# peeterjoot/physicsplay

modified: basicStatMechLecture19.tex

 @@ -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