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modified: ../../bin/latexRegex.pl

new file:   basicStatMechLecture17.tex
new file:   kittelCh3Zipper.tex
new file:   kittelCh3ZipperFig1.png
new file:   kittelZipperFig2.pdf
new file:   kittelZipperFig2pn.png
modified:   renumber
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1 parent b36c40a commit 38439d1c23cad81c4c4390151750b2ef90f0db70 @peeterjoot committed Mar 21, 2013
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@@ -26,6 +26,7 @@
#s/_div/_{\\mathrm{div}}/g ;
#s/_E/_{\\mathrm{E}}/g ;
#s/_eff/_{\\mathrm{eff}}/g ;
+#s/_F/_{\\mathrm{F}}/g ;
#s/_final/_{\\mathrm{final}}/g ;
#s/_full/_{\\text{full}}/g ;
#s/_G/_{\\mathrm{G}}/g ;
@@ -57,5 +58,4 @@
#s/pidotalpha/\\dot{p}_{i_\\alpha}/g ;
#s/xialpha\b/x_{i_\\alpha}/g ;
#s/xidotalpha/\\dot{x}_{i_\\alpha}/g ;
-s/_F/_{\\mathrm{F}}/g ;
s/\\ee/\\epsilon/g ;
@@ -0,0 +1,23 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\input{../blogpost.tex}
+\renewcommand{\basename}{basicStatMechLecture17}
+\renewcommand{\dirname}{notes/phy452/}
+\newcommand{\keywords}{Statistical mechanics, PHY452H1S}
+\input{../peeter_prologue_print2.tex}
+
+\beginArtNoToc
+\generatetitle{PHY452H1S Basic Statistical Mechanics. Lecture 17: XXX. Taught by Prof.\ Arun Paramekanti}
+%\chapter{XXX}
+\label{chap:basicStatMechLecture17}
+
+\section{Disclaimer}
+
+Peeter's lecture notes from class. May not be entirely coherent.
+
+\section{XXX}
+
+%\EndArticle
+\EndNoBibArticle
@@ -0,0 +1,169 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\input{../blogpost.tex}
+\renewcommand{\basename}{kittelCh3Zipper}
+\renewcommand{\dirname}{notes/phy452/}
+\newcommand{\keywords}{Statistical mechanics, PHY452H1S}
+
+\input{../peeter_prologue_print2.tex}
+
+\beginArtNoToc
+
+\generatetitle{Kittel Zipper problem}
+%\chapter{Kittel Zipper problem}
+\label{chap:kittelCh3Zipper}
+
+\makeoproblem{Zipper problem}{pr:kittelCh3Zipper:7}{\citep{kittel1980thermal} 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.
+\makesubproblem{}{pr:kittelCh3Zipper:7:a}
+Find (and sum) the partition function.
+\makesubproblem{}{pr:kittelCh3Zipper:7:b}
+In the low temperature limit $\kB T \ll \epsilon$, find the average number of open links. The model is a very simplified model of the unwinding of two-stranded DNA molecules.
+} % makeoproblem
+
+\makeanswer{pr:kittelCh3Zipper:7}{
+
+The system is depicted in \cref{fig:kittelCh3Zipper:kittelCh3ZipperFig1}, in the $E = 0$ and $E = \epsilon$ states.
+
+\imageFigure{kittelCh3ZipperFig1}{Zipper molecule model in first two states}{fig:kittelCh3Zipper:kittelCh3ZipperFig1}{0.2}
+
+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{dmath}\label{eqn:kittelCh3Zipper:20}
+P_s =
+\frac{e^{- \beta s \epsilon}}{Z},
+\end{dmath}
+
+with
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:40}
+Z = \sum_{s = 0}^N \frac{e^{- \beta s \epsilon}}{Z}.
+\end{dmath}
+
+\makeSubAnswer{}{pr:kittelCh3Zipper:7:a}
+
+We can sum this geometric series immediately
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:60}
+\myBoxed{
+Z =
+\frac{e^{-\beta (N+1) \epsilon} - 1}
+{e^{-\beta \epsilon } - 1}.
+}
+\end{dmath}
+
+\makeSubAnswer{}{pr:kittelCh3Zipper:7:b}
+
+The expectation value for the number of links is
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:80}
+\expectation{s}
+= \sum_{s = 0}^N s P_s
+= \inv{Z} \sum_{s = 1}^N s e^{- \beta s \epsilon}
+= -\inv{Z} \PD{(\beta \epsilon)}{} \sum_{s = 1}^N e^{- \beta s \epsilon}.
+\end{dmath}
+
+Let's write
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:100}
+a = e^{-\beta \epsilon},
+\end{dmath}
+
+and make a change of variables
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:120}
+-\PD{(\beta \epsilon)}{}
+=
+\PD{\ln a}{}
+=
+\PD{\ln a}{a}
+\PD{a}{}
+=
+\PD{(-\beta \epsilon)}{e^{-\beta \epsilon}}
+\PD{a}{}
+=
+a
+\PD{a}{}
+\end{dmath}
+
+so that
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:140}
+-\PD{\ln a}{} \sum_{s = 1}^N a^s
+=
+a \frac{d}{da}
+\lr{ \frac{a^{N+1} - a}{a - 1} }
+=
+a
+\lr{
+\frac{(N+1) a^N - 1}{a - 1}
+-
+\frac{a^{N+1} - a}
+{ (a - 1)^2 }
+}
+=
+\frac{a}{(a-1)^2}
+\lr{
+\lr{(N+1) a^N - 1}(a - 1)
+-
+a^{N+1} + a
+}
+=
+\frac{a}{(a-1)^2}
+\lr{
+N a^{N+1}
+-(N+1) a^N + 1
+}
+=
+\frac{a}{(a-1)^2}
+\lr{
+a^N ( N (a - 1) - 1 ) + 1
+}.
+\end{dmath}
+
+The average number of links is thus
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:160}
+\expectation{k} =
+\frac
+{a - 1}
+{a^{N+1} - 1}
+\frac{a}{(a-1)^2}
+\lr{
+a^N ( N (a - 1) - 1 ) + 1
+},
+\end{dmath}
+
+or
+\begin{dmath}\label{eqn:kittelCh3Zipper:180}
+\myBoxed{
+\expectation{k} =
+\frac{1}
+{1 - e^{-\beta \epsilon(N+1)} }
+\frac{1}{e^{\beta \epsilon} - 1}
+\lr{
+e^{-\beta \epsilon N} ( N (e^{-\beta \epsilon} - 1) - 1 ) + 1
+}.
+}
+\end{dmath}
+
+In the very low temperature limit where $\beta \epsilon \gg 1$ (small $T$, big $\beta$), we have
+
+\begin{dmath}\label{eqn:kittelCh3Zipper:200}
+\expectation{k} \approx
+\frac{1}{e^{\beta \epsilon}}
+=
+e^{-\beta \epsilon},
+\end{dmath}
+
+showing that on average no links are open at such low temperatures. An exact plot of $\expectation{s}$ for a few small $N$ values is in \cref{fig:kittelZipper:kittelZipperFig2}.
+
+\imageFigure{kittelZipperFig2}{Average number of open links}{fig:kittelZipper:kittelZipperFig2}{0.3}
+
+} % makeanswer
+
+\EndArticle
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@@ -5,5 +5,9 @@
#~/bin/lgrep pipeFlowConstPressureGradient.tex | tee o ; . ./o
#~/bin/lgrep t.tex | tee o ; . ./o
#perl -p -i ./p energyProbabilityPathriaQuestion.tex
-~/bin/lgrep basicStatMechLecture16.tex | tee o ; . ./o
-perl -p -i ./p basicStatMechLecture16.tex
+#~/bin/lgrep basicStatMechLecture16.tex | tee o ; . ./o
+#~/bin/lgrep basicStatMechLecture17.tex | tee o ; . ./o
+#perl -p -i ./p basicStatMechLecture16.tex
+#perl -p -i ./p basicStatMechLecture17.tex
+~/bin/lgrep kittelCh3Zipper.tex | tee o ; . ./o
+perl -p -i ./p kittelCh3Zipper.tex

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