%% ================================================================
\section{Introduction}

[xxx]  The problem at hand is to determine the effects of interparticle
interactions on the critical temperature of Bose-Einstein condensation; a
so-called random-cycle model is employed to this end.

%A large fraction of this paper is occupied by the appendices.  This reflects
%the fact that much of my work on this project has been in filling the gap
%between my coursework and the research papers being presented.  The experienced
%reader may wish to skip the appendices entirely.

The plan of this paper is as follows:

\begin{itemize}

\item A non-technical discussion of the historical context of the project; a
mention of alternative approaches.

\item The random-cycle approach to the BEC problem requires a model of spatial
permutations.  This model is developed mathematically --- it is of
probabilistic interest on its own --- without reference to the Bose gas.  (Main
results about the latter are summarized in the appendices.)

\item The model of spatial permutations is connected to the Bose gas:

	\begin{itemize}

	\item One begins with a Hamiltonian for particles with two-body
	interactions.

	\item Using a multi-body Feynman-Kac approach involving permutation
	symmetry of bosonic wave functions, one obtains a Hamiltonian in which
	permutation jumps rather than particles interact.

	\item A cluster expansion, to first order in the scattering length of the
	particles, yields a Hamiltonian with only jump-pair interactions.  At this
	point, one may employ the random-cycle model of spatial permutations.

	\end{itemize}

\item xxx
\item xxx
\item xxx

\end{itemize}

%% ================================================================
\newpage
\section{Historical context}
\label{sec:history}

%% ----------------------------------------------------------------
\subsection{Theory}

In 1924, the physicist Satyendra Nath Bose\index{Bose,~S.N.} examined the
quantum statistics of \plainidx{photons}.  In 1925, collaborating with Bose,
Albert Einstein\index{Einstein,~A.} realized that the same could be done with
non-interacting massive particles.  He also discovered the
\plainidx{condensation} phenomenon:  a macroscopic occupation of the
(single-particle) \plainidx{ground state} of the external potential
\cite{bib:lssy}.  Moreover, Einstein predicted a \plainidx{critical
temperature} for the phenomenon.  This temperature was so low --- at the
nanokelvin scale --- that \plainidx{Bose-Einstein condensation} attracted
little interest in the physics community.

Feynman\index{Feynman,~R.P.} in 1953, along with Penrose and Onsager in 1956
\cite{bib:feynman,bib:penons}, developed the theoretical notion of
\plainidx{long permutation cycles} in the \plainidx{Feynman-Kac representation}
of the Bose gas.  Feynman claimed that long cycles correspond to Bose-Einstein
condensation.

Andr\'as S\"ut\H{o} referred to the existence of long permutation cycles as
\emphidx{cycle percolation}.  He proved in 1993 that BEC implies cycle
percolation in the non-ideal (interacting) gas\index{non-ideal gas}
\cite{bib:suto1}, and proved the converse in 2002 for the ideal
(non-interacting) gas\index{ideal gas}.  S\"ut\H{o} moreover proved in the 2002
paper that there are infinitely many macroscopic cycles in the condensation of
the non-ideal Bose gas.

For the ideal Bose gas, BEC is defined as the macroscopic occupation of the
single-particle ground state of the external potential.  For an interacting
Bose gas, Hamiltonian eigenfunctions do not factor and thus there are no
single-particle ground states.  BEC is carefully defined for interacting
systems \cite{bib:lssy} in terms of the largest eigenvalue of a
\plainidx{density-matrix operator}.  The 1983 work of Buffet and Pul\`e
\cite{bib:buffetpule} examines the macroscopic occupation of the \plainidx{zero
Fourier mode}.

%% ----------------------------------------------------------------
\subsection{Experiments}

Liquid helium was produced in the laboratory by Kammerlingh Onnes
\index{Onnes,~H.K.} in 1908; Fritz London\index{London,~F.} in 1938
\cite{bib:london} connected superfluidity of \plainidx{liquid helium} with
Bose-Einstein condensation.  Here, however, atoms of liquid helium are strongly
interacting --- they attract only weakly, due to helium being a noble gas, but
there are strong repulsive effects due to the high density of the liquid.
Thus, Einstein's non-interacting theory could not explain the phenomenon.

Several groups attempted during the 1990s to produce BECs in vapors of
spin-polarized hydrogen, but were not able to achieve low enough temperatures.
The group of Cornell and Wieman \cite{bib:boulder}, using hybrid cooling
methods, successfully brought \plainidx{rubidium} atoms to well below the
critical temperature and made numerous measurements on the resulting
condensates.  (The group received the 2001 Nobel prize in physics for this
work.)

Interested in BECs was sparked by this experimental success:  thousands of
papers, both theoretical and experimental, have been published on BECs in the
years since.  The work of Cornell and Wieman was of interest for several
reasons:

\begin{itemize}

\item Condensates were directly imaged.  Measurments were taken of temperature,
density, position, velocity, particle number, and the fraction of the
condensate occupying the ground state of the 3D harmonic trapping potential.

\item The method was able to vary temperature and density through wide ranges;
the \plainidx{condensate fraction} was varied from zero to 100 percent.

\item The gaseous rubidium condensate was \emphidx{weakly interacting} ---
permitting a perturbative analysis which liquid helium, with its strong
interactions, did not allow.  (Note in particular that the work of Ueltschi et
al.  is a weak-interaction theory; it is valid to first order in the
interparticle interaction strength.)

\end{itemize}

%% ----------------------------------------------------------------
\subsection{Critical temperature}
\label{sec:critical_temperature}

Recall that Einstein predicted a critical temperature $T_c$ for the ideal Bose
gas.  It is a long-standing question to discover the effects of interbosonic
\plainidx{interaction strength} \plainidx{$a$} on the critical temperature.
Moreover, one may fix the density $\rho_c^{(a)}$ and obtain a critical
temperature $T_c^{(a)}=1/\beta_c^{(a)}$ or vice versa; also, both of these
critical parameters depend on the interaction strength $a$.  One expects the
critical combination of parameters to be a manifold in $(\rho,\beta,a)$ space.
(See figure \ref{fig:critical_manifold}.)

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics{figures/critical_manifold.eps}
\caption{Critical manifold in $(\rho,\beta,a)$ for small $a$.
	\label{fig:critical_manifold}}
\end{center}
\end{figure}

Much is known about the $a=0$ line of this critical manifold; off $a=0$, even
the crude shape has been under debate.  The following findings are described by
\cite{bib:baymetal}:  The superfluid transition temperature of liquid helium is
lower than that of an ideal gas of the same density.  Thus, assuming that
helium superfluidity is a strongly interacting BEC, one would expects
interactions to decrease the critical temperature for the strongly interacting
case.  Various theoretical work (tabulated below) suggested either an increase
or a decrease in critical temperature; path-integral simulations for low
density (i.e. weak interactions) showed an increase in critical temperature.
The emerging consensus is that
$$
	\frac{\Delta T}{T_c} = \frac{T_c^{(a)}-T_c^{(0)}}{T_c^{(0)}}
$$
is linear in $a$ for small $a$.

Ueltschi makes the following summary of the theoretical work on this question.
(See also \cite{bib:arnoldmoore,bib:kps,bib:nholandau} for a review on the
widely varying experimental results on $T_c^{(a)}$; see \cite{bib:baymetal} for
a thorough listing of the progress up to 2001.)

\begin{itemize}

\item 1964:
\emph{Huang}:
$\frac{\Delta T}{T_{\rm c}} \sim (a \rho^{1/3})^{3/2}$, increases

\item 1971:
\emph{Fetter \& Walecka}:
$\frac{\Delta T}{T_{\rm c}}$ decreases

\item 1982:
\emph{Toyoda}:
$\frac{\Delta T}{T_{\rm c}}$ decreases

\item 1992:
\emph{Stoof}:
$
\frac{\Delta T}{T_{\rm c}} = c \, a \rho^{1/3} + o(a \rho^{1/3}), \quad c>0
$

\item 1996:
\emph{Bijlsma \& Stoof}:
$c=4.66$

\item 1997:
\emph{Gr\"uter, Ceperley, Lalo\"e}:
$c=0.34$

\item 1999:
\emph{Holzmann, Gr\"uter, Lalo\"e}:
$c=0.7$;
{\emph{ Holzmann, Krauth}}: $c=2.3$;

\item 1999:
\emph{Baym et.\ al.}: 
$c=2.9$

\item 2000:
\emph{Reppy et.\ al.}:
$c=5.1$

\item 2001:
\emph{Kashurnikov, Prokof'ev, Svistunov}:
$c=1.29$

\item 2001:
\emph{Arnold, Moore}:
$c=1.32$

\item 2004:
\emph{Kastening}:
$c=1.27$

\item 2004:
\emph{Nho, Landau}:
$c=1.32$

\end{itemize}

The work of Ueltschi et al.  \cite{bib:gru,bib:bu,bib:ducycles,bib:qmath}
extends the permutation point of view originated by Feynman, Penrose, and
Onsager, drawing on the work of S\"ut\H{o}, Buffet, and Pul\`e
\cite{bib:feynman,bib:penons,bib:suto1,bib:suto2,bib:buffetpule}.  The main
goal of the project is to quantify $\Delta T/T_c$ for non-ideal Bose gases in
the small-scattering-length regime.  As is often the case in statistical
mechanics, the study of this interacting system necessitates the use of
computational methods.  The papers \cite{bib:bu,bib:qmath}, as well as my
comprehensive examination, develop the theoretical basis for the model which
this work explores computationally and statistically.

%% ================================================================
\newpage
\section{Literature review}


%% ----------------------------------------------------------------
\subsection{BEC}

\begin{itemize}
\item Feynman.
\item S\"ut\H{o}.
\end{itemize}

%% ----------------------------------------------------------------
\subsection{RCM}

The usual \ldots.

%% ----------------------------------------------------------------
\subsection{MCMC}

\begin{itemize}
\item Landau
\item Berg
\end{itemize}

%% ----------------------------------------------------------------
\subsection{FSS}

\begin{itemize}
\item Ceperley, Landau, Nho et al.:  MC path-int and FSS.
\item HK 1999
\item GCL97
\item KPS01
\item PGP08
	Pilati and Georgini.
\item *Barber v. 8
\item *NL04
	Nho and Landau.
\item *PC87
\item *PR92.
	Pollock and Runge.
	PRB 46 3535 1992.
\end{itemize}

%% ================================================================
\newpage
\section{Derivation?}

If so, follow comp --- or, use an It\=o approach.  But make it a bit more
abbreviated.

%% ================================================================
\newpage
\section{Jump interactions}

%% ----------------------------------------------------------------
\subsection{Cycle weights}

xxx type up notes ... hard to choose the correct $\alpha_\ell$'s to match the
Bose-gas model.

For probabilistic interest, we can use various $\alpha_\ell$'s anyway.

%% ----------------------------------------------------------------
\subsection{Bridge simulations}

Bridge simulations with $N_b$ and $N_p$.  Draw from rcm1 slides.

Bridge testing from rcm1.tex.

%% ----------------------------------------------------------------
\subsection{Integral approximation}

BU's integral approximation.  Splines or some such?

%% ================================================================
\newpage
\section{MCMC}
\label{sec:mcrcm}

Include an MCMC appendix with a distillation of mhcoin?

Choice of $(N,\rho,\beta,a)$.

$L$ not $N$; $T$ not $\beta$; $\alpha$ not $a$; $\rho=1$.

Choice of $(L,T,\alpha)$.

%% ----------------------------------------------------------------
\subsection{$\Delta E$ computations}

Include $\Delta N_2$, $\Delta N_\ell$, and point-motion $\Delta E$.

%% ----------------------------------------------------------------
\subsection{Detailed balance}

%% ----------------------------------------------------------------
\subsection{Thermalization detection}

xxx xref/paste in stuff from mhcoin about stabilization of probability
distributions.

%% ----------------------------------------------------------------
\subsection{Winding cycles and the superfluid fraction}

xxx this needs a theory section, and an MCMC section.

\textbf{Theory:}

\begin{align*}
	f_S &= \frac{\langle \vecW^2 \rangle L^2}{3 \beta N}
	= \frac{\langle \vecW^2 \rangle}{3 \beta L}.
\end{align*}

Derivation:  PC87.

Why it scales as $1/L$ --- this is counterintuitive.

\textbf{MCMC:}
Describe the algorithm tweak which allows even parity.
Prove this is the case.

Describe experimentally that $\pi_0 \sim U$ gets all odd or all even.

Different model \ldots not the one we want.  Is this merely an annoyance?

Xref fwd to worm section.

%% ----------------------------------------------------------------
\subsection{Importance sampling?}

%% ----------------------------------------------------------------
\subsection{Pseudocode}

%% ----------------------------------------------------------------
\subsection{Random variables}

$\pig{\rho_{1,k}}$; detection of $f_\infty$.  Diagonal $1 \times 1$ plots;
\texttt{test\_therm rhov=1}.

Superfluid fraction.  This is an ``interesting order parameter.''
Current experiments suggest this doesn't scale nicely in $L$ on the
lattice \ldots .

$\pig H$:  for thermalization detection.

$\pig{N_2}$, $\pig{N_\ell}$, $\pig{\ell_0}$, $\pig{\lmax}$:  misc.

%% ================================================================
\newpage
\section{Worm algorithm}
\label{sec:worm}

xxx see \cite{bib:pst98,bib:bps06}.

\input{mcmcbody.tex}

%% ================================================================
\newpage
\section{$T_c$ and extrapolation}

Fisher 1971; Nho and (David P.) Landau; Landau PRB 1976.  Pseudocode as well?

xxx steps:

\begin{itemize}

\item A single MCMC $f_\infty(L,T,\alpha)$ is as in section
\ref{sec:mcrcm}.

\item Triplication \ldots.

\item Loops over $L$, $T$, and $\alpha$, with triplicated runs, are in
\texttt{n2.sh}.  Files are created in the \texttt{data} subdirectory, one for
each $(L,T,\alpha)$ triple.  Each file contains an estimate of
$f_\infty(L,T,\alpha)$.

\item \texttt{n2extract.sh} reads the \texttt{data} subdirectory and writes
\texttt{analysis/n2rpt\_alpha\_*.txt}: for each $\alpha$, $f_\infty(T,L)$.

\item $f_\infty/T$ plots with $L$ series; $f_\infty/L$ plots with $T$ series.

\item The former:  power-law fit doesn't work.  Show why not.  Suspect $L \to
\infty$ FSS might be appropriate.

\item Log-log plots of the latter permit determination of $T_c^\alpha$.

\item Try to plot $f_\infty/T$ for $L\to\infty$.  Detect slope at critical
point and compare to known value.

\item
$\lim_{a \to 0} \lim_{L \to \infty} \Delta T_c$
vs.
$\lim_{L \to \infty} \lim_{a \to 0} \Delta T_c$?

\end{itemize}


\begin{figure}[pth]
\begin{center}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_0.01_L.eps}
}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_1.0_L.eps}
}
\caption
	[$f_\infty$ vs. $T$ for various $L=10$ to $40$, with $\alpha=0.01,1.0$.]
	{$f_\infty$ vs. $T$ for various $L=10$ to $40$, with $\alpha=0.01,1.0$.
	Lower curves are for larger $L$.
	\label{fig:tca_0.01_1.0_L}}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_0.01_xpose.eps}
}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_1.0_xpose.eps}
}
\caption
	[$f_\infty$ vs. $L$ for various $T=4.00$ to $7.52$, with $\alpha=0.01,1.0$.]
	{$f_\infty$ vs. $L$ for various $T=4.00$ to $7.52$, with $\alpha=0.01,1.0$.
	Lower curves are for larger $T$.
	\label{fig:tca_0.01_1.0_xpose}}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_0.01_loglog.eps}
}
\subfigure{
	\psfragscanon
	\includegraphics[scale=0.5]{figures/tca_1.0_loglog.eps}
}
\caption
	[$\log_{10}(f_\infty)$ vs. $\log_{10}(L)$ for various $T=4.00$ to $7.52$, with $\alpha=0.01,1.0$.]
	{$\log_{10}(f_\infty)$ vs. $\log_{10}(L)$ for various $T=4.00$ to $7.52$, with $\alpha=0.01,1.0$.
	Lower curves are for larger $T$.
	\label{fig:tca_0.01_1.0_loglog}}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\psfragscanon
\includegraphics[scale=0.6]{figures/tca_40_alpha.eps}
\caption
	[$f_\infty$ vs. $T$ for various $\alpha=0.01$ to $1.0$, with $L=40$.]
	{$f_\infty$ vs. $T$ for various $\alpha=0.01$ to $1.0$, with $L=40$.
	Higher curves are for larger $\alpha$.
	\label{fig:tca_40_alpha}}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\psfragscanon
\includegraphics[scale=0.6]{figures/tca_40_xpose.eps}
\caption
	[$f_\infty$ vs. $\alpha$ for various $T=4.00$ to $7.52$, with $L=40$.]
	{$f_\infty$ vs. $\alpha$ for various $T=4.00$ to $7.52$, with $L=40$.
	Lower curves are for larger $T$.
	\label{fig:tca_40_xpose}}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\psfragscanon
\includegraphics[scale=0.6]{figures/tca_40_loglog.eps}
\caption
	[$\log_{10}(f_\infty)$ vs. $\log_{10}(\alpha)$ for various $T=4.00$ to $7.52$, with $L=40$.]
	{$\log_{10}(f_\infty)$ vs. $\log_{10}(\alpha)$ for various $T=4.00$ to $7.52$, with $L=40$.
	Lower curves are for larger $T$.
	\label{fig:tca_40_loglog}}
\end{center}
\end{figure}
\clearpage

%% ----------------------------------------------------------------
\subsection{Power-law fit}

\begin{align*}
	f &= A \left(T-T_c\right)^\gamma \\
	&= A T_c^\gamma \left(\frac{T-T_c}{T_c}\right)^\gamma \\
	&= \left(-A T_c^\gamma\right) \left(1-\frac{T}{T_c}\right)^\gamma \\
	&= B \left(1-\frac{T}{T_c}\right)^\gamma.
\end{align*}
How to get
\begin{align*}
	f &= C \left(1- \left(\frac{T}{T_c}\right)^\gamma\right)?
\end{align*}

%% ================================================================
\newpage
\section{$T_c$ and finite-size scaling}

\input{fssbody.tex}

%% ================================================================
\newpage
\section{Propagation of uncertainties}

xxx appendix?

Denote the \bfidx{uncertainty} and \bfidx{relative error} in a quantity $a$ by
\begin{align*}
	\delta a \qand \frac{\delta a}{a},
\end{align*}
respectively.

Formula for \bfidx{error propagation}:
\begin{align*}
	Q &= Q(a,b) \\
	\delta Q &= \frac{\D Q}{\D a}\,\delta a + \frac{\D Q}{\D b}\,\delta b \\
	\frac{\delta Q}{Q} &= \frac{1}{Q} \left(
		\frac{\D Q}{\D a}\,\delta a + \frac{\D Q}{\D b}\,\delta b \right).
\end{align*}

Relative error is unchanged under a linear change of variable:
\begin{align*}
	y &= kx \\
	\delta y &= k \, \delta x \\
	\frac{\delta y}{y} &= \frac{k \,\delta x}{y}
	= \frac{\delta x}{x}.
\end{align*}

xxx reconcile this with \texttt{edeltatca.py}.

%% ================================================================
\newpage
\section{Acknowledgements}

Daniel Ueltschi,
Tom Kennedy,
Bill Faris,
Janek Wehr,
Joe Watkins,
Volker Betz,
Kevin Lin,
[\ldots elaborate].

NSF for spring 2008 and spring 2009 grants.  [xxx include grant number.]
Summer and fall 2009.

ICE cluster at UA HPC.

%% ================================================================
\appendix

% Flush the figure queue.
\afterpage{\clearpage}

%% ================================================================
\newpage
\section{More figures}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.0.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.0$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.01.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.01$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.1.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.1$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.2.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.2$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.5.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.5$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_0.8.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=0.8$.}
\end{center}
\end{figure}

\begin{figure}[pth]
\begin{center}
\includegraphics{figures/fork_alpha_1.0.eps}
\caption{$f_\infty$ vs. $T$ for $\alpha=1.0$.}
\end{center}
\end{figure}

\clearpage

%% ================================================================
% convince audience of usefulness/importance.

% For sims w/ lattice model:  replace alpha N2 with 2a/||x-y||?

% Oral Q's:
% * Tr(A^2); mx mult ...
% * Defn of SDE in terms of integrals: step functions:  what kind of convergence
%   of random variables?
% * What about measure-zero modification of diagonal of G(x,x)?

% <<< MHCOIN >>>

%% ================================================================
\newpage
\section{To be filed}

mem/cpu reqts table

error bars.

autocorr.

Tools:  copy in from rcm1.tex.

Visualization:
\begin{itemize}
\item Dot plots
\item $\rho$ plots
\item Tca plots with increasing $N$
\item Transposed trend diagrams
\end{itemize}

Things needing substantiating:
\begin{itemize}
\item Detailed balance etc. for MCMC.
\item Importance sampling.
\item Cluster expansion.
\item BU formula for symmetric TJI.
\item Verify $N_2$ sims against theory.
\end{itemize}

Issues:
\begin{itemize}
\item Bridge complexity for each TJI:  $N_b$ and $N_p$.
\item Bubble size for number of TJI's to be computed:  importance sampling?
\item Point motion.
\item Hierarchical partitioning of $\Lambda$.
\end{itemize}

Merge in compbib.

mhcoin!

Spec-heat plots:  $\Delta H / \Delta T$ having the appropriate form.
Not apparently sharp enough to determine $T_c$ \ldots .

lmax quot ... do what with it?

J Chem Phys 70 Aziz --- (choice of $a$)?

$\Delta V$ for $i<j$ case (involving $u$'s \ldots):  type it up.
