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Posts Tagged ‘zipper DNA model’

An updated compilation of notes, for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 27, 2013

Here’s my second update of my notes compilation for this course, including all of the following:

March 27, 2013 Fermi gas

March 26, 2013 Fermi gas thermodynamics

March 26, 2013 Fermi gas thermodynamics

March 23, 2013 Relativisitic generalization of statistical mechanics

March 21, 2013 Kittel Zipper problem

March 18, 2013 Pathria chapter 4 diatomic molecule problem

March 17, 2013 Gibbs sum for a two level system

March 16, 2013 open system variance of N

March 16, 2013 probability forms of entropy

March 14, 2013 Grand Canonical/Fermion-Bosons

March 13, 2013 Quantum anharmonic oscillator

March 12, 2013 Grand canonical ensemble

March 11, 2013 Heat capacity of perturbed harmonic oscillator

March 10, 2013 Langevin small approximation

March 10, 2013 Addition of two one half spins

March 10, 2013 Midterm II reflection

March 07, 2013 Thermodynamic identities

March 06, 2013 Temperature

March 05, 2013 Interacting spin

plus everything detailed in the description of my first update and before.

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Kittel Zipper problem

Posted by peeterjoot on March 20, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Question: Zipper problem ([1] pr 3.7)

A zipper has N links; each link has a state in which it is closed with energy 0 and a state in which it is open with energy \epsilon. we require, however, that the zipper can only unzip from the left end, and that the link number s can only open if all links to the left (1, 2, \cdots, s - 1) are already open. Find (and sum) the partition function. In the low temperature limit k_{\mathrm{B}} T \ll \epsilon, find the average number of open links. The model is a very simplified model of the unwinding of two-stranded DNA molecules.

Answer

The system is depicted in fig. 1.1, in the E = 0 and E = \epsilon states.

Fig 1.1: Zipper molecule model in first two states

 

The left opening only constraint simplifies the combinatorics, since this restricts the available energies for the complete molecule to 0, \epsilon, 2 \epsilon, \cdots, N \epsilon.

The probability of finding the molecule with s links open is then

\begin{aligned}P_s =\frac{e^{- \beta s \epsilon}}{Z},\end{aligned} \hspace{\stretch{1}}(1.0.1)

with

\begin{aligned}Z = \sum_{s = 0}^N \frac{e^{- \beta s \epsilon}}{Z}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

We can sum this geometric series immediately

\begin{aligned}\boxed{Z =\frac{e^{-\beta (N+1) \epsilon} - 1}{e^{-\beta \epsilon } - 1}.}\end{aligned} \hspace{\stretch{1}}(1.0.3)

The expectation value for the number of links is

\begin{aligned}\left\langle{{s}}\right\rangle &= \sum_{s = 0}^N s P_s \\ &= \frac{1}{{Z}} \sum_{s = 1}^N s e^{- \beta s \epsilon} \\ &= -\frac{1}{{Z}} \frac{\partial {}}{\partial {(\beta \epsilon)}} \sum_{s = 1}^N e^{- \beta s \epsilon}.\end{aligned} \hspace{\stretch{1}}(1.0.4)

Let’s write

\begin{aligned}a = e^{-\beta \epsilon},\end{aligned} \hspace{\stretch{1}}(1.0.5)

and make a change of variables

\begin{aligned}-\frac{\partial {}}{\partial {(\beta \epsilon)}} &= \frac{\partial {}}{\partial {\ln a}} \\ &= \frac{\partial {a}}{\partial {\ln a}}\frac{\partial {}}{\partial {a}} \\ &= \frac{\partial {e^{-\beta \epsilon}}}{\partial {(-\beta \epsilon)}}\frac{\partial {}}{\partial {a}} \\ &= a\frac{\partial {}}{\partial {a}}\end{aligned} \hspace{\stretch{1}}(1.0.6)

so that

\begin{aligned}-\frac{\partial {}}{\partial {\ln a}} \sum_{s = 1}^N a^s &= a \frac{d}{da} \left( \frac{a^{N+1} - a}{a - 1} \right) \\ &= a\left( \frac{(N+1) a^N - 1}{a - 1} - \frac{a^{N+1} - a} { (a - 1)^2 } \right) \\ &= \frac{a}{(a-1)^2}\left( \left( (N+1) a^N - 1 \right) (a - 1) - a^{N+1} + a \right) \\ &= \frac{a}{(a-1)^2}\left( N a^{N+1} -(N+1) a^N + 1 \right) \\ &= \frac{a}{(a-1)^2}\left( a^N ( N (a - 1) - 1 ) + 1 \right).\end{aligned} \hspace{\stretch{1}}(1.0.7)

The average number of links is thus

\begin{aligned}\left\langle{{k}}\right\rangle = \frac{a - 1}{a^{N+1} - 1}\frac{a}{(a-1)^2}\left( a^N ( N (a - 1) - 1 ) + 1 \right),\end{aligned} \hspace{\stretch{1}}(1.0.8)

or

\begin{aligned}\boxed{\left\langle{{k}}\right\rangle = \frac{1}{1 - e^{-\beta \epsilon(N+1)} }\frac{1}{e^{\beta \epsilon} - 1}\left( e^{-\beta \epsilon N} ( N (e^{-\beta \epsilon} - 1) - 1 ) + 1 \right).}\end{aligned} \hspace{\stretch{1}}(1.0.9)

In the very low temperature limit where \beta \epsilon \gg 1 (small T, big \beta), we have

\begin{aligned}\left\langle{{k}}\right\rangle \approx\frac{1}{e^{\beta \epsilon}}= e^{-\beta \epsilon},\end{aligned} \hspace{\stretch{1}}(1.0.10)

showing that on average no links are open at such low temperatures. An exact plot of \left\langle{{s}}\right\rangle for a few small N values is in fig. 1.2.

Fig 1.2: Average number of open links

 

References

[1] C. Kittel and H. Kroemer. Thermal physics. WH Freeman, 1980.

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