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\begin{document}
\title{VIGRE Application Part II $\cdot$ Fall 2008}
\author{John Kerl}
\date{\today}
\maketitle

%% ----------------------------------------------------------------
\section{Timeline}

I am currently in my third year of a five-year PhD program.  The period of
support (fall 2008) will begin my fourth year.  During that time, I will take
two courses, continue my research, and complete my comprehensive examination.

\begin{itemize}

\item In fall 2008, I will take Mathematical Physics (Math 541), which will
emphasize classical statistical mechanics, and Quantum Mechanics (Physics
570A), which will fulfill half of my out-of-department coursework requirement.
Both courses will provide key grounding for my research area (described in
detail below).

\item I will continue to participate in the mathematical physics seminar.  I
presented my spring 2008 work this April; this fall I will describe ongoing
progress.

\item My comprehensive examination will take place in the fall; I will exposit
a paper of Volker Betz and Daniel Ueltschi \cite{bib:bu} which is the basis for
my dissertation research.

\end{itemize}

The combination of coursework, comprehensive exams, and continued research will
be challenging; hence my request for VIGRE support for the fall.

This summer, before the period of support, I will be attending the summer
school on Current Topics in Mathematical Physics at the Erwin Schr\"odinger
Institute.  I will spend the remainder of the summer preparing for the
comprehensive examination.

In the longer term, I will take second-semester Quantum Mechanics (Physics
570B) and Theory of Statistics (Math 566) in spring of 2009.  The former will
complete my out-of-department requirement; the latter will complete my PhD
minor in Statistics.  I will then be well prepared for continued research and
summer school(s)/conference(s) in summer 2009, followed by research and
dissertation in the academic year of 2009-2010 (my fifth year) with timely
graduation in May of 2010.

%% ----------------------------------------------------------------
\section{Professional development and outreach activities}

I will be organizing the departmental weekly Graduate Colloquium for the
upcoming year; I have finalized this agreement with Dan Champion, the current
coordinator.  Duties include scheduling speakers and publicizing the series.

Tom Kennedy has approved my request to serve as super TA for his graduate
probability (564) course for fall 2008.  I took this course in spring of 2007
and enjoyed it thoroughly.  In particular, Tom's pedagogy as well as the course
content changed the course of my career.  I am excited about probability and
equally eager to work with and learn from Tom; I will convey this excitement to
the students.  Math 564 has for decades been cross-listed with the
undergraduate 464 course; now, for the first time, 564 will be convened
separately.  The course is non-measure-theoretic:  it bridges an important gap
between the undergraduate curriculum and the fully abstract treatment in Math
563.  In particular, the course plays a key role as a core course for the new
Graduate Interdisciplinary Program in Statistics, and is expected to have high
graduate enrollment.  Thus, my contributions can positively influence many
students as well as the Mathematics and Statistics programs in general.

As a supplementary plan, I have consulted with Dorin Dumitrascu, who has led
high-school visitations this year.  Since Dorin will not be coordinating these
visits next fall, we will be under new leadership.  According to Bill Velez,
that leadership will not be chosen until the fall.  I will be working with that
as-yet-unnamed faculty member to help schedule speakers and provide
transportation.  As well, I will conduct my own presentations.

I had been a bit reluctant to give such talks in the past.  Although I am a
frequent and avid contributor to the Graduate Colloquium (see Part I of this
application), I did not know what I could say to high-school students.  About a
month ago, though, I realized that I do have an appropriate topic.  Before
graduate school, when working at Lockheed Martin in an engineering role, I
heard that the Cassini probe's newly acquired orbit around Saturn would be
circularized by firing reverse thrust at periapsis, rather than apoapsis as my
(errant) intuition would have suggested.  To help me visualize this, I coded up
a Runge-Kutta solver for the two-body problem (and later, I implemented a
three-body solver) with graphical controls for thrust:  namely, thrust was
applied only during the duration of a mouse press, in a specified direction
with respect to the velocity vector.  As a result of that experiment, it became
clear to me why periapsis is the appropriate location for a circularizing burn:
it has everything to do with the fact that non-thrusted trajectories in the
two-body case are ellipses.  The subsequent orbit must pass through the point
at which thrust was ceased on the previous orbit; yet, kinetic energy was lost
and so the semimajor axis of the orbit must shorten.  This topic combines
theoretical, computational, and graphical results, in a way that gave me (and
will give students) a yes-I-can attitude toward problem solving, mathematics,
and the universe beyond our planet.

%% ----------------------------------------------------------------
\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.

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

Bose-Einstein condensation is the macroscopic occupation of a lowest-energy
quantum state by a system of bosons.  The effect was theoretically predicted in
the mid 1920s,  but it occurs at such a low temperature --- at the nanokelvin
scale! --- that it received relatively little attention.  Only in 1995 were
Bose-Einstein condensates produced in the laboratory by E.~Cornell and
C.~Wiemann, using sophisticated cooling techniques.  They, along with
W.~Ketterle, were awarded the 2001 Nobel Prize for this work.  BECs have now
enjoyed a resurgence of interest by theorists as well as experimentalists.
Their practical applications at this point might be compared to
electromagnetism in the mid 1800s:  they contain unknown possibilities, but it
is certain that there is new physics to be discovered.

The work of S.~Bose and A.~Einstein \cite{bib:bose,bib:einstein} predicts a
critical temperature $T_c$ for the so-called ideal Bose gas consisting of $N$
non-interacting particles in a domain $\Lambda \subset \R^d$.  R.~Feynman's
approach \cite{bib:feynman} is as follows.  The Hamiltonian for the
$N$-particle system is
\begin{align}
\label{eqn:bosonic_H}
	\bfH &= -\sum_{i=1}^N \Delta_i + \sum_{i<j} U(x_i-x_j)
\end{align}
where $\Delta_i$ is the Laplacian and $U$ is a multiplication operator modeling
the interaction potential.  This operator acts in the space
$L^2(\Lambda^N)_{\textrm{sym}}$ of symmetric, square-summable wavefunctions
with Dirichlet boundary conditions.  Feynman then expands the partition fuction
$Z_{\Lambda,N} = \Tr(e^{-\beta \bfH})$
in the Feynman-Kac representation \cite{bib:faris}:
\begin{equation}
\label{eqn:Z}
\begin{aligned}
	Z_{\Lambda,N} = \frac{1}{N!} &\int_{\Lambda_N} \dd x_1, \cdots \dd x_N
		\sum_{\pi\in\caS_N} \int
			\dd W^{2\beta}_{x_1, x_{\pi(1)}}(w_1) \cdots
			\dd W^{2\beta}_{x_N, x_{\pi(N)}}(w_N) \\
	& \cdot
	\exp\left\{ -\frac{1}{2}
	\int_0^{2\beta} \dd \tau \sum_{i<j} U(w_i(\tau) - w_j(\tau))
	\right\}.
\end{aligned}
\end{equation}
Here, $\caS_N$ is the group of permutations on $N$ symbols and $\beta$ is
proportional to inverse temperature as usual; $\dd W^{2\beta}_{x_i,x_{\pi(i)}}$
is the Wiener measure for trajectories $w_i$ running from $x_i$ to $x_{\pi(i)}$
in time $2\beta$.  Feynman then considers the lengths of cycles formed by
permutations in $\caS_N$, claiming that Bose-Einstein condensation occurs if
and only if there are infinite cycles in the thermodynamic limit, namely,
$N,|\Lambda|\to\infty$ with fixed density $N/|\Lambda|$.

Ueltschi considers a cube $\Lambda \in \R^d$ of size $L$ and volume $L^d$.  The
interparticle potential $U$ is in terms of a scattering length $a$, where $a$
is nominally the radius of a hard-core potential.  The state space of random
permutations in the Feynman-Kac representation is $\Omega_{\Lambda,N} =
\Lambda^N \times \caS_N$.  Letting $\ell_x(\pi)$ denote the length of the
permutation cycle containing a site $x$, the density of sites in cycles of
specified length --- the quantitative setting for Feynman's infinite cycles in
the thermodynamic limit --- is defined to be
\begin{equation}
	\rho_{m,n}(\pi) = \frac{1}{V} \,
	\#\{i=1, \ldots, N: m \le \ell_{x_i}(\pi) \le n\}.
\end{equation}
The expected value of this density is taken with respect to permutation
probabilities:
\begin{equation}
\label{eqn:E_rho}
	E_{\Lambda,N}[\rho_{m,n}] = \frac{1}{Z_{\Lambda,N} N!}
		\int_{\Lambda^N} \dd x_1 \cdots x_N
		\sum_{\pi\in\caS_N} \rho_{m,n}(\pi) e^{-H(x_1, \ldots, x_N, \pi)}.
\end{equation}
That is, energy (of a permutation) is the logarithm of probability (of a
permutation), as is usual in statistical mechanics, and the denominator is a
normalizing factor.  A thermodynamic limit of this expectation quantifies the
probability of the occurrence of infinite cycles.  The new $H$ appearing here
is a Hamiltonian for positions and permutations.  Applying a cluster expansion
in the small parameter $a$ to the bosonic Hamliltonian (equation
\ref{eqn:bosonic_H}), one obtains
\begin{equation}
\label{eqn:perm_H}
	H = \frac{1}{4\beta} \sum_{i=1}^N \|x_i - x_{\pi(i)}\|^2
		+ \sum_{i<j} V(x_i, x_{\pi(i)}, x_j, x_{\pi(j)}).
\end{equation}
The central point of Ueltschi's approach is that the energy has been recast in
terms of permutations:  the new Hamiltonian $H$ is more amenable to analysis
than the original Hamiltonian $\bfH$.  Terms in the first sum count lengths of
permutation jumps; terms in the second sum count interactions between pairs of
permutation jumps.  (Higher-order interactions between triples of jumps, etc.,
are omitted.)  For the interjump potential $V$, one obtains
\begin{equation}
\label{eqn:V}
	V(x,y,x',y') = \int \left[ 1 - \e^{-\frac{1}{4\beta}
		\int_0^{4\beta} U(w(s)) \,ds}
	\right] \dd \hat{W}^{4\beta}_{x-x', y-y'}(w)
\end{equation}
where $x$ jumps to $y$, $x'$ jumps to $y'$, and $d\hat{W}$ is a unit-normalized
Wiener measure on Brownian bridges $w$ running from $x-x'$ to $y-y'$ in time
$4\beta$.
It can be shown that $V(x, y, x', y')$ is the probability that a Brownian
bridge from $x-x'$ to $y-y'$ intersects the ball of radius $a$.

In the non-interacting case, one has scattering length 0 which corresponds to
$V=0$ in equation \ref{eqn:perm_H}.  This model was treated in \cite{bib:gru}.
A detailed exposition of the interacting model is in \cite{bib:bu}, with a
summary in \cite{bib:qmath}.  Intermediate between the non-interacting and
interacting models is the $r_2$ model of \cite{bib:bu}, wherein the only
interactions that are retained in equation \ref{eqn:perm_H} are those between
the jump pairs comprising two-cycles.  For technical reasons, the $r_2$ model
is significantly more efficient to simulate than the interacting model, and so
merits continued attention.

%% ----------------------------------------------------------------
\subsection{Current results}

I have worked under Daniel Ueltschi in the spring of 2008.  To date, the
following progress has been obtained.

The non-interacting results of \cite{bib:gru} were first reproduced.  This gave
me confidence in my Monte Carlo simulations.  The computational approach is as
follows:  Given an inverse temperature $\beta$ and a lattice configuration of
$N=L^d$ points, for $L$ ranging from 10 to 50 and $d=1,2,3$, generate a
sequence of permutations on the $N$ points.  These permutations are generated
using the Metropolis-Hastings algorithm (a particular algorithm of Monte Carlo
type), sampling from the non-uniform probability distribution which is implicit
in the right-hand side of equation \ref{eqn:E_rho}.  Compute the sample means
$\rho_{0,k}(\pi)$ over all the permutations $\pi$ generated.  Using this
averaged $\rho_{0,k}$, estimate the probability that the origin is in a
long-range cycle.  Repeat this process, varying $\beta$, to find the critical
$\beta_c$ (and thus the critical temperature $T_c$) at which the transition to
long-range cycles occurs.

A key problem at the start of the project was the detection of thermalization.
The Metropolis-Hastings algorithm is justified theoretically by an invariant
distribution of a particular Markov chain.  However, the full Markov matrix is
never written down:  indeed, for this problem, with a system of 1000 particles,
the matrix would have dimensions 1000! by 1000!  In practice, it is generally
difficult to detect when the Metropolis-Hastings simulation has reached an
invariant distribution, or has \emph{thermalized} in statistical-mechanics
jargon.  The work of Gandolfo and Ruiz \cite{bib:gru} used manually detected
thermalization.  I implemented an automatic thermalization criterion based on
counting the number of turning points of a time-smoothed system energy
function.  The result matches the manual detection results across the full
range of parameter values.

The $r_2$ model was simulated soon thereafter, although I deferred tabulated
computations of critical temperature until recently, as Daniel and I were
pressing hard toward progress on the fully interacting model.  Currently, $r_2$
simulations are in progress on a departmental minicluster.  The computational
approach is as outlined at the start of this section, except that now one also
varies the scattering length $a$ in order to estimate the critical temperature
$T_c(a)$.  It is already clear that $r_2$ interactions do raise the critical
temperature; continued simulations will allow me to quantify the interaction
dependence of the critical temperature.

The main difficulty in simulating the interacting model has been computation of
the interjump potential $V(\cdot)$.  The defining equation (equation
\ref{eqn:V}) is in terms of Brownian bridges:  for each proposed permutation
modification in a step of the Metropolis-Hastings algorithm, one must sum over
all affected interactions between pairs of permutation jumps.  For each
interaction pair, one must generate $N_b$ Brownian bridges, each sampled at
$N_p$ points on a time-step mesh from $0$ to $4\beta$.  For each bridge, one
must detect whether the trajectory of the bridge has intersected the ball of
radius $a$ centered at the origin.  None of these steps is conceptually
difficult, yet the product of the number of operations is quite large.  Thus,
we assessed the parameters $N_b$ and $N_p$ and considered techniques to avoid
redundant computation.

It is well known that that the standard deviation of Brownian motion is
$\sqrt{\Delta t}$ for a time-step mesh of $\Delta t$.  Since the time step is
$\Delta t = 4 \beta / N_p$, one must increase $N_p$ by a factor of 100 in order
to reduce the motion's standard deviation by a factor of 10.  The standard
deviation of a Brownian bridge is different:  it scales as $t(T-t)/T$.
Nonetheless, the standard deviation of a Brownian bridge is of order
$\sqrt{\Delta t}$.  Thus, it is perhaps no surprise that the bridge size $N_p$
must be exceedingly large (in experiments, on the order of 500,000) in order
for $V(\cdot)$ to stabilize.  Interestingly, increasing $N_b$ only decreases
the error bars around the stochastic value $V(\cdot)$; bias can only be removed
by increasing $N_p$.

One may reduce the excessive time burden of these computations by precomputing
a database of Brownian bridges which run from the origin in $\R^d$ back to the
origin in time 1.  Then, one may draw from that database to rescale, using a
simple change of variable, to an arbitrary bridge.

One may further reduce the computational burden by taking advantage of
symmetries of $V(\cdot)$.  The potential function $V(\cdot)$ is rotationally
invariant; likewise, the probability that a Brownian bridge from $x-x'$ to
$y-y'$ intersects the ball of radius $a$ centered at the origin will not change
if the bridge is run backward.  Thus
$$
	V(x,y,x',y') = V(x',y',x,y).
$$
As well, since only the differences $x-x'$ and $y-y'$ matter, we have, for all
real $b,c$, the double translation invariance
$$
	V(x+b, y+c, x'+b, y'+c) = V(x, y, x', y').
$$
Taking advantage of all these symmetries, we may write the potential in terms
of $r_1=\|x'-x\|$, $r_2=\|y'-y\|$, and $\theta$ being the angle between $x-x'$
and $y-y'$.  Then, one may tabulate equation \ref{eqn:V} over a range of $r_1$,
$r_2$, $\theta$, $\beta$, and $a$, and then use interpolation at run time.

Creation of the bridge database, with $N_p=500,000$ and $N_b=1000$, is
feasible:  one needs many hours of compute time and several hundred megabytes
of disk space, but the database, once computed, is reusable.  Creation of the
$V(\cdot)$ database appeared more daunting.  Fortunately, Betz and Ueltschi
have recently developed a Riemann-integral formula for $V(\cdot)$, which is
valid to lowest order in $a$.  Now, the Brownian-bridge database is
unnecessary.  The $V$ database has been generated and is in computational use
at present.  (By change-of-variable tricks we may yet be able to obtain a
closed-form expression for $V(\cdot)$.  If so, we will certainly use it.)

As with the $r_2$ case, it is clear that the interactions raise the critical
temperature as expected from recent consensus within the physics community.
Currently-running simulations are providing the data needed to quantify this
dependence of critical temperature on scattering length.

%(The $a$ dependence in the integral equation is trivial and may be factored in at run time.)

%% ----------------------------------------------------------------
\subsection{Future work}

The results obtained up to this point may be extended in theoretical and
experimental directions.  We will begin pursuing these directions during the
period of support.

\begin{itemize}

\item The first order of business is to unfreeze particle positions:  they were
held frozen only as a convenience.  The positions follow a point process;
however, the process it is not currently known.  One may conjecture a Poisson
process, but it has been shown \cite{bib:deviations} that the positions for
bosons in a condensate do not satisfy a large-deviation principle which would
be expected for Poisson.  Thus, one may choose to nonetheless simulate a
Poisson process --- in terms of initial placement of particles, as well as
particle motion during Monte Carlo simulations --- and learn whatever lessons
are to be had.  Also, though, one would like to characterize the true point
process at the theoretical level.

\item It is known that Bose-Einstein condensation occurs only in dimension
$d=3$.  Yet one could replace the $\|x_i-x_{\pi(i)}\|^2$ term in the
Hamiltonian by a more general $\xi(x_i,x_{\pi(i)})$ to study effects in
dimensions 1 and 2.

\item The cluster expansion used to obtain equation \ref{eqn:perm_H} is
non-rigorous and needs mathematical justification.  It is expected to work for
$a$ much smaller than interparticle spacing (e.g. on the cubic lattice).  For
the point process, where interparticle spacing may approach the scattering
length $a$, further work will be required.

\end{itemize}

% other random variables?

% non-gaussian weights for $d=2$?


%% ----------------------------------------------------------------
%\section{References}
\begin{thebibliography}{}

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

\bibitem{bib:faris}
Faris,~W.
\newblock {\em The Feynman-Kac formula}.
\newblock Notes for the Arizona Mathematical Physics Seminar (2004).

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

\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:deviations}
Lebowitz,~J., Lenci,~M., and Spohn,~H.
\newblock {\em Large deviations for ideal quantum systems.}
\newblock J. Math. Phys. 41, 1224-1243 (2000).


\end{thebibliography}

\end{document}
