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Homework Statement
A rubber band at temperature ##T## is fastened at one end to a peg and supports from its other end a weight ##W##. Assume as a simple microscopic model of the rubber band that it consists of a linked polymer chain of ##N## segments joined end to end; each segment has length ##a## and can be oriented either parallel or antiparallel to the vertical direction. Find an expression for the resultant mean length ##\bar{l}## of the rubber band as a function of ##W##. (Neglect the kineti energies or weights of the segments themselves, or any interaction between the segments.)
The Attempt at a Solution
Let the positive vertical direction be pointing upwards and let the zero of gravitational potential energy be set at the plane of the peg from which the rubber band hangs. The problem is quite easy at face value but I keep getting the wrong answer. Presumably when the problem says that the polymer chains "can be oriented either parallel or antiparallel to the vertical direction" they mean with the linked ends of the polymer chains kept fixed, otherwise the problem would be trivial. For any given microstate of this system, let ##n## be the number of polymer chains oriented antiparallel to the vertical direction so that the length of the rubber band for this given microstate is ##l_n = (2n - N)a##; the total energy of the system for this microstate is then ##E_n = -Wl_n##. The canonical partition function will thus be ##Z = \sum_{n = 0}^N e^{\beta W l_n}## and the average length will be ##\bar{l} = \sum_{n = 0}^N l_n e^{\beta W l_n}/Z = \sum_{n = 0}^N \frac{1}{\beta}\partial_{W}e^{\beta W l_n}/Z = \frac{1}{\beta}\partial_W \ln Z##.
All that's left then is a calculation of ##Z = e^{-\beta WNa}\sum_{n = 0}^{N}e^{2\beta Wna}## which is a geometric series
hence ##Z = e^{-\beta WNa}(\frac{e^{2\beta W Na}e^{2\beta Wa} - 1}{e^{2\beta W a}-1} )= \frac{e^{\beta W a(1+N)} - e^{-\beta Wa(1 + N)}}{e^{\beta Wa} - e^{-\beta Wa}} = \frac{\sinh \beta Wa(1 + N)}{\sinh \beta W a}##
therefore ##\bar{l} =\frac{1}{\beta}\partial_W \ln Z= (1 + N)a\coth \beta Wa(1 + N) - a\coth \beta Wa##.
Incidentally this is also the partition function for a single atom in an external uniform magnetic field ##B## if we replace ##N## with spin ##J## and ##Wa## with ##\mu B## where ##\mu## includes the gyromagnetic factor.
However, according to the book, the answer is supposed to be ##\bar{l} = Na \tanh \beta Wa## which makes much more sense than the answer I got because my answer doesn't have the correct behavior in the appropriate limiting cases e.g. it is divergent when ##W\rightarrow 0## even though it should give ##\bar{l}\rightarrow 0## as ##W\rightarrow 0##, which the book's answer does, since without any weight present the most probable microstate is the one for which there are equal numbers of polymer chains parallel and antiparallel to the vertical direction and as we know the average will agree with the most probable microstate.
However I cannot for the life of me figure out where the mistake in my solution is. Could anyone help me in finding my error? Thank you in advance.