User:Brian P. Josey/Notebook/2010/07/08

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Focusing on One Task at a Time
As I'm sure my notebook portrays, I have the habit of bouncing around between various different ideas and working on a couple of different projects at a time without focusing on just one. This works for me because it keeps things fresh and interesting, but the down side is that it hurts my productivity and gives me a periods where it feels like I have very little to show. So I'm trying a new approach today, and I am just setting down to do one thing at a time and not three. I have three main projects that I want to do, and that will point me in the right direction:


 * 1) Understanding Brownian Motion I still do not quite comfortable with Brownian motion yet, and I still have a lot of questions about it that I want to answer. While I understand the basic idea behind it, I don't know enough to tackle all of the questions that pop up while working with the emulsions.
 * 2) FEMM model I started to create a FEMM model of the mounted magnet and objective  last Tuesday, but I haven't finished it yet. I need to have a working one soon to serve as a theoretical basis for my LabVIEW project. When I calculated the force on ferritin by hand, I returned results that where smaller than I had calculated in my previous models, so I want to account for the unique set up that I have begun to work with.
 * 3) LabVIEW As I stated before, I want to create a tracking software that measures the force on ferritin on my own. Larry offered to build it for me, but with his help I am going to do most of the work on my own. I've been wanting to learn how to use LabVIEW for a couple of months now, and what I did yesterday showed me that I enjoy working on it, plus it would be beneficial for myself to be familiar with it. Because I'm still learning the language, this task will take the longest out of the three, but will probably pay off the most in the long run.

I am going to attack them in the same order as above. To have my LabVIEW project working, I need to account for Brownian motion and have some calculated forces from the FEMM model to compare too. So I am going to start with Brownian motion today, and hopefully finish it before the weekend. I am going to post my notes here, even though they are nothing new, but for myself. I'll be following the Biological Physics book and some articles from online so that I can refer back to them if I need to later.

Ch. 3 The Molecular Dance
Normalizing Condition

$$ \sum_{i} P(x_i)= \frac {N_1+N_2+...} {N}= \frac {N} {N} = 1 $$

P is the probability of observing xi, and Ni is the number of instances measured at xi with N being the total number of measurements. The normalizing condition means that the probability of observing some value is 100%. If a probability distribution satisfies this, then it is properly normalized.

Continuous Distributions For a continuous distribution, the normalizing condition is:

$$ \int P(x) dx =1 $$

For example, you have a Gaussian distribution of probability:

$$ P(x) = A e^{ \frac {-(x-x_0)^2} {2 \sigma^2}} $$

Which is not normalized, and using the normalization condition, you get the probability distribution as:

$$ P(x) = \frac {1} {\sigma \sqrt {2 \pi}} e^{ \frac {-(x-x_0)^2} {2 \sigma^2}} $$

Graphing the Gaussian distribution, also know as the normal distribution creates a bell-curve that is centered on x0, and σ is the variance of the graph and controls both the height and width of the graph. The expectation value,  is the average value that you would measure going across the whole set of data. It generally will not equal x0, and can be calculated from:

$$ \langle x \rangle = \begin{cases} \sum x_i P(x_i), discrete \\ \int x P(x) dx, continuous \end{cases} $$

There is then some spread in the values that is measured by the root-mean-square deviation, or standard deviation, given by:

$$ deviation = \sqrt { \langle (x-\langle x \rangle )^2 \rangle } $$

There are some rules on how to calculate the probability of multiple events occurring. The first is the addition rule, which states that for exclusive events, like being taller then six feet or shorter than five, the probability of being either is simple the sum of the probabilities of the two (or more) cases. This works for both the discrete and continuous cases. For two discrete cases, xi and xj the probability is given by the sum, P(xi) + P(xj). For the continuous case over the intervals, a to b and c to d, the probability is:

$$ \int_a^b P(x) dx + \int_c^d P(x) dx $$

This of course can be extended and generalized for n cases.

The multiplication rule accounts for cases that are independent, like the chance that a coin would land on heads while a die rolls a six. The joint probability distribution for two independent events is the product of the two cases. With the coin and die example, there are twelve different cases: two for each number on the die corresponding to the heads and tails cases.

Ideal Gas Law and Boltzmann
Given by:

$$ p V = N k_B T $$

Where the product of kBT has units of about 4.1 pN nm.

Imagine that there is a box containing N molecules that are bouncing around and not colliding with each other. They exert a pressure on the wall that comes from the elastic collision of the molecule and the wall. Even if the molecules are all moving at different rates, and even glancing at different angles, the pressure can be determined by the changes in momentum that are perpendicular with the wall in question. This gives a pressure, p:

$$ p= m \langle v_x ^2 \rangle \frac {N} {V} $$

where x is measured as perpendicular to the wall. Plugging this into the ideal gas law, the relationship gets simplified to:

$$ m \langle v_x ^2 \rangle = k_B T $$

Then this can be used with the fact that there are three coordinate directions to show that the average kinetic energy of the air molecule is equal to three halves of kB T.

Boltzmann Distribution

The Boltzmann distribution is:

$$ P(state) \propto exp{ \frac {-E} {k_B T}} $$

There are some interesting points that come into play after the bit on the Boltzmann distribution. The first is a discussion of evaporating water in a pan. When you are heating up a pan of water, all of the molecules have a distribution of energies, and in turn velocities that follow the Boltzmann distribution. Because most of the water molecules are moving slower than the speed they need to move at to evaporate, the water doesn't instantly turn into vapor. Only the water molecules that are near the surface and have an energy that can compensate for the binding energy can escape. This is called the activation barrier. Any water molecules that overcome this barrier rob the system of just a little energy, and actually cool it off. Adding more and more heat to the system moves the peak of the distribution towards the activation barrier and evaporation occurs more rapidly.

The second interesting fact is when a molecule with a very high energy is added to the system. Because of its interactions with the other particles, it eventually slows down and moves into the large portion of the bell-curve.


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