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modified: basicStatMechLecture19.tex

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1 parent 5d59637 commit a0ca58e466e7e53dc48eb38f783bc547dfcb33e0 @peeterjoot committed Mar 28, 2013
Showing with 45 additions and 17 deletions.
  1. +45 −17 notes/blogit/basicStatMechLecture19.tex
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@@ -5,7 +5,7 @@
\input{../blogpost.tex}
\renewcommand{\basename}{basicStatMechLecture19}
\renewcommand{\dirname}{notes/phy452/}
-\newcommand{\keywords}{Statistical mechanics, PHY452H1S}
+\newcommand{\keywords}{Statistical mechanics, PHY452H1S, Boson, Bose condensate, fugacity, occupation number, density, zeta function}
\input{../peeter_prologue_print2.tex}
\beginArtNoToc
@@ -75,7 +75,7 @@ \section{Bosons}
Observe that at large energies we have
\begin{dmath}\label{eqn:basicStatMechLecture19:240}
-n_{\mathrm{B}}(\text{large} \Bk) \sim z e^{-\beta \epsilon_\Bk}
+n_{\mathrm{B}}(\text{large} \, \Bk) \sim z e^{-\beta \epsilon_\Bk}
\end{dmath}
For small energies
@@ -92,19 +92,46 @@ \section{Bosons}
\begin{dmath}\label{eqn:basicStatMechLecture19:140}
\rho = \frac{N}{V}
= \int \frac{d^3 \Bk}{(2 \pi)^3} \inv{z^{-1} e^{\beta \epsilon_\Bk} -1 }
-= \frac{2}{(2 \pi)^2} \int_0^\infty k^2 dk \inv{z^{-1} e^{\beta \hbar^2 k^2/2m} -1 }.
+= \frac{2}{(2 \pi)^2}
+\int_0^\infty k^2 dk \inv{z^{-1} e^{\beta \hbar^2 k^2/2m} -1 }
+= \frac{2}{(2 \pi)^2}
+\lr{
+\frac
+{2 m}
+{\beta \hbar^2}
+}
+^{3/2}
+\int_0^\infty
+\lr{
+\frac
+{\beta \hbar^2}
+{2 m}
+}
+^{3/2}
+k^2 dk \inv{z^{-1} e^{\beta \hbar^2 k^2/2m} -1 }
\end{dmath}
-With
+With the substitution
\begin{dmath}\label{eqn:basicStatMechLecture19:160}
x^2 = \beta \frac{\hbar^2 k^2}{2m},
\end{dmath}
-this leads to (FIXME: work out.)
+we find
\begin{dmath}\label{eqn:basicStatMechLecture19:180}
-\rho \lambda^3 = \frac{4}{\sqrt{\pi}} \int_0^\infty dx \frac{x^2}{z^{-1} e^{x^2} - 1 }
+\rho \lambda^3
+= \frac{2}{(2 \pi)^2}
+\lr{
+\frac
+{2 \cancel{m}}
+{\cancel{\beta \hbar^2}}
+}
+^{3/2}
+\lr{ \frac{ 2 \pi \cancel{\hbar^2 \beta}}{\cancel{m}} }^{3/2}
+\int_0^\infty
+x^2 dx \inv{z^{-1} e^{x^2} -1 }
+= \frac{4}{\sqrt{\pi}} \int_0^\infty dx \frac{x^2}{z^{-1} e^{x^2} - 1 }
\equiv g_{3/2}(z).
\end{dmath}
@@ -136,8 +163,6 @@ \section{Bosons}
\zeta(s) = \sum_{ n = 1 } \inv{n^s}.
\end{dmath}
-F5
-
\begin{dmath}\label{eqn:basicStatMechLecture19:340}
g_{3/2}(z) = \rho \lambda^3
\end{dmath}
@@ -150,7 +175,7 @@ \section{Bosons}
(as $T$ does down, $\rho \lambda^3$ goes up)
-Looking at $g32(z = 1) = \rho \lambda^3(T_{\mathrm{c}})$ leads to
+Looking at $g_{3/2}(z = 1) = \rho \lambda^3(T_{\mathrm{c}})$ leads to
\begin{dmath}\label{eqn:basicStatMechLecture19:380}
\myBoxed{
@@ -162,13 +187,14 @@ \section{Bosons}
\paragraph{How do I satisfy number conservation?}
-Worked out by
+We have a problem here since as $T \rightarrow 0$ the $1/\lambda^3 \sim T^{3/2}$ term in $\rho$ above drops to zero, yet $g_{3/2}(z)$ cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this was worked out by
+
\begin{itemize}
\item Bose (1924) for photons (examining statistics for symmetric wave functions).
\item Einstein (1925) for conserved particles.
\end{itemize}
-We have a problem here since as $T \rightarrow 0$ the $1/\lambda^3 \sim T^{3/2}$ term in $\rho$ above drops to zero, yet $g_{3/2}(z)$ cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this is to introduce a non-zero density for $\Bk = 0$. We'll adjust the approximation so that we have
+To deal with this issue, we (somewhat arbitrarily, because we need to) introduce a non-zero density for $\Bk = 0$. This is an adjustment of the approximation so that we have
\begin{dmath}\label{eqn:basicStatMechLecture19:400}
\sum_{\Bk} \rightarrow \int \frac{d^3 \Bk}{(2 \pi)^3} \qquad \mbox{Except around $\Bk = 0$},
@@ -180,14 +206,14 @@ \section{Bosons}
\begin{dmath}\label{eqn:basicStatMechLecture19:420}
\sum_\Bk
-= \mbox{Contribution at $\Bk = 0$} + V \int \frac{d^3 \Bk}{(2 \pi)^3}.
+= \lr{ \mbox{Contribution at $\Bk = 0$} } + V \int \frac{d^3 \Bk}{(2 \pi)^3}.
\end{dmath}
Given this, we have
\begin{dmath}\label{eqn:basicStatMechLecture19:420b}
N
-= N(\Bk = 0)
+= N_{\Bk = 0}
+ V \int \frac{d^3 \Bk}{(2 \pi)^3} n_{\mathrm{B}}(\Bk)
\end{dmath}
@@ -197,15 +223,17 @@ \section{Bosons}
\begin{dmath}\label{eqn:basicStatMechLecture19:420c}
\rho
-= \rho(\Bk = 0)
-+ \inv{\lambda^3} g32(z)
-= \rho(\Bk = 0)
+= \rho_{\Bk = 0}
++ \inv{\lambda^3}
+g_{3/2}(z)
+= \rho_{\Bk = 0}
+
\frac{ \lambda(T_{\mathrm{c}}) }{ \lambda(T)}
\inv{ \lambda^3(T_{\mathrm{c}})}
+g_{3/2}(z)
\end{dmath}
-At $T > T_{\mathrm{c}}$ we have $\rho_\Bk = 0$, whereas at $T < T_{\mathrm{c}}$ we must introduce a non-zero density if we want to be able to keep a constant density constraint.
+At $T > T_{\mathrm{c}}$ we have $\rho_{\Bk = 0}$, whereas at $T < T_{\mathrm{c}}$ we must introduce a non-zero density if we want to be able to keep a constant density constraint.
%\EndArticle
\EndNoBibArticle

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