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I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present.

There is a section that i can not understand:


$S= \sum_{j=1}^{N} F_{j}$

$<S>=< \sum_{j=1}^{N} F_{j}> = \sum_{j=1}^{N} <F_{j}>$

$\sigma^{2}_{S} =<S^{2}>-<S>^{2}$

line a:

$=\sum_{j=1}^{N}\sum_{k=1}^{N} <F_{j}F_{k}> - \sum_{j=1}^{N} <F_{j}>\sum_{k=1}^{N}<F_{k}>$

line b:

$=\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} <F_{j}><F_{k}> +\sum_{j=1}^{N} <F_{j}^{2}> - \sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$

line c:

$=\sum_{j=1}^{N} (<F_{j}^{2}>-<F_{j}>^{2})$ $=\sum_{j=1}^{N} \sigma_{j}^{2}$

My question is what happened after line a to line b and after that to line c?

My other question is, i have a little math, what should i study to understand such thermodynamics root math studies, calculus 1 or 2 or what else, can you specify a math topic?


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up vote 1 down vote accepted

I'll use a much simpler notation for starters, going to drop $\langle$ and $\rangle$. So the first term in line a is


and if you write it explicitly you have

$\sum_{i}\sum_{j}A_iA_j=(A_1A_1+A_2A_2+\dots+A_nA_n)+(A_1A_2+A_1A_3+\dots+A_1A_n)+\dots+(A_nA_1+A_nA_2+\dots+A_nA_{n-1})=\sum_iA_{i}^2+A_1\sum_{i\ne1}A_i+A_2\sum_{i\ne2}A_i+\dots+A_n\sum_{i\ne n}A_i=\sum_i A_{i}^2+\sum_i\sum_{j\ne i}A_iA_j$

So this gives you term one and two in line b. The third term in line b stays the same. Now for the last line when you take the following difference

$\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} <F_{j}><F_{k}> - \sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$ you get

$\sum_{j=1}^{N} <F_{j}>^2$

this is because the first double sum contains only terms like $F_iF_j$ and the second sum cointains terms like $F_iF_i$ and $F_iF_j$. So when you take the difference all the terms $F_iF_j$ will cancel out and your left with $\langle F_i\rangle\langle F_i\rangle=\langle F_i\rangle^2$. Thus you find your final result.

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thanks very much. i understood very well – merveotesi Apr 4 '13 at 7:31
Hello, i have a question, what is the difference between $=\sum_{j=1}^{N}\sum_{k=1}^{N} <F_{j}F_{k}>$ and $\sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$ ? – merveotesi Jun 14 '13 at 11:32

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