[haz658]

Hello again. I have two additional questions stemming from the chapter concerning the Boltzmann distribution from Engel and Reid's Thermodynamics, Statistical Thermodynamics, and Kinetics. This time, my questions refer to problems given at the end of the chapter.
        The first problem with which I am struggling  provides an expression that relates the weight of a configuration other than the dominant configuration with that of the dominant configuration. It states:

$$ log \left (\frac {W} {W_{max}} \right) = -H log \left ( \frac {H} {\frac {N} {2}} \right ) - T log \left (\frac {T} {\frac {N} {2}} \right ) /$$

where H is the number of coin tosses that resulted in heads, N is the total number of coin tosses, and T is the number of coin tosses that resulted in tails. N/2 is value for heads and tails in the dominant configuration, where the resulting total number of tosses is split 50/50 for heads and tails. The problem then gives a deviation index for a configuration other than the occurrence when the number of tosses is split 50/50 heads/tails, which describes how far the configuration under inspection deviated from that ratio.

$$ \alpha = \frac {(H - T)} {N} /$$

It follows by giving equations to solve for the number of tosses resulting in heads or tails when one knows the total number of tosses and the deviation index.

$$ H = \left (\frac {N} {2} \right ) (1 + \alpha ) /$$

$$ T = \left (\frac {N} {2} \right ) (1 - \alpha) /$$

Using this information, I assume, the problem then asks to demonstrate that:

$$ \frac {W} {W_{max}} = e^{-N \alpha ^{2}} /$$

If anyone would be willing to provide their insight, I would greatly appreciate it. It might also be helpful if I disclose my method of tackling this problem, and where I got stuck. To start, I took the first equation given, with the adjustment of replacing log (base 10) with natural log, since the answer incorporates the use of e, and then plugged in the expressions for H and T provided in the 3rd and 4th equations above, respectively. Please note that I also attempted to replace only H with these expressions, by redefining T as (N - H); as might be obvious to some of you, this didn't change the result as I proceeded. Anyway, by reexpressing the first equation using the third and fourth equations, I was able to cancel (N/2) from the numerator and denominator of natural logs of both terms, leaving me with this:

$$ ln \left (\frac {W} {W_{max}} \right ) = - \left (\frac {N} {2} \right ) (1 + \alpha) ln (1 + \alpha) - \left (\frac {N} {2} \right ) (1 - \alpha) ln (1 - \alpha) /$$

I went one more step before I could not see any means by which to obtain the final answer nor manipulate the terms any further. This step was taken by recognizing that, if I changed (1 - alpha) (before the natural log function) to (1 + alpha - 2*alpha), I could separate this into two terms, (N/2)(1 + alpha)ln(1 - alpha) and (N/2)(2*alpha)ln(1 - alpha) (the (N/2)(2*alpha) would yield (N*alpha)), and subsequently merge the first of these two terms with (N/2)(1 + alpha)ln(1 + alpha).

$$ ln \left (\frac {W} {W_{max}} \right ) = - \left (\frac {N} {2} \right) (1 + \alpha) ln (1 - \alpha ^{2}) + N \alpha ln (1 - \alpha) /$$

Again, any insight, even for a single step that I should change or add in my derivation, would be greatly appreciated.

My second problem is a little less derivationally involved. The book gives the scenario where there are 10 oscillators with 8 quanta of energy. It instructs me to list all of the configuration, or distributions, of energy, determine the dominant configuration, based on its having the largest weight, and calculate its probability. It seems pretty straightforward to illustrate all of the configurations, where 1 oscillator has all 8 quanta of energy, 1 oscillator has 7 and another has 1 quantum, etc. I then calculated the corresponding weight of each configuration using the equation:

$$ W = \frac {N!} {\prod_{n} a_{n}!} /$$

where N is the total number of oscillators over which to distribute the energy, which, in this case, is 10, and a_n is the occupation number, or number of oscillators occupying energy level n. The dominant configuration was that with 5 oscillators with 0 quanta of energy, 3 with 1, 1 with 2, and 1 with 3, with a weight of 5,040, meaning that there are 5,040 microstates or different ways in which this configuration can occur. I divided this weight by the sum of all the weights to obtain its probability:

$$ P_{dc} = \frac {W_{dc}} {\sum_{n} W_{n}} /$$

where dc stands for the dominant configuration. I obtained the value of .21, or 21%, but the answer in the back of the book is .25. I checked my work several times at each step, so I am pretty confident in my numbers. However, it is always possible that I missed something. If someone could give me some advice, or if you would like me to type out the numbers that I crunched for confirmation, I would greatly appreciate it. I haven't discounted the possibility that the book could be incorrect, but I found a copy of the errata, and it is not listed there. This is not a guarantee, though. Anyway, thanks in advance (hopefully).