "climb" up the wall to a certain height. This is called a Meniscus, and is caused by a physical property called surface tension.



You can put some water in an ice tray, and freeze it, and you will be able to see the frozen meniscus as a raised lip on the top surface of the ice cube.

Estimate how high the meniscus is for water.

Here is a hint for you: 
The surface tension for water is about 0.07 N/m

------------ End Fermi Problem --------------

At the end of the day, you can do an experiment to help refine your model. Freeze some ice, and measure the height of the meniscus.

Monday, January 25, 2010

Liquid Crystals and the Buckingham Pi Method





Physicists have at their disposal many mathematical tools to help them solve problems. However, some of the most powerful are dimensional analysis methods.

Dimensional analysis not only helps you check if a formula you derived (or copied down to your test sheet if you are a student) is correct at least dimensionally, but it can help you estimate the answer to many physical systems.

For example, say you are a physicist who is trying to understand how their watch liquid crystal display works.
They have done some reading in the popular press and have found that a watch LCD is probably made of a twisted nematic liquid crystal cell, which can be switched with a magnetic field. (Or electric field).
The physicist reads up on Wikipedia about liquid crystals, how energy efficient they can be, how they are switched on and off, how polarizes are used and that nematic is derived from the Greek nema, which means thread. After some more reading our physicist starts wondering how strong of a magnetic field is needed to switch the LC cell.



Of course one can simply look up the answer online, but our intrepid physicist decides to use dimensional analysis, and more specifically the Buckingham Pi method to determine the general form of the strength of the magnetic field.

Lets start with an aside on the Buckingham Pi method.

Suppose you are attempting to describing a physical system with a physically meaningful equation of n measurable variables.

f\left( {{x_1},{x_2}, \ldots ,{x_n}} \right) = 0

where each {x_n} is expressed in terms of k independent physical units.

The above equation can be restated in terms of p = k - n dimensionless parameters constructed from {x_n} of the form

\pi  = x_1^{{m_1}}x_2^{{m_2}} \ldots x_n^{{m_n}}

where each {m_n} are rational numbers.

From this we can construct the new equation:

F\left( {{\pi _1},{\pi _2}, \ldots ,{\pi _p}} \right) = 0

Note that this solution is not unique, and is only one of a set of dimensionless parameters you can generate with your measurable variables in terms of physical constants. This method does not choose the best form, or even the most physically meaningful form. But it is very useful none the less.
Note that independent physical units are those such as length, time, mass etc. The fundamental SI units (of which there are seven) represent such a set of independent units.

New, lets see how we can use this method to determine a physically meaningful form for an equation to determine the magnetic field strength needed to switch the LCD cell in question.

Consider that we have a cell full of a nematic liquid crystal. The liquid has an elastic constant K, which is measured in Newtons. The LC is also diamagnetic in that it will anti-align with the magnetic field. How strongly the little "threads" or "nemas" in the LC will interact with an external magnetic field has to do with the diamagnetic susceptibility of the LC, {\Delta \mu } .
The "threads" in the LC are twisted, which means that the "threads" in each layer of the LC is rotated, or "twisted" a bit from the direction the previous layer is facing. Picture a DNA molecule, or a helix spiral staircase. This means that the thickness, l, of the cell that is holding the LC is an important variable, as each layer will require a stronger or weaker magnetic field to anti-align the "threads".

In general terms, the little threads like to align along the direction (actually opposite) with the external magnetic field. But the restoring force of the liquid surrounding the threads want to keep this from happening. Sort of like trying to rotate raisins in a bowl of jello. Unless you hold the raisin in place, they spring back to their old alignment due to the elastic constant of the jello.

These little threads act as a polarizer, and you can rotate this polarizer using an applied magnetic field. Crossing the LC polarizer with another fixed polarizer stops light from being transmitted through the LC cell. This is switching the LCD ON.




So, using dimensional analysis, the general form of a dimensionful equation is:

H \propto {l^\alpha }{K^\beta }\Delta {\mu ^\gamma }


We then construct a set of dimensionless quantities from this equation, and equate the exponents to determine the final form.
Lets examine each variable, and determine its dimensional quantities.

H -> magnetic field -> Amperes/meter -> [A][L^-1]

K -> elastic constant -> Newtons -> [M][L][T^-2]

delta_mu -> diamagnetic susceptibility -> Henry/meter -> [M][L^2][T^-2][A^-2][L^-1]

So, equating the fundamental SI units we get:




Equating exponents we get a set of 3 equations and 3 unknowns.




Solving this set of equations we get:




Therefore the dimensionful equation is:





Now, lets do a sanity check; Does this equation make sense?

First, the units check out. So the equation is dimensionally correct. Check.

As the elastic constant K increases ("jello" force increases), it makes sense that you would need a stronger magnetic field to rotate the "threads" in the LC. Check.

If a larger {\Delta \mu } means a greater interaction with the magnetic field. This means that a weaker magnetic field can couple to the "threads" and overcome the "rubberyness" of the LC, and therefore the magnetic field does not have to be as strong to rotate the "threads". Check.

As the thickness of the cell increases, there are more "threads" that are already nearly aligned to the ON position. The more threads that are aligned, the less light makes it through the cross polarizers, and therefore a weaker field is needed to switch ON the LCD. Check.

So, the equation makes physical sense, and is dimensionally correct.

Our physicist uses their google-fu and determines that indeed this is the correct form, up to a multiplicative constant.

Of course, there may be a reliance on other variables, such as temperature, on how the LC behaves. More than likely these are subsumed into K, but can be put into our model later.

This is a very simplistic treatment of the Buckingham Pi theorem, as well as of twisted nematic liquid crystals. However, I hope the power of dimensional analysis is evident in this example.