\documentclass[10pt]{amsart}
\parindent=0pt \parskip=8pt

\usepackage{amssymb,amsmath,amsthm}
\usepackage[mathscr]{eucal}

\usepackage{verbatim}
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{psfrag}

\input{isomath}
\input{mathenv}
\input{syms}

%% ================================================================
\begin{document}
\title{VIGRE Application Part II $\cdot$ Summer and Fall 2009}
\author{John Kerl}
\date{\today}
\maketitle

%% ----------------------------------------------------------------
\section{Timeline, professional development, and outreach activities}

I am finishing my fourth year of a five-year PhD program.  The period of
support (summer and fall 2009) will begin my fifth year.  The spring of 2009
saw the following:

\begin{itemize}

\item I successfully completed my comprehensive examination in January 2009.
I exposited a recent paper of Daniel Ueltschi \cite{bib:qmath}, which my
dissertation research is extending.

\item Ueltschi was present in Tucson on a sabbatical semester; he and I are
collaborating closely on my dissertation research.  In particular, my funding
this semester is from Ueltschi's NSF grant.  (In spring of 2008 I was also
funded under Ueltschi's grant.  The work I did at that time preceded my
comprehensive examination, but that work has led seamlessly into my
dissertation.)

\item I obtained access to the ICE cluster in the University of Arizona's High
Performance Computing center.  This reduces compute time, relative to my
previous use of the math department's chivo cluster, by an order of magnitude.

\item The local March 2009 workshop on Entropy and the Quantum, organized by
Daniel Ueltschi and Robert Sims, drew several visitors from the international
mathematical-physics community.  Among these was Volker Betz, Ueltschi's
co-worker at Warwick and chief collaborator.  During the workshop (as I write
this proposal), Betz and I are exploring a question in bulk flow of permutation
cycles.

\item Also during the workshop, John LaPeyre and I are sketching out a joint
paper which follows on work of Perseguers, Wehr, et al. \cite{bib:entdist}.  My
portion is an independent study project under Jan Wehr in spring of 2008.  I
obtained interesting results that spring, and presented them to the UA
mathematical physics seminar in October 2008; it would benefit me
professionally to write these results up nicely and submit them for
publication.

\item In the spring of 2009, I am taking Bill Faris' course on Discrete-Time
Stochastic Processes, which reinforces my current knowledge of Markov chains.
I am also taking Theory of Statistics from Walter Piegorsch.  This is a core
course in the statistics program; it requires no small amount of effort, but it
completes coursework for my PhD minor in statistics.

\item For fall 2008 and spring 2009, I am organizing the weekly Graduate
Student Colloquium.  I am also serving as this year's student representative to
the department's graduate committee.  In addition to the light committee work
per se, I am organizing the weekly grad tea as well as weekly lunches with the
departmental colloquium speaker.  I also contributed significantly to the
department's recruitment workshop in March 2009.

\end{itemize}

The period of support, summer and fall 2009, will include the following:

\begin{itemize}

\item Principally, I will continue my research.  This is discussed in detail
below.

\item I will deliver a contributed talk (already accepted) at the Conference on
Stochastic Processes and Their Applications in July in Berlin.  I will present
my worm algorithm for the random cycle model.

\item As time permits, I may visit Daniel Ueltschi in Marseille before that
conference.

\item Working with John LaPeyre, I will write up the above-described
percolation results for publication.

\item I have spoken with the department's graduate director about my
participation in the department's August integration workshop for incoming
graduate students; he was very enthusiastic.  I will work side by side with
incoming grads throughout the five-day workshop, guiding them through
team-based discovery-method projects which help them transition to advanced
mathematics.

\item Tom Kennedy currently has a group consisting of three graduate students,
two undergraduates, and himself making a multi-tiered examination of
self-avoiding random walks.  I have received his agreement about my
participation, which will include regular team meetings and the bridging of
computational and theoretical results.

\item I will present the progress of my research to the UA mathematical physics
seminar in the fall.

\item I will complete my final course --- Probability and Random Processes in
Engineering, in the Electrical Engineering department --- in fall 2009,
satisfying an out-of-department requirement.

\end{itemize}

In my final semester at the UA I will continue my dissertation work, with
timely graduation in May 2010.  I will, of course, be spending much of the year
applying for employment.  Thus, it is my desire to accomplish as much during
the summer and fall as possible.  VIGRE support will help me to achieve that
goal.

%% ----------------------------------------------------------------
\section{Plan of study and research}

My research is under Daniel Ueltschi, formerly of the University of Arizona,
currently at the University of Warwick.  We are studying the effects of
interparticle interactions on the critical temperature of Bose-Einstein
condensation.  Ueltschi is spending a sabbatical semester at the UA for spring
2009; by the time of his return to Warwick, I will be working largely
independently.  We will continue to communicate; as well, my local advisor, Tom
Kennedy, is up to date on my research and I will be working with him to bring
the dissertation to completion.

%% ----------------------------------------------------------------
\subsection{Historical context}

Bose and Einstein \cite{bib:bose,bib:einstein} predicted in the 1920s that
non-interacting particles with integer spin may collapse into a macroscopic
occupation of the ground state of the external potential.  Einstein predicted a
critical temperature for the phenomenon.  Thousands of research papers have
investigated BEC theoretically, as well as experimentally following the 1995
Nobel-prize-winning successes of Anderson et al. \cite{bib:boulder}.  Of the
many questions that could be asked about BEC, we restrict attention to that of
the effects of interbosonic interaction strength $a$ on the condensate's
critical temperature $T_c^{(a)}$.  Several research groups working over the
last three decades have obtained widely varying results; see
\cite{bib:baymetal} for a survey.  Current consensus within the theoretical
physics community \cite{bib:baymetal} is that
\begin{align}
	\Delta T_c &= \frac{T_c^{(a)}-T_c^{(0)}}{T_c^{(0)}}
\end{align}
is linear in $a$ for small $a$.  The main goal of this project is to determine
the constant of linearity with a high degree of certainty, for various
interaction models.

%% ----------------------------------------------------------------
\subsection{Background}

One begins with a Hamiltonian for particles with two-body interactions.
Specifically, consider a system of $N$ bosons in a cube $\Lambda \in \R^d$ of
size $L$ and volume $L^d$.  The positions of the particles are either on a unit
lattice, so $N=L^d$ with density $\rho=1$, or continuously distributed
according to a point process with variable density $\rho=N/V$.  The
interparticle potential $U$ is in terms of a scattering length $a$, where $a$
is nominally the radius of a hard-core potential.  The Hamiltonian for
pair-interacting particles is
\begin{align}
\label{eqn:bosonic_H}
	H &= -\sum_{i=1}^N \Delta_i + \sum_{i<j} U(\vecx_i-\vecx_j)
\end{align}
where $\Delta_i$ is the Laplacian and $U$ is a multiplication operator.  This
operator acts in the space $L^2(\Lambda^N)_{\textrm{sym}}$ of symmetric,
square-summable wavefunctions with periodic boundary conditions.

One may write down the partition function $Z_{\Lambda,N} = \Tr(e^{-\beta H})$
and apply a multi-particle Feynman-Kac formula, involving permutation symmetry
of bosonic wave functions, to obtain a Hamiltonian in which permutation jumps
rather than particles interact.  (The derivation is sketched in
\cite{bib:qmath,bib:bu}, and was worked out in full detail in my written
comprehensive paper.) A cluster expansion, to first order in the scattering
length of the particles, yields a Hamiltonian on $\SN$ with only jump-pair
interactions:
\begin{equation}
\label{eqn:perm_H}
	H_P(\vecx_1, \ldots, \vecx_N, \pi)
	= \frac{1}{4\beta} \sum_{i=1}^N \|\vecx_i - \vecx_{\pi(i)}\|^2
		+ \sum_{i<j} V(\vecx_i, \vecx_{\pi(i)}, \vecx_j, \vecx_{\pi(j)}).
\end{equation}
The jump-pair interaction $V(\vecx_i, \vecx_{\pi(i)}, \vecx_j, \vecx_{\pi(j)})$
may be interpreted as the collision probability for a pair of Brownian bridges
running from $\vecx_i$ and $\vecx_j$ to $\vecx_{\pi(i)}$ and $\vecx_{\pi(j)}$,
respectively, in time $2\beta$.

%The jump-pair interaction $V(\vecx_i, \vecx_{\pi(i)}, \vecx_j, \vecx_{\pi(j)})$
%may be interpreted as the probability that a Brownian bridge from
%$\vecx_i-\vecx_j$ to $\vecx_{\pi(i)}-\vecx_{\pi(j)}$ intersects the ball of
%radius $a$ centered at the origin.

Equation \ref{eqn:perm_H} gives rise to a Gibbs distribution $P(\pi) =
e^{-H(\pi)}/Z$ (point positions are either held fixed on the lattice, or
integrated out on the continuum) on the $N!$ permutations of the $N$ bosons.
Then, one may compute the expectation of various random variables in the finite
system.  For \cite{bib:qmath,bib:bu,bib:gru} as well as my computational work
in spring 2008, the random variable of interest was the fraction $f_I$ of sites
in macroscopic cycles.  As suggested by figure \ref{fig:fI_fS}, one may use
$f_I$ as an order parameter namely, a quantity which is zero on one side of a
phase transition and non-zero on the other.  For finite systems, the $f_I(L,T)$
is smooth in $T$; as $L$ increases, the graph approaches a non-analyticity at
$T_c$.

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics[scale=0.5]{figures/vigre_f_I_alpha_0.0.eps}
\includegraphics[scale=0.5]{figures/vigre_f_S_alpha_0.0.eps}
\caption{Order parameters $f_I$ and $f_S$ for finite systems.
	\label{fig:fI_fS}}
\end{center}
\end{figure}

Direct use of the Brownian bridge jump-pair interaction is prohibitive both
analytically and numerically.  The following approaches have been taken:

\begin{itemize}

\item One may drop interactions altogether.  This model was examined using
Markov chain Monte Carlo methods in \cite{bib:gru}, with particles distributed
on a cubic unit lattice.  (To date, \cite{bib:gru} is the only work in the
literature on the lattice case:  my dissertation extends this territory.) One
recovers the critical density for the Bose gas in the non-permutation
Hamiltonian (equation \ref{eqn:bosonic_H}), and finds experimentally that the
occurrence of infinite macroscopic cycles corresponds to condensation as
predicted by Feynman \cite{bib:feynman}.  An analogous result is proved in
\cite{bib:bu} for non-interacting particles distributed on a continuum.

\item One may drop all jump-pair interactions except the most strongly
interacting ones, namely, two-cycles \cite{bib:bu}.  Letting $N_2(\pi)$ denote
the number of two-cycles in the permutation $\pi$, one has a simple Hamiltonian
of the form
\begin{align}
	\HP(\vecx_1, \ldots, \vecx_N, \pi) &= \sum_{i=1}^N
		\frac{1}{4\beta}\|\vecx_i-\vecxpii\|^2
		+ \alpha N_2(\pi).
\end{align}
In \cite{bib:bu08}, the $N_2$ concept is extended to the form
\begin{align}
	\HP(\vecx_1, \ldots, \vecx_N, \pi)
	&= \sum_{i=1}^N \frac{1}{4\beta}\|\vecx_i-\vecxpii\|^2
	+ \sum_{\ell=2}^N \alpha_\ell N_\ell(\pi)
\end{align}
where $N_\ell(\pi)$ counts the number of $\ell$-cycles in the permutation
$\pi$.  Here, it is not known how to compute the $\alpha_\ell$'s to match the
Brownian-bridge interactions.  Nonetheless, one may examine this model for its
intrinsic interest --- say, with $\alpha_\ell$ constant in $\ell$.

\item Recent, unpublished collaboration between Betz and Ueltschi obtains lower
and upper bounds on the Brownian-bridge interactions; these can be computed
easily.  Collision probability for Brownian bridges is increased when paths
cross.  Thus, Betz expects permutation jumps to be preferentially aligned or
anti-aligned.

\end{itemize}

%% ----------------------------------------------------------------
\subsection{Current work}

I am presently pursuing two threads:  (1) re-working individual MCMC
simulations, each of which produces an order-parameter data point, and (2)
extrapolation and finite-size scaling which permit determination of the
critical temperature from order-parameter data.

In spring 2009, Ueltschi and I began to consider the superfluid fraction $f_S$
rather than $f_I$ as an order parameter.  As figure \ref{fig:fI_fS} suggests,
the two order parameters have the same critical temperature but different
critical exponents.  The choice of $f_S$ as order parameter was motivated by
consideration of path-integral Monte Carlo (PIMC) studies \cite{bib:ceperley,
bib:nl04, bib:bps06, bib:pst98, bib:gcl97, bib:hk99, bib:pc87, bib:pgp08,
bib:pr92}.  (One may think of PIMC as sampling from Brownian-bridge
interactions.)  The superfluid fraction is a simple function of the
\emph{winding number}: for a given permutation, one counts the number of
windings around the 3-torus (with topology emplaced by periodic boundary
conditions) in each direction.

We soon discovered that the Metropolis sampling algorithm of \cite{bib:gru}
suffers from a serious defect:  it creates winding cycles only with opposite
signs, so that permutations are created with zero winding number.  Another
concern (as yet unexplained) is that the superfluid fractions we are seeing
experimentally on the lattice exceed the unit interval.

To sample permutations with non-zero winding number, I reviewed the above-cited
PIMC studies, where the same problem arises, then adapted the PIMC concept of a
\emph{worm algorithm} to our random-cycle model.  Namely, one opens,
rearranges, and closes permutation cycles.  The open cycles are free to wrap
around the torus before closing.  As shown in figure \ref{fig:worm1}, an open
cycle on $N$ points may be viewed as a cycle on $N+1$ points.  I take the
$(N+1)$st point to be \emph{non-spatial}: it lies at no distance from any
lattice point.  This degree of freedom allows Metropolis udpates to explore the
full configuration space of lattice permutations on the 3-torus.

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics[scale=0.7]{figures/worm1.eps}
\caption{Open cycles as permutations on $N+1$ points.
	\label{fig:worm1}}
\end{center}
\end{figure}

I have proved several results for Metropolis updates on the extended
$(N+1)$-point system, including the aperiodicity and detailed-balance
conditions.  Full details will be provided in my dissertation.

The second thread of my current efforts, given the ability to correctly
determine the finite order parameter $f_S(T,L)$, is to take the $L \to \infty$
limit.  One technique, which I am developing ad hoc, is to extrapolate
pointwise in $T$ from data such as those in figure \ref{fig:fI_fS} to the true
$f_S$ curve for $L=\infty$.  At this point, it becomes easy, using log-log
plots and linear regression, to read off the critical temperature with error
bars.  In parallel, I am employing a finite-size scaling technique adapted from
several PIMC studies \cite{bib:nl04,bib:pc87,bib:pgp08,bib:pr92}.  Here one
assumes that the finite system's $f_S(T,L)$ is of the form
\begin{align}
	f_S(T,L) &=
		\frac{1}{L}\;
		Q\left( \left(\frac{T-T_c}{T_c}\right) L^{1/\nu}, \zeta L^{-\delta/\nu}
		\right)
\end{align}
for some analytic function $Q$ and some parameters $\nu$, $\zeta$, and
$\delta$.  Taylor-expanding to first order in the small arguments, followed by
some algebra, permits a two-step regression analysis of the MCMC data resulting
in a determination of $T_c$ with error bars.

My research goals for the summer and fall, in addition to other activites as
detailed as the start of this proposal, are quite precise:

\begin{itemize}

\item Write the worm-algorithm, superfluid-fraction, extrapolation, and
finite-size-scaling chapters of my dissertation.

\item Quantify $\Delta T_c$ for the $N_2$ model on the lattice; compare to the
continuum result in \cite{bib:qmath}.

\item Quantify $\Delta T_c$ for the $N_\ell$ model on the lattice with constant
$N_\ell$.  This topic is completely open in the literature.

\item Investigate angle correlation and $\Delta T_c$ for the Betz-Ueltschi
lower-bound and upper-bound interactions.  This topic is also completely open
in the literature.

\item All computations this summer will be done on the lattice.  Point-process
simulations may be performed in the fall, if we deem that to be of interest for
the dissertation.  A Poisson point process is a tractable approach; the true
point process for the Bose gas is very difficult to obtain.

\item Let $\ell_{\textrm{max}}$ be the length of the longest cycle in a
permutation; let $N f_I$ be the number of cycles in macroscopic cycles.  For
non-interacting and $N_2$-interacting spatial permutations, $\langle
\ell_{\textrm{max}}\rangle / N f_I$ is empirically found to be the same
(approximately 63\%) as for uniformly distributed permutations.  For $N_\ell$
interactions, this is no longer the case.  This phenomenon needs to be
characterized.

\end{itemize}

The above is the current list of priorities, although other questions will
certainly suggest themselves through the seven-month period of support.

%% ----------------------------------------------------------------
\begin{thebibliography}{}

% bib:qmath
% bib:entdist
% bib:bose
% bib:einstein
% bib:boulder
% bib:baymetal
% bib:bu
% bib:gru
% bib:feynman
% bib:bu08
% bib:ceperley
% bib:nl04
% bib:bps06
% bib:pst98
% bib:gcl97
% bib:hk99
% bib:pc87
% bib:pgp08
% bib:pr92

\bibitem{bib:qmath}
Ueltschi,~D.
\newblock {\em The model of interaction spatial permutations and its relation
to the Bose gas.}
\newblock \texttt{arXiv:0712.2443v3}.

\bibitem{bib:entdist}
Perseguers, Cirac, Ac\'{i}n, Lewenstein, and Wehr.
\newblock \emph{Entanglement Distribution in Pure-State Quantum Networks.}
\newblock Phys. Rev. A, vol. 77, issue 2, 2008.
\newblock \texttt{arXiv:0708.1025v2}.

\bibitem{bib:bose}
Bose,~S.N.
\newblock{Z. Phys. 26, 178 (1924)}.

\bibitem{bib:einstein}
Einstein,~A.
\newblock{Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 1, 3 (1925)}.

\bibitem{bib:boulder}
Anderson,~M.H., Ensher,~J.R., Matthews,~M.R., Wieman,~C.E., and Cornell,~E.A.
\newblock {\em Observation of Bose-Einstein Condensation in a Dilute Atomic
	Vapor}.
\newblock \emph{Science}, vol. 269, issue 5221 (Jul. 14 1995), 198-201.

\bibitem{bib:baymetal}
Baym,~G., Blaizot,~J.-P., Holzmann,~M., Lalo\"e,~F., and Vautherin,~D.
\newblock {\em Bose-Einstein transition in a dilute interacting gas}.
\newblock \texttt{arXiv:cond-mat/0107129v2}.

\bibitem{bib:bu}
Betz,~V. and Ueltschi,~D.
\newblock{\em Spatial random permutations and infinite cycles}.
\newblock \texttt{arXiv:0711.1188}.

\bibitem{bib:gru}
Gandolfo,~D., Ruiz,~J., and Ueltschi,~D.
\newblock {\em On a model of random cycles}.
\newblock \texttt{arXiv:cond-mat/0703315}.
\newblock Statist. Phys. 129, 663-676 (2007).

\bibitem{bib:feynman}
Feynman,~R.P.
\newblock {\em Atomic theory of the $\lambda$ transition in Helium}.
\newblock Phys. Rev. 91, 1291–1301, (1953).

\bibitem{bib:bu08}
Betz,~V. and Ueltschi,~D.
\newblock {\em Spatial random permutations with small cycle weights.}
\newblock \texttt{arXiv:0812.0569v1}.

\bibitem{bib:ceperley}
Ceperley,~D.M.
\newblock {\em Path integrals in the theory of condensed helium}.
\newblock Rev. Mod. Phys. vol. 67, no. 2, April 1995.

\bibitem{bib:nl04}
Nho,~K. and Landau,~D.P.
\newblock {\em Bose-Einstein condensation temperature of a homogeneous weakly
	interacting Bose gas:  PIMC study}.
\newblock \texttt{arXiv:cond-mat.stat-mech/0404238v1}.

\bibitem{bib:bps06}
Boninsegni,~M., Prokof'ev,~N.V., and Svistunov,~B.V.
\newblock {\em Worm algorithm and diagrammatic Monte Carlo:  A new approach
	to continuous-space path integral Monte Carlo simulations.}
\newblock Physical Review E \textbf{74}, 036701 (2006).

\bibitem{bib:pst98}
Prokof'ev,~N.V., Svistunov,~B.V., and Tupitsyn,~I.S.
\newblock {\em Exact, complete, and universal continuous-time worldline Monte
	Carlo approach to the statistics of discrete quantum systems.}
\newblock Journal of Experimental and Theoretical Physics, vol. 87, no. 2
	(1998).

\bibitem{bib:gcl97}
Gr\"uter,~P. and Ceperley,~D.
\newblock {\em Critical temperature of Bose-Einstein condensation of
	hard-sphere gases.}
\newblock Physical Review Letters, vol. 79, no. 19 (1997).

\bibitem{bib:hk99}
Holzmann,M. and Krauth,~W.
\newblock {\em Transition temperature of the homogeneous, weakly interacting
	Bose gas.}
\newblock Physical Review Letters, vol. 83, no. 14 (1999).

\bibitem{bib:pc87}
Pollock,~E.L. and Ceperley,~D.M.
\newblock {\em Path-integral computation of superfluid densities.}
\newblock Physical Review B, vol. 36, no. 16 (1987).

\bibitem{bib:pgp08}
Pilati,~S., Giorgini,~S., and Prokof'ev,~N.V.
\newblock {\em Critical temperature of interacting Bose gases in two and
	three dimensions.}
\newblock Physical Review Letters \textbf{100}, 140405 (2008).

\bibitem{bib:pr92}
Pollock,~E.L. and Runge,~K.J.
\newblock {\em Finite-size-scaling analysis of a simulation of the
	${}^4$He superfluid transition}.
\newblock Physical Review B, vol. 46, no. 6 (1992).

\end{thebibliography}

\end{document}
