Boltzmann and Fourier for biologist

[nucleargirl]

Hello,

Please could someone explain to me about Boltzmann distribution and Fourier transformation? or point me in the direction of some really really easy-to-understand guide?

I need to understand it for biology - to understand NMR and mass spectrometry.

Thanks.

 

[Bacle2]

Have you tried the Wikipedia pages? They are usually a good intro., and you

can come back here with more specific questions which makes it easier both

foryou, and for us to help you.

 

[nucleargirl]

I have tried wikipedia, it was confusing and complicated! But I'll try to read it again.
I just need a basic understanding of what they are and what they do.

 

[I like Serena]

Hi.

Is this the sort of thing you're looking for?

http://www.cs.unm.edu/~brayer/vision/fourier.html

It shows how an image can be transformed with Fourier to a spectrum.
Then some filtering can be done.
And then it can be converted back into an image.
The link explains more about how it works and what it does.

It's about applying the Fourier transform to images, which may be what you'd do in NMR.



The other typical application of Fourier is to signals and their conversion to a frequency spectrum.
This applies to spectrometry that you also want to know about.
But they don't have such cool pictures.

 

[Muphrid]

The Boltzmann distribution says that if you have a system with a well defined temperature and a series of energy states, the probability that the system is in a given energy state is proportional to $\exp(-E/kT)$. Lower energy states are always preferred, but only on a statistical level, and as temperature rises, all states become nearly equally populated.

The Fourier transform is a way of breaking down an arbitrary function (signal, wave, whatever) into a continuous set of sine or cosine functions. The transformed function is the amplitude of the sine or cosine at any particular frequency.

 

[nucleargirl]

The other typical application of Fourier is to signals and their conversion to a frequency spectrum.
This applies to spectrometry that you also want to know about.
But they don't have such cool pictures.

yes, this is what I need to know - conversion of a signal into some kind of graph. what are the important aspects of this?

 

[nucleargirl]

The Fourier transform is a way of breaking down an arbitrary function (signal, wave, whatever) into a continuous set of sine or cosine functions. The transformed function is the amplitude of the sine or cosine at any particular frequency.

hm, interesting, I vaguely remember sines and cosines from school... how does it do this?
did not understand the Boltzmann explanation at all...

 

[Muphrid]

Sines and cosines have a special property that, for two frequencies $\omega_1$ and $\omega_2$,

$$\int_{-\infty}^{\infty} \sin(\omega_1 t) \sin(-\omega_2 t) \; dt = 0$$

unless $\omega_1 = \omega_2$. The Fourier transform takes advantage of this "orthogonality" to decompose a signal into a bunch of sines and cosines, each with their own amplitude.

As for Boltzmann, here's a simpler idea: you have some system that can be in a bunch of energy states. This could be a molecule, for example, that can have different configurations. Some configurations are more stable (lower energy); others are less stable (higher energy).

The Boltzmann distribution says that lower energy states are always more likely than higher energy states, but with higher temperatures, the difference in these likelihoods shrinks.

 

[I like Serena]

yes, this is what I need to know - conversion of a signal into some kind of graph. what are the important aspects of this?

Didn't you like the pictures?

As for spectrometry, consider the light of the sun - it is white.
But it is not really white, it contains all the colors of the rainbow and many invisible colors all mixed together.
If you put it through a prism it disperses into a spectrum.

Basically a Fourier transform does the same thing.
It's a complex mathematical method that converts a signal into a spectrum.
Then you can see how much of each color is in it.

Specifically in NMR and mass spectometry an object is treated in such a way that it emits (or absorbs) radiation.
The radiation is caught in sensors that record it as a signal in a computer.
With a Fourier transform the signal is converted to a spectrum so you can see how much of each wavelength of radiation is in there.

Different atoms will emit different radiation at specific wavelengths.
From the spectrum you can deduce how much of each atom is present in the object.

Here's a page that explains in more detail:
http://en.wikipedia.org/wiki/Fourier_transform_spectroscopy

 

[Bacle2]

Excellent answers by both; better than anything I would have come up with.

 

[nucleargirl]

Thanks for trying to help :)
I think I will never really understand... but its ok. its like knowing to use the tv without understanding how it works...

 

[Edgardo]

The basic idea behind a Fourier transformation is to represent a function f(x) as a sum of other functions b₁(x), b₂(x), ..., b_n(x):
f(x) = c₁*b₁(x) + c₂*b₂(x) +...+ c_n*b_n(x)
The c's are the coefficients.


Let's take a very easy example:
Suppose you are given the functions
b₁(x) = sin(x)
b₂(x) = sin(2x)
b₃(x) = sin(3x)

Is it possible to build the following function
f(x) = 7*sin(3x) + 5*sin(2x)

by using b₁(x), b₂(x), b₃(x)?

It is possible:
f(x) = 0*b₁(x) + 5*b₂(x) + 7*b₃(x)

The numbers 0, 5 and 7 are the coefficients. They tell you "how much" of each b-function you need in order to represent f(x). These coefficients are what you see in the Fourier-spectrum. In other words the Fourier transform determines the numbers 0,5 and 7 for you.

E.g. you have a signal and observe how it it behaves in time (perhaps from NMR). Suppose for this signal you want to know if it is possible to represent it as a sum of the b-functions. Then you apply the Fourier transform which gives you the numbers 4, 1, 3.

These appear as peaks in the Fourier spectrum:

x
x       x
x       x
x   x   x
----------------------------------------------------

These peaks just tell you how much of each b-function you need to construct your signal.