Differential Geometry

I have been enjoying the last couple of days preparing for my Statistical Mechanics class. My new angle is to throw in a bit of differential geometry to expose students to its beauty as well as to give them a new and powerful tool to solve problems.

This morning, I was going over my notes on canonical transformations - a topic that I also covered in Classical Mechanics last semester. One concept that I still find a tad uncomfortable is in the manipulations of the quantities q, the coordinate, and q dot, its associated velocity. Central to the calculus of variations is the assumption that these two variables are independent. I can logically understand how this is the case, but my gut protests.

One of my colleagues, who shares an interest in the beauty of differential geometry, gave me the book Applied Differential Geometry, by William L. Burke. I noticed recently the most magnificent dedication on its inner cover, which reads, "To all those who, like me, have wondered how in the hell you can change q dot without changing q."

I can't wait to dive into this wonderful book; but, it will have to wait until I meet a looming proposal deadline and have prepared/delivered an invited talk in a few days. I must get back to preparing for my class, which meets in 45 minutes. Mornings before class are one of my favorite times...

Recent paper accepted

All papers are written with expectations that they will be appreciated by the scientific community. That's why authors are indignant when a manuscript gets rejected by reviewers before it can see the light of day in a scientific journal.

Recently, Shoresh submitted a manuscript to JOSA B (see http://lanl.arxiv.org/PS_cache/arxiv/pdf/1006/1006.1320v2.pdf for a preprint). In this work, he used Monte Carlo calculations, which were first implemented by my son, to let the computer role the dice to randomly determine transition moments and matrix elements of a hypothetical quantum system. To make the results consistent with quantum mechanics, we used a procedure that constrains the choices to be consistent with the sum rules. By rolling the dice millions of times, we can get a feeling for the properties of a much broader range of systems than one could synthesize in the laboratory.

In a paper that we published three years ago, we used this approach to study the hyperpolarizability. In the recent work, we aplied the approach to the second hyperpolarizability. The calculations yielded the same kind of surprising result, which this time around were a tad bit less unexpected, and that is that the Monte Carlo approach seems to give certain properties that are not observed experimentally nor predicted theoretically using standard Hamiltonians. The upshot is that some very exotic systems with a large second hyperpolarizabilities may be lurking out there - yet to be discovered.

Every time I submit a new paper, I fret that my long string of acceptances will come to an end. Sure, I have had manuscripts that got rejected, but eventually they get published once I fix some relatively minor error. W were delighted to have another acceptance. The reviewers' summaries follow this post. The detailed comments are technical in nature and have been omitted. The bottom line is that the paper is slated to be published in the next couple of months. While we are pleased, we are already working on the next two papers, each which extend our work to the next level. The more we learn, the longer term our goals become.

Reviewer comments appear here:

Reviewer 1
This manuscript is closely related to similar work on the first
hyperpolarizability by two of the authors (ref. 13) which sought
to understand the gap between the theoretical maximum
hyperpolarizability and the highest values measured experimentally.
The present paper extends this approach to the second
hyperpolarizability, γ, whose theoretical maximum value was derived
in ref. 2. In theory, the second hyperpolarizability in the zero-
frequency limit depends only upon the energies of the excited
states of the system and the transition dipole moments connecting
these states with each other and with the ground state. (Actually
the usual expression involves both transition dipole moments and
permanent dipole moments, but it can be transformed to one that
involves only transition dipoles as shown in ref. 19.)  There are
also fundamental sum rules that relate these quantities.  In this
manuscript, the authors use a Monte Carlo method to randomly sample
different combinations of excited state energies and transition
dipole moments, always requiring that they be constrained by the
general sum rules. The goal is to gain some insight into the
physical parameters needed to produce γ values near the theoretical
limit. The largest second hyperpolarizabilities found by this method
do approach the analytically calculated theoretical maximum.
Furthermore, these results indicate that the largest γ values are
found when only three states—the ground state and two excited
states dominate.

The manuscript is clearly written and well motivated.  While the
derivations refer to much previous work from the corresponding
author and his co-workers, it is not necessary to have read those
papers to understand the results presented here.  The results do
not provide much help to those trying to design real molecules with
large second hyperpolarizabilities.  However, they are certainly
interesting and this is a worthwhile addition to the literature of
this field from one of its
most original thinkers.

There are a few typos/misspellings/grammatical errors that do not
interfere with the readability of the paper.

Reviewer 2
This article presents very interesting new results and should
be published as soon as possible provided that some minor
changes/clarifications are addressed...