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Lecture 18 plus 17 tweaking.

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1 parent 27ba7c9 commit dbd8776997874df10bd4ef0f30bbbf219a551ed7 @peeterjoot committed Mar 26, 2013
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2 bin/latexRegex.pl
@@ -58,5 +58,5 @@
#s/xidotalpha/\\dot{x}_{i_\\alpha}/g ;
s/\\ee/\\epsilon/g ;
#s/\\pp(.)/{($1)}/g ;
-#s/_F/_{\\mathrm{F}}/g ;
+s/_F/_{\\mathrm{F}}/g ;
#s/_T/_{\\mathrm{T}}/g ;
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2 bin/movetex
@@ -128,7 +128,7 @@ if ( $spellcheck )
#}
push( @newDirCommands, "~/bin/filtertex $filebase.tex" ) ;
-push( @newDirCommands, "echo '\%\\include{'$filebase'}' >> chapters.tex" ) ;
+push( @newDirCommands, "echo '\%\\input{'$filebase'.tex}' >> chapters.tex" ) ;
#push( @newDirCommands, "echo 'BOOKDEPENDENCIES +=' $filebase.tex >> make.dep" ) ;
push( @newDirCommands, "
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20 env/.aspell.en.pws
@@ -7,8 +7,8 @@ confocal
ParametericPlot
Indistinguishability
Exponentiating
-Ficks
Fick's
+Ficks
cgs
phasor
linestyle
@@ -87,8 +87,8 @@ Eikonal
Goldstein's
GPS
colinear
-hoc
Hmm
+hoc
Peeter's
Benard
ijk
@@ -160,8 +160,8 @@ Hestenes's
parametrizations
imaginaries
reparametrize
-n'l'm
nlm
+n'l'm
PDE
Lut
quantized
@@ -188,8 +188,8 @@ arctan
entropic
invertible
pion
-outermorphism
OuterMorphism
+outermorphism
QFT
rescaling
spinors
@@ -310,8 +310,8 @@ df
LIGO
spacetime
dH
-BT
dj
+BT
Routhian
dk
dL
@@ -339,9 +339,9 @@ Prandtl
ia
isync
inferometer
-eV
ib
iB
+eV
orthonormalization
ic
ie
@@ -355,16 +355,16 @@ variates
im
Dekker's
ji
-jj
KE
+jj
elastostatics
amino
-ip
jk
+ip
indices
kj
-iu
kk
+iu
mc
mE
iz
@@ -377,8 +377,8 @@ Strang's
xyz
mk
resistive
-mn
kx
+mn
Eulerian
anticommutes
n'l
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9 notes/METADATA
@@ -3061,6 +3061,14 @@ my @phy452 =
URL => 'http://sites.google.com/site/peeterjoot2/math2013/relStatMechExploration.pdf',
WHAT => qq()
}
+,{
+ SOURCE => 'basicStatMechLecture17',
+ TITLE => qq(Fermi gas thermodynamics),
+ DATE => 'March 26, 2013',
+ REF => 'basicStatMechLecture17',
+ URL => 'http://sites.google.com/site/peeterjoot2/math2013/basicStatMechLecture17.pdf',
+ WHAT => qq()
+}
) ; # @phy452
my @phy454 =
@@ -5263,3 +5271,4 @@ sub printHistory
}
+
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2 notes/TODO.txt
@@ -138,6 +138,8 @@ PHY485
PHY452
------
+- fixme's in lecture 17 (and perhaps elsewhere)
+
- check signs for all the n^b n_f results in lectures 15,16.
- introduce \EF and \kF macros, like \kB and search out the hand expanded stuff.
View
255 notes/blogit/basicStatMechLecture18.tex
@@ -9,15 +9,264 @@
\input{../peeter_prologue_print2.tex}
\beginArtNoToc
-\generatetitle{PHY452H1S Basic Statistical Mechanics. Lecture 18: XXX. Taught by Prof.\ Arun Paramekanti}
-%\chapter{XXX}
+\generatetitle{PHY452H1S Basic Statistical Mechanics. Lecture 18: Fermi gas thermodynamics. Taught by Prof.\ Arun Paramekanti}
+%\chapter{Fermi gas thermodynamics}
\label{chap:basicStatMechLecture18}
\section{Disclaimer}
Peeter's lecture notes from class. May not be entirely coherent.
-\section{XXX}
+\paragraph{Review}
+
+Last time we found that the low temperature behaviour or the chemical potential was quadratic
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:20}
+\mu =
+\mu(0) - a \frac{T^2}{T_{\mathrm{F}}}
+%^
+%\kB T ?
+\end{dmath}
+
+%\epsilon(k) = \frac{\hbar^2 k^2}{2m}
+
+F1
+
+\paragraph{Specific heat}
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:40}
+E = \sum_\Bk n_{\mathrm{F}}(\epsilon_\Bk, T) \epsilon_\Bk
+\end{dmath}
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:60}
+\frac{E}{V}
+= \inv{(2\pi)^3} \int d^3 \Bk n_{\mathrm{F}}(\epsilon_\Bk, T) \epsilon_\Bk
+= \int d\epsilon N(\epsilon) n_{\mathrm{F}}(\epsilon, T) \epsilon,
+\end{dmath}
+
+where
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:80}
+N(\epsilon) = \inv{4 \pi^2}
+\lr{\frac{2m}{\hbar^2}}
+^{3/2}
+\sqrt{\epsilon}.
+\end{dmath}
+
+\paragraph{Low temperature $\CV$}
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:100}
+\frac{\Delta E(T)}{V}
+=
+\int_0^\infty d\epsilon N(\epsilon)
+\lr{ n_{\mathrm{F}}(\epsilon, T) - n_{\mathrm{F}}(\epsilon, 0)}
+\end{dmath}
+
+The only change in the distribution
+
+F2
+
+that is of interest is over the step portion of the distribution, and over this range of interest $N(\epsilon)$ is approximately constant as in
+
+F3
+
+\begin{subequations}
+\begin{dmath}\label{eqn:basicStatMechLecture18:120}
+N(\epsilon) \approx N(\mu)
+\end{dmath}
+\begin{dmath}\label{eqn:basicStatMechLecture18:140}
+\mu \approx \epsilon_{\mathrm{F}},
+\end{dmath}
+\end{subequations}
+
+so that
+\begin{dmath}\label{eqn:basicStatMechLecture18:160}
+\Delta e \equiv
+\frac{\Delta E(T)}{V}
+\approx
+N(\epsilon_{\mathrm{F}})
+\int_0^\infty d\epsilon
+\lr{ n_{\mathrm{F}}(\epsilon, T) - n_{\mathrm{F}}(\epsilon, 0)}
+=
+N(\epsilon_{\mathrm{F}})
+\int_{-\epsilon_{\mathrm{F}}}^\infty d x (\epsilon_{\mathrm{F}} + x)
+\lr{ n_{\mathrm{F}}(\epsilon + x, T) - n_{\mathrm{F}}(\epsilon_{\mathrm{F}} + x, 0)}.
+\end{dmath}
+
+Here we've made a change of variables $\epsilon = \epsilon_{\mathrm{F}} + x$, so that we have near cancelation of the $\epsilon_{\mathrm{F}}$ factor
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:180}
+\Delta e
+=
+N(\epsilon_{\mathrm{F}})
+\epsilon_{\mathrm{F}}
+\int_{-\epsilon_{\mathrm{F}}}^\infty d x
+\mathLabelBox{
+\lr{ n_{\mathrm{F}}(\epsilon + x, T) - n_{\mathrm{F}}(\epsilon_{\mathrm{F}} + x, 0)}
+}{almost equal everywhere}
++
+N(\epsilon_{\mathrm{F}})
+\int_{-\epsilon_{\mathrm{F}}}^\infty d x x
+\lr{ n_{\mathrm{F}}(\epsilon + x, T) - n_{\mathrm{F}}(\epsilon_{\mathrm{F}} + x, 0)}
+\approx
+N(\epsilon_{\mathrm{F}})
+\int_{-\infty}^\infty d x x
+\lr{
+\inv{ e^{\beta x} +1 }
+-
+\evalbar{\inv{ e^{\beta x} +1 }}{T \rightarrow 0}
+}.
+\end{dmath}
+
+Here we've extended the integration range without changing much. FIXME: justify for self. Taking derivatives with respect to temperature we have
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:200}
+\frac{\delta e}{T}
+=
+-N(\epsilon_{\mathrm{F}})
+\int_{-\infty}^\infty d x x
+\inv{(e^{\beta x} + 1)^2}
+\frac{d}{dT} e^{\beta x}
+=
+N(\epsilon_{\mathrm{F}})
+\int_{-\infty}^\infty d x x
+\inv{(e^{\beta x} + 1)^2}
+e^{\beta x}
+\frac{x}{\kB T^2}
+\end{dmath}
+
+With $\beta x = y$, we have for $T \ll T_{\mathrm{F}}$
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:220}
+\frac{C}{V}
+=
+N(\epsilon_{\mathrm{F}})
+\int_{-\infty}^\infty \frac{ dy y^2 e^y }{ (e^y + 1)^2 \kB T^2} (\kB T)^3
+=
+N(\epsilon_{\mathrm{F}}) \kB^2 T
+\mathLabelBox{
+\int_{-\infty}^\infty \frac{ dy y^2 e^y }{ (e^y + 1)^2 }
+}{$\pi^2/3$}
+=
+\frac{\pi^2}{3} N(\epsilon_{\mathrm{F}}) \kB (\kB T).
+\end{dmath}
+
+Using \eqnref{eqn:basicStatMechLecture18:80} at the Fermi energy and
+
+\begin{subequations}
+\begin{dmath}\label{eqn:basicStatMechLecture18:240}
+\frac{N}{V} = \rho
+\end{dmath}
+\begin{dmath}\label{eqn:basicStatMechLecture18:260}
+\epsilon_{\mathrm{F}} = \frac{\hbar^2 \kF^2}{2 m}
+\end{dmath}
+\begin{dmath}\label{eqn:basicStatMechLecture18:280}
+\kF = \lr{6 \pi \rho}
+^{1/3},
+\end{dmath}
+\end{subequations}
+
+we have
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:320}
+N(\epsilon_{\mathrm{F}})
+= \inv{4 \pi^2}
+\lr{\frac{2m}{\hbar^2}}
+^{3/2}
+\sqrt{\epsilon_{\mathrm{F}}}
+= \inv{4 \pi^2}
+\lr{\frac{2m}{\hbar^2}}
+^{3/2}
+\frac{\hbar \kF}{\sqrt{2m}}
+= \inv{4 \pi^2}
+\lr{\frac{2m}{\hbar^2}}
+^{3/2}
+\frac{\hbar }{\sqrt{2m}} \lr{6 \pi \rho}^{1/3}
+= \inv{4 \pi^2}
+\lr{\frac{2m}{\hbar^2}}
+\lr{6 \pi \frac{N}{V}}^{1/3}
+\end{dmath}
+
+FIXME: ...
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:300}
+\frac{C}{N} =
+\frac{\pi^2}{2} \kB \frac{ \kB T}{\epsilon_{\mathrm{F}}}.
+\end{dmath}
+
+This is illustrated in
+
+F4
+
+\paragraph{Relativisitic gas}
+
+\begin{itemize}
+\item Relativisitic gas
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:340}
+\epsilon_\Bk = \pm \hbar v \Abs{\Bk}.
+\end{dmath}
+\begin{dmath}\label{eqn:basicStatMechLecture18:360}
+\epsilon = \sqrt{(m_0 c^2)^2 + c^2 (\hbar \Bk)^2}
+\end{dmath}
+
+\item graphene
+
+\item massless Dirac Fermion
+
+F5
+
+We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition.
+
+\end{itemize}
+
+Considering graphene, a 2D system, we want to determine the density of states $N(\epsilon)$,
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:380}
+\int \frac{d^2 \Bk}{(2 \pi)^2} \rightarrow \int_{-\infty}^\infty d\epsilon N(\epsilon),
+\end{dmath}
+
+We'll find
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:400}
+N(\epsilon) = \text{constant factor} \frac{\Abs{\epsilon}}{v},
+\end{dmath}
+
+F7
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:420}
+C \sim \frac{d}{dT} \int N(\epsilon) n_{\mathrm{F}}(\epsilon) \epsilon d\epsilon,
+\end{dmath}
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:440}
+\Delta E
+\sim
+\mathLabelBox
+{
+T}{window}
+\times
+\mathLabelBox
+[
+ labelstyle={below of=m\themathLableNode, below of=m\themathLableNode}
+]
+{
+T}{energy}
+\times
+\mathLabelBox
+[
+ labelstyle={xshift=2cm},
+ linestyle={out=270,in=90, latex-}
+]
+{
+T}{number of states}
+\sim T^3
+\end{dmath}
+
+so that
+
+\begin{dmath}\label{eqn:basicStatMechLecture18:460}
+C_{\mathrm{Dimensionless}} \sim T^2
+\end{dmath}
%\EndArticle
\EndNoBibArticle
View
3 notes/blogit/renumber
@@ -14,6 +14,5 @@
#perl -p -i ./p basicStatMechLecture16.tex
#perl -p -i ./p energyProbabilityPathriaQuestion.tex
#~/bin/lgrep basicStatMechProblemSet6Problem1.tex | tee o ; . ./o
-perl -p -i ./p basicStatMechLecture17.tex
-~/bin/lgrep basicStatMechLecture17.tex | tee o ; . ./o
~/bin/lgrep basicStatMechLecture18.tex | tee o ; . ./o
+perl -p -i ./p basicStatMechLecture18.tex
View
1 notes/phy452/basicStatMechLecture14.tex
@@ -18,7 +18,6 @@
This lecture had a large amount of spoken content not captured in these notes. Reference to \S 4 \citep{pathriastatistical} was made for additional details.
-\section{Grand partition function}
%\cref{fig:lecture14:lecture14Fig1}.
\imageFigure{figures/lecture14Fig1}{Ensemble pictures}{fig:lecture14:lecture14Fig1}{0.3}
View
1 notes/phy452/basicStatMechLecture15.tex
@@ -17,7 +17,6 @@
%
%Peeter's lecture notes from class. May not be entirely coherent.
%
-\section{Fermions and Bosons}
Was mentioned that three dimensions confines us to looking at either Fermions or Bosons, and that two dimensions is a rich subject (interchange of two particles isn't the same as one particle cycling around the other ending up in the same place -- how is that different than a particle cycling around another in a two dimensional space?)
View
41 notes/blogit/basicStatMechLecture17.tex → notes/phy452/basicStatMechLecture17.tex
@@ -2,22 +2,21 @@
% 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, Fermi gas, chemical potential, density, Fermi energy, Fermi temperature, occupancy, delta function}
-\input{../peeter_prologue_print2.tex}
-
-\beginArtNoToc
-\generatetitle{PHY452H1S Basic Statistical Mechanics. Lecture 17: Fermi gas thermodynamics. Taught by Prof.\ Arun Paramekanti}
-%\chapter{Fermi gas thermodynamics}
+%\input{../blogpost.tex}
+%\renewcommand{\basename}{basicStatMechLecture17}
+%\renewcommand{\dirname}{notes/phy452/}
+%\newcommand{\keywords}{Statistical mechanics, PHY452H1S, Fermi gas, chemical potential, density, Fermi energy, Fermi temperature, occupancy, delta function}
+%\input{../peeter_prologue_print2.tex}
+%
+%\beginArtNoToc
+%\generatetitle{PHY452H1S Basic Statistical Mechanics. Lecture 17: Fermi gas thermodynamics. Taught by Prof.\ Arun Paramekanti}
\label{chap:basicStatMechLecture17}
-
-\section{Disclaimer}
-
-Peeter's lecture notes from class. May not be entirely coherent.
-
-\section{Fermi gas thermodynamics}
+%
+%\section{Disclaimer}
+%
+%Peeter's lecture notes from class. May not be entirely coherent.
+%
+\paragraph{Review}
\begin{enumerate}
\item Energy was found to be
@@ -28,11 +27,11 @@ \section{Fermi gas thermodynamics}
\item Pressure was found to have the form \cref{fig:lecture17:lecture17Fig1}
-\imageFigure{lecture17Fig1}{Pressure in Fermi gas}{fig:lecture17:lecture17Fig1}{0.3}
+\imageFigure{figures/lecture17Fig1}{Pressure in Fermi gas}{fig:lecture17:lecture17Fig1}{0.3}
\item The chemical potential was found to have the form \cref{fig:lecture17:lecture17Fig2}.
-\imageFigure{lecture17Fig2}{Chemical potential in Fermi gas}{fig:lecture17:lecture17Fig2}{0.3}
+\imageFigure{figures/lecture17Fig2}{Chemical potential in Fermi gas}{fig:lecture17:lecture17Fig2}{0.3}
We found that
@@ -173,8 +172,8 @@ \section{Fermi gas thermodynamics}
Another example is graphene, a carbon structure of the form \cref{fig:lecture17:lecture17Fig3}. The energy and momentum for such a structure is related in roughly as shown in \cref{fig:lecture17:lecture17Fig4}, where
-\imageFigure{lecture17Fig3}{Graphene bond structure}{fig:lecture17:lecture17Fig3}{0.3}
-\imageFigure{lecture17Fig4}{Graphene energy momentum dependence}{fig:lecture17:lecture17Fig4}{0.3}
+\imageFigure{figures/lecture17Fig3}{Graphene bond structure}{fig:lecture17:lecture17Fig3}{0.3}
+\imageFigure{figures/lecture17Fig4}{Graphene energy momentum dependence}{fig:lecture17:lecture17Fig4}{0.3}
\begin{dmath}\label{eqn:basicStatMechLecture17:260}
\epsilon_\Bk = \pm v_{\mathrm{F}} \Abs{\Bk}.
@@ -271,7 +270,7 @@ \section{Fermi gas thermodynamics}
Assuming a quadratic form for the chemical potential at low temperature as in \cref{fig:lecture17:lecture17Fig5}, we have
-\imageFigure{lecture17Fig5}{Assumed quadratic form for low temperature chemical potential}{fig:lecture17:lecture17Fig5}{0.3}
+\imageFigure{figures/lecture17Fig5}{Assumed quadratic form for low temperature chemical potential}{fig:lecture17:lecture17Fig5}{0.3}
\begin{dmath}\label{eqn:basicStatMechLecture17:360}
1 =
@@ -310,4 +309,4 @@ \section{Fermi gas thermodynamics}
\mu = \epsilon_{\mathrm{F}} - \frac{\pi^2}{12} \frac{(\kB T)^2}{\epsilon_{\mathrm{F}}}.
\end{dmath}
-\EndArticle
+%\EndArticle
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8 notes/phy452/chapters.tex
@@ -87,8 +87,12 @@ \part{Lecture Notes}
\input{pathriaHarmonicOscPertubation.tex}
\input{pathriaCh3pr30.tex}
\chapter{Grand canonical ensemble}
- \input{basicStatMechLecture14.tex}
- \input{basicStatMechLecture15.tex}
+ \section{Grand partition function}
+ \input{basicStatMechLecture14.tex}
+ \section{Fermions and Bosons}
+ \input{basicStatMechLecture15.tex}
+ \section{Fermi gas thermodynamics}
+ \input{basicStatMechLecture17.tex}
\section{Problems}
\input{entropyProbabilityForm.tex}
\input{varianceN.tex}
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0 notes/blogit/lecture17Fig1.png → notes/phy452/figures/lecture17Fig1.png
File renamed without changes
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0 notes/blogit/lecture17Fig2.png → notes/phy452/figures/lecture17Fig2.png
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0 notes/blogit/lecture17Fig3.png → notes/phy452/figures/lecture17Fig3.png
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0 notes/blogit/lecture17Fig4.png → notes/phy452/figures/lecture17Fig4.png
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0 notes/blogit/lecture17Fig5.png → notes/phy452/figures/lecture17Fig5.png
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2 notes/phy452/renumber
@@ -1,4 +1,6 @@
#~/bin/lgrep 1dRandomWalk.tex | tee o ; . ./o
+#perl -p -i ./p basicStatMechLecture17.tex
+#~/bin/lgrep basicStatMechLecture17.tex | tee o ; . ./o
#~/bin/lgrep 3nParticlePhaseSpaceVolume.tex | tee o ; . ./o
#~/bin/lgrep basicStatMechLecture1.tex | tee o ; . ./o
#~/bin/lgrep basicStatMechLecture10.tex | tee o ; . ./o

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