qw098
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Boltzmann distribution
Hi guys,
I have a system of particles in different energy levels. I know how many particles are in each energy level.
I was wondering how I could find out if the distribution of these particles was in a Boltzmann distribution or not?
Thanks!
In other words, how do I determine if a distribution is a Boltzmann distribution or not?
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qw098
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All I am given as data is: That the system contains 38 particles with three equally spaced energy levels (0 J, a J, and 2a J) and I know that the
population distribution is: A(18,12,8).
How do I determine if this is a Boltzmann distribution or not?
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watson.fawkes
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Wikipedia on the Boltzmann distribution has enough of a description in its first equation for you to determine yourself that your problem statement is ill-posed.
Is this a homework problem?
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qw098
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Thank you for the response Watson.
No, the question is not a homework question per se, just a question I found on the internet when reading about the Boltzmann distribution, and
probablity.
Well I know that equation on Wikipedia quite well Watson. However, to know if my data fits that equation/distribution, would I not have to know the
partition function and other quantities like temperature?
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qw098
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If I find the ratios of the adjacent energy levels and they are the same then it is a Boltzmann distribution. So if given N0,N1.N2. Then N1/N0 should
= N2/N1 . Is my thinking correct?
Yes, I am about 100% sure this is correct!!! Thanks everyone! I understand this now!! 
[Edited on 4-3-2012 by qw098]
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AJKOER
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Construct a Chi-Square Goodness of Fit Test
1. Fit the distribution based on data (method of moments or maximum likelihood).
2. Per the fit, calculate the expected # in a multi-cell partition of the distribution.
3. Tabulate the actual observed in these cells.
4. Form the sum of squares of the differences between actual and fitted. Divide by the degrees of freedom (function of the data set and the # of
parameters estimated for the postulated distribution). This is the Ch-Square variable. See if significant at the 5% level (so called selected Type I
error). I would also be interested in the Power of the Test (1- Type II error).
Possibly more powerful modification: Calculate a statistic of the distribution (like the ratio of adjacent energy levels) for which you can compute an
expected value to compare to your observed statistic. Perform the same Chi-Square Test, but adjust the degrees of freedom as you are no longer
consuming degrees of freedom to estimate parameters of the distribution.
Good luck.
[Edited on 4-3-2012 by AJKOER]
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watson.fawkes
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Quote: Originally posted by qw098  | | If I find the ratios of the adjacent energy levels and they are the same then it is a Boltzmann distribution. So if given N0,N1.N2. Then N1/N0 should
= N2/N1 . Is my thinking correct? |
This is true only because the energy levels you specified are an
arithmetic sequence, that is, E<sub>2</sub> - E<sub>1</sub> = E<sub>1</sub> - E<sub>0</sub>. In that
case, the occupation numbers for each level must then be in geometric sequence, as you've observed. Only for this case, they form a Boltzmann
distribution for any temperature.
If you want to really make sure you understand this, show that any three energy levels and occupation numbers form a Boltzmann distribution at
some temperature, as long as the occupation numbers decrease with increasing energy. Hint: express a family of functions with parameter T,
then use the zero-crossing lemma.
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watson.fawkes
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Statistical tests have absolutely nothing to do with the stated
problem, which is posed with exact data and asks if it matches an given functional form for some values of the parameters of the form.
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qw098
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Thanks Watson!
I will indeed try that!
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Poppy
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I don't believe something will ever follow the boltzmann distribuition perfectly, besides confined bunches of electrons or protons.
Sorry its meant that atoms will not come into any even closely sharp boltzmann distribuition because of lasing effects, therefore you can't tell how
many particles of the ensemble are given for a determined energy level. You will need a spectrometer of some kind.
[Edited on 3-6-2012 by Poppy]
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