2024-06-13

Reflection: Leaving UC Davis

This week is my last week as a postdoctoral researcher at the UC Davis Institute of Transportation Studies (ITS-Davis). I am glad that I was able to transition from physics to transportation policy within the setting of academia and to particularly to so at ITS-Davis, which is renowned for having multidisciplinary transportation research & education that has included an increasing focus on issues of equity & accessibility in transportation planning. I learned so much, not just about transportation per se (which, in a professional context, was totally new to me when I started this job) but also about hiring, advising, and managing graduate students, applying for grants, managing grant-funded projects, communicating with different audiences beyond academia in many different forms, working on projects that are not just academic research ending in a peer-reviewed journal article, forming & managing relationships with stakeholders from government agencies, community-based organizations, and other organizations, and expanding my professional network on my own. This ultimately became the right time for me to leave ITS-Davis, but I will be grateful for the experiences & opportunities that I had in it and for the people that I got to work with.

2024-05-02

Finite Determinants of Linear Operators in Continuous Vector Spaces

Recently, I wondered whether it is possible for a linear operator in a continuous (infinite-dimensional) vector space to have a finite determinant. By "continuous vector space", I mean that the identity operator can be resolved for a complete orthonormal basis \( |\phi(x) \rangle \) for all \( x \) such that \( \langle \phi(x), \phi(x') \rangle = \delta(x - x') \) as \( \hat{1} = \int |\phi(x)\rangle\langle \phi(x)|~\mathrm{d}x \). If an operator \( \hat{A} \) has continuous matrix elements \( A(x, x') = \langle \phi(x), \hat{A}\phi(x') \rangle \), then it is easy to see that the conditions for its trace \( \operatorname{trace}(\hat{A}) = \int A(x, x)~\mathrm{d}x \) to be finite are that the integral must converge, so the "function" \( A(x, x) \) must asymptotically approach 0 strictly faster than \( 1/x \) as \( |x| \to \infty \) and must at most have singularities at finite points \( x_{0} \) that diverge strictly slower than \( 1/|x - x_{0}| \). This can be seen as the continuum limit of a sum over the diagonal. However, the determinant is harder to express in this way because it involves products over diagonals & subdiagonals that are harder to express in a continuum space.

For this post, I will only consider Hermitian positive-definite operators. The conditions that I will list for which the determinant exists for such operators are sufficient for the determinant to exist, but I am not convinced that they are necessary. If such operators have an eigenvalue decomposition \( \hat{A} = \int a(x) |\phi(x)\rangle\langle \phi(x)|~\mathrm{d}x \) where the vectors \( \{ |\phi(x) \rangle \} \) form a complete orthonormal basis and the eigenvalues satisfy \( a(x) > 0 \) for all \( x \), then one can make use of the identity \( \ln(\det(\hat{A})) = \operatorname{trace}(\ln(\hat{A})) \) to say that \( \ln(\det(\hat{A})) = \int \ln(a(x))~\mathrm{d}x \). For the right-hand side to converge, then \( \ln(a(x)) \) must asymptotically approach 0 with \( x \) as \( |x| \to \infty \) strictly faster than \( 1/x \), which means that \( a(x) \) must asymptotically 1 with \( x \) as \( |x| \to \infty \) strictly faster than \( \exp(1/x) \) (which is not the same as \( e^{-x} \)), and \( \ln(a(x)) \) can at most have singularities at finite points \( x_{0} \) that diverge strictly slower than \( 1/|x - x_{0}| \), which means that \( a(x) \) must either diverge to \( \infty \) strictly slower than \( \exp(1/|x - x_{0}|) \) or drop to 0 strictly slower than \( \exp(-1/|x - x_{0}|) \). For example, \( a(x) = \exp(1/(x^{2} + x_{0}^{2})) \) fits the bill; note that this is not the same as the Gaussian kernel \( \exp(-(x^{2} + x_{0}^{2})) \). Intuitively, this condition makes sense, because for a finite-dimensional diagonal matrix as the dimension becomes arbitrarily large, the diagonal elements must mostly be exactly or very close to 1 for the determinant to not grow arbitrarily large with the dimension.

In finite-dimensional vector spaces, it is also easy to compute the determinants of triangular matrices simply as the products of the diagonal elements. (This is why the determinant is most often computed by an algorithm like first computing the LU decomposition and then taking the product of the diagonal elements of the upper-triangular matrix, which for an \( N \times N \) matrix involves \( O(N^{3}) \) operations, as opposed to the Leibniz formula involving every permutation which involves \( O(N!N) \) operations.) In infinite-dimensional vector spaces, a matrix that is triangular in a countable basis can have the determinant computed similarly as in finite-dimensional vector spaces; if an operator \( \hat{A} \) in that basis has elements \( A_{ij} \), then using the definition \( \ln(|\det(\hat{A})|) = \prod_{i} \ln(|A_{ii}|) \), the determinant converges as long as the diagonal elements \( |A_{ii}| \) are mostly exactly or very close to 1, specifically such that as \( |i| \to \infty \), \( \ln(|A_{ii}|) \) decays to 0 strictly faster than \( 1/i \). (Note that \( i \) is an integer index written in slanted font, not the imaginary unit \( \operatorname{i} \) written in upright font.) However, I am not sure how to generalize this to operators that are expressed as triangular matrices in continuous bases.

2024-04-01

Transitioning from microscopic to macroscopic and quantum to classical regimes

I recently read two things that were of interest to me having previously worked in physics. One was an article in The New Yorker magazine [LINK], in which the author does a good job of going over the successes of mathematical modeling in the physical sciences and contrasting this with the limitations of mathematical modeling in public health (showing, for example, how many models of the spread of contagions fail when governments & societies take fast & drastic collective actions to limit the spread), the failures of mathematical models in social sciences where the outputs of those models can create feedback loops with public sentiment (for example in political polling), and the way that many people who use machine learning models in different domains expect the fancy curve-fitting of those models to represent fundamental understanding when that might not really be so. The other was a journal article published in Physical Review Letters [LINK] about how it can be possible to test the extent to which a massive (as opposed to massless) object which exhibits the dynamics of a simple harmonic oscillator and prepared in a quantum coherent state can be tested for deviations from classical behavior using a protocol that does not depend on the mass of the object (although I question this given that the protocol depends on timed measurements that depend on the frequency of oscillation, and in many physics contexts the frequency does depend on the mass as \( \omega = \sqrt{k/m}\), but this is somewhat of a quibble). These two things got me to think about something that I realized I never got out of many years of formal undergraduate & graduate education in physics. This can be illustrated with the following example.

In introductory physics classes that focus on Newtonian mechanics, a prototypical problem involves a block, modeled as a point mass, sliding (with or without friction) down a fixed triangular incline in the constant gravitational field of the Earth. In the context of those classes, instructors will be careful to note that this is merely a model, and corrections could come from the inclusion of the variation of the Earth's gravitational field & surface curvature, the technical possibility of moving the triangular incline (which must be much more massive than the block in question), the shape of the block, variations in the touching surfaces, air resistance, et cetera. In later classes, instructors may point out corrections due to special relativity (i.e. the speed of light) and general relativity (as it relates to the Earth's gravitational field).

However, in later classes about quantum mechanics & statistical mechanics, instructors explain how different the models are from models of Newtonian mechanics at human scales, but they often promise that appropriate treatments of aggregates of microscopic constituents can consistently recover results from Newtonian mechanics, yet this promise is almost never fulfilled. In particular, wavefunctions that describe pure states of single microscopic particles are quite far removed from the simple dynamical variables describing blocks on inclined planes, although statistical mechanics can probabilistically describe the solid states of the block & inclined plane as well as the gaseous state of the surrounding air, it is not usually extended to describe the dynamics of the block sliding down the inclined plane. For example, if a block sliding down a fixed inclined plane of horizontal angle \( \theta \) in a uniform gravitational field is described as having equations of motion \( m\ddot{x} = mg\sin(\theta) \) where the \( x \)-axis is defined as pointing downward parallel to the slope of the inclined plane for increasing \( x \) and the \( y \)-axis points outward in the normal direction from the inclined plane, then I wish to see corrections of the form \( m\ddot{\vec{x}} = \sum_{\mu = 0}^{\infty} \sum_{\nu = 0}^{\infty} \hbar^{\mu} k_{\mathrm{B}}^{\nu} \vec{f}^{(\mu, \nu)} \) where the lowest-order term is \( \vec{f}^{(0, 0)} = mg\sin(\theta)\vec{e}_{x} \). I have never seen these sorts of quantum or statistical corrections to Newtonian equations of motion in simple (in the context of Newtonian mechanics) systems. Similarly, it is rare to see how quantum or statistical mechanical systems can, in appropriate limits, reproduce classical systems; I can only think of the quantum coherent state of the simple harmonic oscillator as well as how the Moyal bracket in the phase space formulation of quantum mechanics reduces to lowest order in \( \hbar \) to the Poisson bracket, and in the latter case, intuitive construction of the quantum phase space quasiprobability function is made more difficult (compared to construction of a classical phase space probability density function, as I did in a post [LINK] from a few years ago) by the fact that unlike the classical phase space probability density function, the quantum phase space quasiprobability function cannot be arbitrarily localized in phase space, it can take on negative values for certain wavefunctions, it is compressible in phase space with respect to its own evolution over time, and it is not obvious how it should look for a system of many particles constituting a macroscopic object like a block (in contrast to a classical phase space probability density function, which for such a system could just be a product of Dirac delta functions localizing each microscopic constituent to a point in phase space).

These considerations reminded me of a discussion I had last year with friends from college, who also did course 8 (physics) with me. We came to a consensus that while people who do not become physics majors should, as usual, get exposure to Newtonian physics and the basics of electricity & magnetism, people who become physics majors should have a curriculum over 3-4 years that exhibits a sensible conceptual progression. In particular, after seeing Newtonian mechanics, such students should then be exposed to Lagrangian & Hamiltonian formulations of classical mechanics. The Lagrangian formulation of classical mechanics should then be used to develop intuitions about mechanical waves, which in turn can lead to introductions to classical field theory and development of classical electromagnetic theory as a rich example of a classical field theory. (I would also personally recommend using the introduction of mechanical waves to introduce the linear algebraic treatment of waves and then reintroduce the linear algebraic treatment of waves into the treatment of linear classical field theories in general & linear classical electromagnetic theory in particular.) The Hamiltonian formulation of classical mechanics should then be used to develop intuitions about probability distributions in classical mechanics, which in turn can be used to develop intuitions about statistical mechanics. Optionally, at this point, the Hamiltonian formulation of classical mechanics can also be used to develop intuitions about nonlinear dynamics & chaos theory, but while this is good for the broader education of physics students, it is less immediately relevant for the introduction of quantum theory to come soon after (because quantum mechanics is linear). Finally, only after these things happen should quantum theory be introduced, such that there are clear connections of the wavefunction formulation of quantum mechanics to mechanical waves, the phase space formulation of quantum mechanics to classical phase space probability distributions, and the linear algebraic framework of quantum mechanics to linear algebraic treatments of classical field theories (including linear classical electromagnetic theory); this will ensure that students understand how ideas like superposition, interference, rotation through a Hilbert space, statistical uncertainty, and related ideas are not unique to quantum mechanics (which is unfortunately too often a consequence of the way quantum mechanics is typically introduced in undergraduate curricula, at least in the US). We also came to a consensus that in each course, there should be clear explanations of what prototypical systems are analytically solvable, what prototypical systems are not analytically solvable, and why (in each case).

2024-03-01

Progression of Winter Storms across the Contiguous US

This winter has featured many winter storms over the contiguous US that have swept from the west coast to the east coast. In previous posts, I have discussed basic intuitions for why different climates occur in different regions [LINK], my assessment of the deficiencies of the Trewartha climate classification system [LINK], what I would change about the Trewartha climate classification system [LINK], how my proposed changes to the Trewartha climate classification system can be applied to understand what climates occur where in middle latitudes [LINK], why popular understanding of the effects of the Gulf Stream over the Atlantic Ocean on the climate of Europe is incorrect in many ways [LINK], and why different climates occur in coastal locations on different coasts at different latitudes [LINK]. These posts have suggested, among other things, that many winter storms on the east coast of the US would come from warm moist air from over the Gulf of Mexico or mild moist air from over the Atlantic Ocean colliding with cold dry air over the continent, but these collisions would be somewhat more sporadic because the prevailing westerlies, which would have dumped moisture primarily over the west coast, would be weak & dry by the time they reach the east coast. Thus, it is somewhat surprising to me that these winter storms seem to be driven by the prevailing westerlies over the continent. The following is my attempt to intuitively explain, based only on sea-/surface-level temperatures, air pressures, and air flows, why this happens. Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.

Why this happens in North America

This happens in North America mainly because of the arrangement of landmasses & seas/oceans. In the winter half of the year in North America, the subtropical ridge is strongest around 30 degrees in latitude (north of the equator) to the west of the continents of North America in the Pacific Ocean & of Africa in the Atlantic Ocean. Prevailing westerlies generated by the subtropical ridge over the Pacific Ocean bring moisture to the west coast of the US and turn clockwise due to the Coriolis force, meaning that around the time the prevailing westerlies reach the Rocky Mountains, they may have turned more toward the Gulf of Mexico, though this is not guaranteed to happen every time. In doing so, the prevailing westerlies, by this point colder & drier, can pick up warm moist air from the Gulf of Mexico. This clockwise turn by the Coriolis force is reversed within the Gulf of Mexico by southerly winds coming from air coming clockwise off of the subtropical ridge over the Atlantic Ocean, so this newly warmed & moistened air turns toward the east coast of the US, bringing moisture there before moving east & turning clockwise (again due to the Coriolis force) over the Atlantic Ocean toward Europe. This is how the subtropical ridge can function like a conveyor belt of moisture. Essentially, the continent of North America & the Atlantic Ocean are both narrow enough (with respect to the ranges of longitudes), and the Gulf of Mexico with warm water is favorably placed, to ensure that this can happen. That said, the prevailing westerlies will not always turn clockwise enough to go over the Gulf of Mexico and then counterclockwise enough to go over the east coast of the US, which is why the prevailing westerlies are more likely to bring moisture to the west coast of the US but only sporadically do so for the east coast of the US.

I should clarify that the storms that sweep across the contiguous US are often localized highly mobile systems of low pressure. They internally turn counterclockwise, but the motion of the centers of these storms is affected by the aforementioned prevailing westerlies coming from the subtropical ridges over the eastern Pacific Ocean & Atlantic Ocean in the northern hemisphere.

Why this does not happen in other continents

This does not happen in other continents because of unfavorable arrangements of landmasses & seas/oceans. I will give details for each continent in turn.

Eurasia

In the northern hemisphere, Eurasia & the Pacific Ocean are much wider (with respect to the range of longitudes) than North America & the Atlantic Ocean, so the conveyor belt effect is lost there; this point is amplified by the much stronger system of high pressure forming due to the settling of cold dry air over the continent in the winter half of the year. Additionally, the Indian Ocean (which would supply warm moist air) is not far enough from the equator and there are too many mountains in between for the Indian Ocean to function analogously to the Gulf of Mexico.

South America

The east coast of South America in the middle latitudes would refer to the east coast of Argentina. There is no major body of water immediately to the north (toward the equator) of Argentina analogous to the Gulf of Mexico, so although the subtropical ridge over the Atlantic Ocean to the west of South Africa is somewhat close by, the prevailing westerlies are largely dry by the time they reach Argentina and have no way of replenishing moisture & warmth before reaching the east coast.

Africa

In the southern hemisphere, Africa does not extend much into the middle latitudes. Thus, this issue is moot there.

Oceania

Oceania does not extend much into the middle latitudes and is surrounded by much more water, keeping the temperatures more moderate anyway (so there is less opportunity for big temperature contrasts between land & water to form, which would lead to stronger winter storms). Additionally, the Pacific Ocean in the southern hemisphere is much wider (with respect to the range of longitudes) than the Atlantic Ocean in the northern hemisphere, so the conveyor belt effect is lost there.

2024-02-02

My time at the TRB 2024 Annual Meeting

Last month, I attended the TRB 2024 Annual Meeting. The conference, which was held in DC, was a lot of fun. The graduate student researcher working with me was able to present our work as a poster. Additionally, although I didn't meet as many people whom I had not met before, this was a good opportunity for me to deepen connections with people with whom I had connected more briefly in past conferences in person or remotely. In particular, these included people working like me at the intersection of transportation & disability as well as people working in the autonomous vehicle industry. Strengthening connections with people in the former group was especially important to me because of the relevance of my work, how few of us there are in the US, and being able to feel like I am part of a group of like-minded academic researchers (given that I lead so much of my work essentially alone in the context of academic researchers). Strengthening connections with people in the latter group was important to me because of recent turbulence in the space of autonomous vehicles and the need for consistent pushes for inclusion of people with disabilities as users of such vehicles.

2024-01-01

Variations of Coastal Monsoon Climates with Latitude

I have learned about different aspects of the Earth's climate and shared what I've learned over 5 posts in 2022, including basic intuitions for why different climates occur in different regions [LINK], my assessment of the deficiencies of the Trewartha climate classification system [LINK], what I would change about the Trewartha climate classification system [LINK], how my proposed changes to the Trewartha climate classification system can be applied to understand what climates occur where in middle latitudes [LINK], and why popular understanding of the effects of the Gulf Stream over the Atlantic Ocean on the climate of Europe is incorrect in many ways [LINK]. Ultimately, my learning about different aspects of the climates of the world was done with the personal aim of understanding why cities on opposite coasts of the US at the same latitudes have such different climates, with those on the west coast having characteristically mild to hot arid rainless summers & cool (but not cold) rainy winters and those on the east coast having typically warm or hot humid rainy summers & cool or cold slightly drier but still rainy or snowy winters. I did learn about that to a great extent, but as I learned more, I started to question whether my previous intuitions (from when I started learning about different climates) were correct. Follow the jump to see more and the resolution to this problem. Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.