Nonlocality and Infinite LDOS in Lossy Media

While I have written many posts on this blog about various topics in physics or math unrelated to my graduate work as well as posts promoting papers from my graduate work, it is rare that I've written direct technical posts about my graduate work. It is even more unusual that I should be doing so 2 years after leaving physics as a career. However, I felt compelled to do so after meeting again with my PhD advisor (a day before the Princeton University 2020 Commencement, which was held in person after a delay of 2 years due to this pandemic), as we had a conversation about the problem of infinite local density of states (LDOS) in a lossy medium.

Essentially, the idea is the following. Working in the frequency domain, the electric field produced by a polarization density in any EM environment is \( E_{i}(\omega, \vec{x}) = \int G_{ij}(\omega, \vec{x}, \vec{x}')P_{j}(\omega, \vec{x}')~\mathrm{d}^{3} x' \) which can be written in bra-ket notation (dispensing with the explicit dependence on frequency) as \( |\vec{E}\rangle = \hat{G}|\vec{P}\rangle \). The LDOS is proportional to the power radiated by a point dipole and can be written as \( \mathrm{LDOS}(\omega, \vec{x}) \propto \sum_{i} \mathrm{Im}(G_{ii}(\omega, \vec{x}, \vec{x})) \). This power should be finite as long as the power put into the dipole to keep it oscillating forever at a given frequency \( \omega \) is finite. However, there appears to be a paradox in that if the position \( \vec{x} \) corresponds to a point embedded in a local lossy medium, the LDOS diverges there.

I wondered if an intuitive explanation could be that loss should properly imply the existence of energy leaving the system by traveling out of its boundaries, so the idea of a medium that is local everywhere (in the sense that the susceptibility operator takes the form \( \chi_{ij}(\omega, \vec{x}, \vec{x}') = \chi_{ij}(\omega, \vec{x})\delta^{3} (\vec{x} - \vec{x}') \) at all positions) and is lossy at every point in its domain may not be well-posed as energy is somehow disappearing "into" the system instead of leaving it. Then, I wondered if the problem may actually be with locality and whether a nonlocal description of the susceptibility could help. This is where my graduate work could come in. Follow the jump to see a very technical sketch of how this might work (as I won't work out all of the details myself).


Book Review: "Algorithms to Live By" by Brian Christian & Tom Griffiths

I've recently read the book Algorithms to Live By by Brian Christian & Tom Griffiths. This book shows how many problems & heuristics in computer science can be applied to explain or improve human decision-making. Each chapter focuses on a certain class of problems or issues. Such classes include the optimal stopping problem, the multi-armed bandit problem, searching & sorting, task scheduling, Bayesian inference, overfitting data, constraint relaxation, random stimulus, communication protocols, and social interaction. Additionally, most chapters try to show how results from computer science can either improve or justify certain human behaviors.

This book was frustrating for me to read. If it had fully met my expectation that it would show, in a unified & consistent way, how these computer science problems apply to human behavior and connect to each other, I would be singing its praises. If it had completely failed, I'd be happy to rhetorically trash this book. Instead, I found that each chapter would be a great vignette on its own, and each chapter showed the great potential of what the book could have been, but the book failed to live up to that potential. First, there was very little connection among the chapters, and any acknowledgment that the authors did make of such connections was almost always superficial instead of deeply insightful. For example, the respective chapters about the optimal stopping problem, caches, and overfitting each could have been so much better with greater discussion about the connection to social pressure & game theory, yet those topics were discussed only in the last chapter, which I think was a mistake. Second, only in the concluding section did the authors make clear that they wanted to either improve or justify human behavior with each class of problems or issues. This because clear over the course of reading the book, yet there was very little guidance in each chapter about whether improvement versus justification would be the goal. Perhaps the worst offender was the chapter about constraint relaxation, as there was little connection to human behavior in a way that would be obvious to lay readers. These problems meant that reading the last numbered chapter (about game theory) and the conclusion felt simultaneously wonderful for finally seeing these concepts discussed clearly and maddening for knowing that the book could have been so much better if these ideas had been more consistently executed through the book.

There are two other minor criticisms I have of the book too. First, the chapter about overfitting seems to use the word "overfitting" to mean too many different things, which is ironic and undermines any clarity that the discussion could have provided. Second, the chapter about randomized algorithms attempts to make a tenuous connection between randomized algorithms used in computer science and the way that random mental stimuli can produce very creative responses in people, but it never makes clear whether the latter result is true at an individual level or only holds statistically for large populations.

Overall, I think the author's goals were laudable and that each chapter is interesting to read in isolation. However, other readers may be disappointed, as I was, in the way that the authors fail to synthesize many of the ideas across chapters in a smooth & unified manner. Thus, I would advise that readers who may be interested in these topics go into this book with lower expectations.


FOLLOW-UP: How to Tell Whether a Functional is Extremized

This post is a follow-up to an earlier post (link here) about how to tell whether a stationary point of a functional is a maximum, minimum, or saddle point. In particular, as I thought about it more, I realized that using the analogy to discrete vectors could help when formulating a more general expression for the second derivative of the nonrelativistic classical action for a single degree of freedom (i.e. the corresponding Hessian operator). Additionally, I thought of a few other examples of actions whose Hessian operators are positive-definite. Finally, I've thought more about how to express these equations for systems with multiple degrees of freedom (DOFs) as well as for fields and about how these ideas connect to the path integral formulation of quantum mechanics. Follow the jump to see more


How to Tell Whether a Functional is Extremized

I happened to be thinking recently about how to tell when a functional is extremized. Examples in physics include minimizing the ground state energy of an electronic system expressed as an approximate density functional \( E[\rho] \) with respect to the electron density \( \rho \) or maximizing the relativistic proper time \( \tau \) of a classical particle with respect to a path through spacetime. Additionally, finding the points of stationary action that lead to the Euler-Lagrange equations of motion is often called "minimization of the action", but I can't recall ever having seen a proof that the action is truly minimized (as opposed to reaching a saddle point). This got me to think more about the conditions under which a functional is truly maximized or minimized as opposed to reaching a saddle point. Follow the jump to see more. I will frequently refer to concepts presented in a recent post (link here), including the relationships between functionals of vectors & functionals of functions. Additionally, for simplicity, all variables and functions will be real-valued.


Google Chrome OS Flex and Broader Adoption of Linux

I recently read [Chin, The Verge (2022); Raphael, Computerworld (2022)] that Google is releasing a version of Chrome OS called Chrome OS Flex which can be copied to a USB storage drive and installed on computers that didn't come with Chrome OS. This seems very similar to how many popular Linux distributions work, so I initially wondered if Chrome OS Flex will succeed with the muscle of Google behind it where similar efforts by Linux distributions backed by smaller not-for-profit organizations have failed. At the same time, it seems clear to me that Google will not hesitate to use this as an opportunity to collect more valuable data from people who use Chrome OS Flex. This got me to think more broadly about how much ordinary people who might consider using Chrome OS Flex really care about their privacy (especially considering that such people would typically use Microsoft Windows 10/11, which are known to collect significant amounts of data from users) even after revelations about Facebook's practices, earlier revelations about government surveillance, and so on.

However, upon closer reading, I noticed that the first article makes clear that the target audience is schools & businesses which have many old computers whose Windows versions may no longer be supported. This makes more sense to me than targeting ordinary individuals, because I get the sense that the learning curve even to copy an ISO file onto a USB storage drive and install it onto a computer is steep for most ordinary individuals (despite the significant progress that distributions like Ubuntu & Linux Mint have made in making the installation process easy). By contrast, it seems more reasonable to expect specialists in schools & businesses to learn these things once and then do them for many different computers. Meanwhile, the second article makes clear that while traditional Chrome OS is capable of running many programs built for Windows, Chrome OS Flex will not have such capabilities. This suggests to me that while many schools may take up this opportunity given that few user-facing applications need to be installed on the computer and most user-facing applications can be accessed through web equivalents, this might not be the case for many businesses, so it is unclear to me which businesses will actually take this up.

Ultimately, I don't expect to see much adoption among ordinary individuals even though they aren't forbidden from installing & using Chrome OS Flex. That said, I would be interested to see how adoption evolves in schools & businesses over time.


Turning off Comments for this Blog Going Forward

This is a quick post to say that I have unfortunately had to turn off comments for this blog. Recently, I had trouble using the comment feature on various posts in this blog (despite being its sole author & maintainer). Furthermore, for the last few years, the vast majority of comments have been unwanted commercial solicitations, while I haven't really gotten much useful feedback about posts themselves. I recognize that there are very few humans who regularly read this blog and that most, if not all, of those readers probably know me personally, so I trust that they can reach out to me if they want to share their thoughts about posts. In any case, going forward, comments will no longer be available on this blog. This also necessarily means that the "Featured Comments" blog post series cannot continue.


Hit by a Car

I was recently hit by a car when crossing the street in Davis, California. This explains why I haven't posted anything on this blog this month until now (late in the month) and why I may be slower to post next month too. I'm sure I will have much more to say about this at a later date, but for now, for legal reasons, this is all I can say.


Book Review: "The Heartbeat of Wounded Knee" by David Treuer

I've recently read the book The Heartbeat of Wounded Knee by David Treuer. This book was a gift to me from a family friend who heard that I like reading about American history and other nonfiction topics, and when that person gave me this book, we further discussed the historical injustices inflicted by white Americans upon Native Americans, the deep connections that Native American religions & spirituality have with land & nature, and the way that Native Americans see themselves as truly indigenous as opposed to being traveling groups of humans like any other group passing through a given land. In that conversation, I also noted that reading this book would be professionally useful to me given that my professional interest in transportation equity would intersect with how Native Americans have been at the forefront of many recent social & environmental justice movements and how Native cultures & issues are much more visible in the Southwest than in the Northeast (due to the history of forced removal). It was with these conversations in mind that I read this book.

This book is a combination of historical accounts, contemporary interviews, and observations by the author. The author is a Native American (from the Ojibwe tribe) who grew up on a reservation, and his stated goal in writing this book was to create a more complete picture of how Native Americans have continued to live (especially but not exclusively under adverse conditions) in every part of the US since the Wounded Knee massacre in 1890, as he felt that too many people (both people sympathetic to Native Americans and those wishing to exterminate them) wrongly framed that massacre as the final death of Native American culture, life, and identity.

Coming into reading this book, I already knew a bit about how some Native people and tribes owned black slaves, the complex loyalties that Native tribes had for or against the US government at different points (especially in the Revolutionary War as well as the Civil War), Indian boarding schools, Native land management practices, the complex results of trade between Native tribes & European settlers, how Native tribal raids were often ceremonial instead of being true acts of war, and how some tribes have gotten guaranteed income from their casinos. However, I learned more about seminal battles & massacres, notable successes in Native governance in the 19th & 20th centuries, land allotments, termination of tribal governments, the way that some Native tribes became much more mobile & spread much more after the introduction of horses, the way that agriculture (especially of corn) spread throughout the Americas, the Native conception of the geography of the Americas (especially centering the Great Lakes, with river networks to the north and south leading away), how varying numbers of Native Americans remained in every part of the US even after forced removal, the Indian Rights movements of the 19th & 20th centuries, and how Native American cultures & identities were mixed & reformed many times in the crucible of shared suffering. It was also interesting to see the author directly compare the situation in Palestine to European settlement of the Americas, implicitly compare white American settlement of Native lands in the 19th & 20th centuries to rich white gentrification of urban neighborhoods which previously mostly had poor people of color, and explain that the fact that some but not all companies that helped white Americans move west in the 19th century had smallpox vaccine mandates for their [white American] employees so those companies that didn't have such mandates could be directly linked to the spread of smallpox in Native tribes (which bears striking parallels to current debates in the US regarding mandates for vaccines against this coronavirus).

Initially, I was a little put off by the meandering nature of the narrative, but I came to appreciate it more as the book progressed. Also, based on my expectations of this book, I wanted to see more of the promised discussion of Native American religions that arose from the crucible of oppression as well as from the broader 1960s American counterculture, as I figured there would be parallels with how Hinduism in North India had to reinvent itself in the crucibles of Mughal conquest and then British colonialism and later found common cause with American & British hippies. I was thus initially disappointed to not see so much discussion of those issues, but as I read further, I thought it was nice to read about what contemporary Native American individuals are actually doing in & for their communities instead of feeling beaten over the head with arcane mysticism. Similarly, I was initially disappointed by the lack of further discussion after the initial mention of how Native Americans see themselves as part of a people truly indigenous to North America, but as I read further, I thought it was nice to read about Native American individuals' & communities' actual modern subsistence practices & knowledge of the land instead of feeling beaten over the head by mysticism of a connection to the land that cannot be explained to people who aren't Native Americans.

I really appreciated that the author, in his desire to make Native Americans feel empowered to tell their own people's story in a way that doesn't necessarily end in 1890 as a tragedy, doesn't shy away from hard truths, like Native Americans owning black slaves, Native Americans helping European and later white American settles massacre other Native Americans, and more recently Native tribal officials not doing their duties by not meeting with executives of the Dakota Access Pipeline soon enough. The author of course doesn't deny the history of oppression and genocide, but he strikes a good balance between acknowledging genocide and acknowledging Native Americans' own agency in their own history as a way of showing that with empowerment comes responsibility. I also appreciated that the author didn't frame Native religions, land practices, or their lack of written records in mystical ways that ultimately fail to connect with people who aren't Native Americans. This is really an anti-cynical, forward-looking, optimistic book that I believe can be a light for all Americans in these bitterly divided & troubled times, so I recommend this book to anyone, especially in the US; I don't claim that this is the best book ever, but I do believe that reading it carefully can slowly plant seeds of cautious optimism about the US in a reader's mind. Furthermore, from my own perspective, while I don't claim that this book is a comprehensive overview of Native American history & culture, I can use it as a starting point to better frame my view of Native Americans, and I can feel a little less intimidated to read more deeply about Native American history, culture, and activism.


Functionals in Probability and Bayesian Inference

My work on transportation policy research in part involves conducting & analyzing surveys of people's travel behaviors & attitudes. Analyzing survey data requires an understanding of basic probability and statistics, which is an area that I previously felt I had just enough knowledge of to get by when learning about statistical physics but that I need to build more practical skills in now. In the process of refreshing my understanding of probability and statistics, I thought more about Bayes's theorem. In the context of hypothesis testing or inference, Bayes's theorem can be stated as follows: given a hypothesis \( \mathrm{H} \) and data \( \mathrm{D} \) such that the likelihood of measuring the data under that hypothesis is \( \operatorname{P}(\mathrm{D}|\mathrm{H}) \), and given a prior probability \( \operatorname{P}(\mathrm{H}) \) associated with that hypothesis, the posterior probability of that hypothesis is \[ \operatorname{P}(\mathrm{H}|\mathrm{D}) = \frac{\operatorname{P}(\mathrm{D}|\mathrm{H})\operatorname{P}(\mathrm{H})}{\operatorname{P}(\mathrm{D})} \] given the data. The key is that the denominator is evaluated as a sum \[ \operatorname{P}(\mathrm{D}) = \sum_{\mathrm{H}'} \operatorname{P}(\mathrm{D}|\mathrm{H}')\operatorname{P}(\mathrm{H}') \] where the label \( \mathrm{H}' \) runs over all possible hypothesis.

In practice, however, the set of hypotheses doesn't literally encompass all hypotheses but encompasses only one particular type of function with one or a few free parameters which then go into the prior probability distribution. For one free parameter, if the hypothesis specifies only the value of the (assumed continuous) parameter \( \theta \), if the prior probability of that hypothesis is given by a density \( f_{\mathrm{H}}(\theta)\), and the likelihood of measuring a (assumed continuous) data vector \( D \) under that hypothesis is the density \( f(D|\theta) \), then Bayes's theorem gives \[ f_{\mathrm{H}}(\theta|D) = \frac{f(D|\theta) f_{\mathrm{H}}(\theta)}{\int f(D|\theta') f_{\mathrm{H}}(\theta')~\mathrm{d}\theta'} \] as the posterior probability density under that hypothesis given the data.

I understood that in most cases, a single class of likelihood functions varied through a single parameter is good enough, and especially for the purposes of pedagogy, it is useful to keep things simple & analytical. Even so, I was more broadly unsatisfied with the lack of explanation for how to more generally consider summing over all possible hypotheses.

This post is my attempt to address some of those issues. Follow the jump to see more explanation as well as discussion of other tangentially related philosophical points.


Moved Closer to UC Davis

This is an update from the last few weeks. I have finally moved, a year later than originally planned, and live closer to my job at UC Davis. This has allowed me to go into the office regularly, which in turn has allowed me to meet my supervisors and colleagues in person after having worked remotely for a year. I was really looking forward to this because I felt that when I and my colleagues were all working remotely, it was much harder to exchange ideas, get feedback on my ideas and progress, and learn about the tricks of the trade. In particular, when everyone was working fully remotely, I would have needed to schedule meetings with colleagues in the absence of serendipitous in-person interactions, and others would feel similarly about each other too. I felt this made it easier (almost like the Nash equilibrium of a prisoner's dilemma game) for each individual to settle for limited scheduled virtual interactions punctuating long stretches of working alone without much feedback instead of making that effort to keep meeting regularly; I certainly felt that the need to schedule each remote meeting and the challenges associated with interacting remotely left me feeling more tired & less motivated than would be the case when working in person. Separately, I've been able to explore the town of Davis a little more: I appreciate the presence of sidewalks on every road, though they could be a little better designed. Plus, the weather right now is quite hot, and as climate change makes summers hotter & longer and fire seasons longer & more intense, I wonder how much longer I'll feel comfortable living here in the long-run, though I can see myself living here comfortably in the short-run.