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My apologies for the increasing lack of schedule to these letters – somehow all of my fine-tuned routines got disturbed this March but I am hoping to get back on track soon. Thanks for bearing with me.
OTOC
For many people that have grown up in the 90s, the phrase ‘butterfly effect’ inevitably conjures up Jeff Goldblum somewhat creepily holding Laura Dern’s hand in a rickety safari vehicle while riding through an island theme park populated with dinosaurs. In contrast to Goldblum’s mathematician character trying to leverage theoretical chaos in order to impress Dern, a more tangible chaos soon ensues and the Spielberg classic re-orients its focus to terrifying reptiles rather than something as dainty as a (theorized) butterfly. Yet, the butterfly effect is a notion that can be made mathematically rigorous, systems that display high sensitivity to initial conditions are still studied and a quantity dubbed ‘the butterfly velocity’ is a common feature of academic talks concerning such systems. First employed by Edward Lorenz in the 1960s, this metaphor is a rather elegant one: to imagine what a strong effect of initial conditions might look like one considers a butterfly flapping its wings and causing a small stream of air which then, weeks later, evolves into a tornado. Had the butterfly not been where it was, the tornado might not have happened – the small difference it caused had an exponential effect on the time evolution of the atmosphere (perceived as a physical system). A related idea when it comes to the quantum version of chaos is that of an out-of-time-order correlator (OTOC) or a measure of how the initial value of one operator* influences another at a much later time. In other words, this is a measure of how much the system ‘remembers’ its detailed initial state. As with Lorenz’s butterfly, systems that remember rather strongly are more chaotic and less likely to assume some equilibrium state in which all quantities have calmly averaged out.
Our intuitive notion of chaos, the one we invoke when hyperbolically describing the state of our apartments, or maybe lives on a particularly busy week, rarely has much to do with memory. We associate chaos with randomness and unpredictability. Often, memory is an anchor that can help weather a particularly chaotic period as remembering that we have made it through rough patches in the past can be both instructive and encouraging. Sometimes we associate chaos with wilderness and a lack of conscious organization or order. Sometimes chaos is just the easiest way to describe an overwhelming feeling growing from the pit of your stomach.
Lorenz first encountered chaos when studying a set of three differential equations meant to describe atmospheric convection i.e. a particular way in which parts of the atmosphere move when cooled in some places and warmed in others. For some parameters, Lorenz’s equations showed standard system dynamics and produced easy to understand trajectories in phase space. If he picked a point on a particular trajectory he could trace it further in time to the next point in phase space and then read off what system parameters will be at that point in the future. If he considered two points that were initially very close, the later time prediction would be similar enough that it looked like their trajectories were the same. However, for another set of system parameters, the equations produced a much more complicated phase space image – one in which starting at very similar points in parameter space took one along very different trajectories and produced very different outcomes. As time moves further forward the two points might come close together again and then diverge but with no clear periodicity. The figure that this process generates in phase space – the Lorenz attractor – resembles, in an act of cosmic irony, a butterfly and has been studied extensively with Lorenz’s work bringing attention to chaos theory as a promising branch of physics and mathematics.
Discussing a trajectory of someone’s life is a use of language akin to expressing a desire to talk about career paths or life journeys; this sort of language is reserved for planning and reflection and, occasionally, self-promotion. However, we do acknowledge our sensitivity to initial conditions when it comes to determining where we might end up in the phase space of life outcomes. The idea that one small thing can change one’s life or, less dramatically, one small interaction can ruin one’s day, has been explored and romanticized in films and on TV. In part, we cling to routines and repetitive cycles in order to protect ourselves of such small changes that could result in our ending up on a very different trajectory. Here as well memory is helpful. Remembering where we started helps us restart at that same point – or avoid it if a radically different outcome is preferable.
The fact that chaotic systems remember so well leads to complex physics, physics that we understand less than that of other dynamical systems and physics that has to deal with disorder and notions other than equilibrium. Physicists are typically excited to find such strong exceptions to theories of ‘nice’ dynamical behaviors they learn about early in their education. Quantizing chaos only makes the problem more appealing, as attractive professionally as the other kinds of, more personal, chaos are intimidating and off-putting off the page or outside of an experimental setup. There is a strange interplay of what we think chaos means in theory and the fact that, in some sense, we cannot personally avoid being at least somewhat chaotic.
In the past few weeks I have been working on a physics-themed theater production my advisor and various collaborators are premiering in a few days, honoring a very round birthday of a Nobel laureate in my department. A large portion of the cast consists of physicists rather than actors and the production has been an experience in learning a different mode of work for many. A couple of nights ago, while trying to hot glue LED lights to a number of paper Chinese lanterns, I overheard an interviewer reporting on the show bring this up to the director. In her response she stressed that the opportunity to bring theater to those who are not formally associated with it delights her as she believes we are all naturally storytellers and could always use more tools to tell stories. The claim rang powerful and true; if nothing else we tell ourselves stories about ourselves to make sense of where we are and how we got there. A few days later I spent a morning in a physics symposium celebrating the same Nobel laureate. Without missing a beat each speaker had some story to tell about their past interactions that transcended work and served to paint a picture of what this great man might be really like beyond his incredible mind. As the morning progressed the talks became more technical and soon it was all business as usual – quantum decoherence, cuprate superconductors, almost superfluids – but the beginning of the symposium was an exercise in storytelling and memory more than anything else.
As yet another old white man launched into a tale about a chance encounter in a corridor of some famous British institution, a friend remarked to me that the similarity of these narratives was boring. I was more struck by the fact that everyone spoke of the Nobel laureate as if he had been the exact same person throughout these many years; as if he was a character whose past self was so heavily correlated with his present that they were almost exactly the same. The irony of this is that, as for most of us, his life must have been way more chaotic than that, way more dependent on small, mundane moments – maybe even some of those anecdotal encounters with colleagues that occurred at just the right time and just the right place – than anyone was willing to acknowledge.
The Lorenz attractor is a fractal. Its Hausdorff dimension is between two integers which means that it is not smooth like a circle but rather ‘rough’ like an intricate coastline whose curves one can follow down to the scale of winding around each individual grain of sand. It is self-similar meaning that ‘zooming in’ on any part of it reproduces the same image. This makes it very complex and difficult to analyze. A physicist with a strong sense of whimsy finds this beautiful, others would be more likely to call the structure rich; both words seem fit to be coupled with the notion of complexity.
You can see where this is going: people are complex and rich and beautiful too. Yet, we run away from the notion of chaos and mostly stick to something akin to personality OTOCs – the idea that we know who we are because we so strongly depend on our past and who we used to be. However, as we engage in our natural urge to tell stories it is worth recalling just how chaotic we can be and how often very small things, seemingly insignificant, take us on completely different trajectories. As a consequence, none of us are as monolithic and predictable as the stories we tell ourselves about ourselves may suggest. Further, even the smallest things we do can have much larger consequences than we envision. That somewhat uncomfortable flirtation scene in Jurassic Park serves to set the stage for the destruction and danger that follows it; it provides a semi-scientific backdrop for dinosaur special effects and vague messaging about capitalism and nature. As usual, chaos gets a bad reputation. However, the potential for goodness at the same scale remains equally possible. For that Nobel winner that even I have had a chance to run into in a corridor, at least some of the small niceties and pleasant interactions others described must have spiraled into wonderful things. Papers being born over drinks and lunches are near legendary in physics circles anyway, despite being an outcome that may not be obvious from the starting point of the interaction. Then, if even the smallest things can take us to unexpected places, we might as well pay attention to them and strive to make them nice and kind.
Best,
Karmela
P. S. I first learned about Lorenz, chaos and fractals by reading a popular science book by J. Gleick that a family friend lent to me on a whim. I was so fascinated by it that I got my grandfather to give me one of his hardcover notebook with yellow wide-ruled pages and took endless color-coded notes. When I moved to the United States at sixteen I brought this notebook with me so that I would not forget about everything I had basically inhaled off of the pages of Gleick’s book. I have not thought about this in years, and probably would not have had I not started seeing OTOCs and the butterfly velocity appear in seminars. My understanding of quantum chaos is severely limited, and it is related to my research only in very tangential ways at best but having the memory of how much I cared for it when I was younger is still heartwarming.
* In quantum mechanics, each measurement of a physical quantity corresponds to an expectation value of an operator acting on the state the system is most accurately described by. Suppose you have a marble rolling around the bottom of a fruit bowl. In classical mechanics you could measure its position at some given time with a ruler and mathematically represent that measurement as just a value of the position as a function of time. In quantum mechanics the analogous process entails writing down the energy state i.e. a wave-function corresponding to ‘rolling around the bottom’ then applying a position operator to it in order to represent a measurement being made. Next, one calculates an expectation value of the position operator that can then be compared to the outcome of a set of repeated, realistic position measurements. Of course, for the quantum mechanical, probabilistic, approach to be necessary the marble and the fruit bowl would have to be, roughly, smaller than a Bucky-ball.
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ABOUT THE PAST FEW WEEKS
LEARNING: In the time since my last letter, I spent a week being on Spring Break with my boyfriend, and I have worked on my advisor’s theater production on top of my two research projects and taking a course on college teaching I enrolled in as a part of a future faculty fellowship. Things have definitely been somewhat chaotic, especially coming on the heels of my participating in a strike and attending an out of state conference in prior weeks.
On the research front, my group used Spring Break as a good time to catch up on work we may have missed while in Los Angeles, so I participated in two Skype discussions related to my project dealing with ultracold atoms on the International Space Station and vortices in such systems. It is always very exciting to hear updates from NASA and I am still sort of floored by the notion that my work is even tangentially related to something that will genuinely make it into space over the course of this year. The issue of vorticity is not likely to be studied in space anytime soon due to technical difficulties that come with implementing a rotating superfluid on an atom chip in a space station (which I am led to believe is as complicated as it sounds) but from a theoretical stand point the problem is rather rich and interesting. While I have been working on non-physics problems for a large fraction of this past week (being the Production Manager on an interdisciplinary, experimental theater piece is also as complicated as it sounds) I have been keeping a long list of ideas about superfluid vortices in the back of my mind and I am hoping to get to devote time to at least some of them soon.
While research work is usually my absolute first priority, as my advisor got more and more swept up in her theater project in these past few weeks so did I. I am writing this portion of the letter a few days after a very successful premiere of her show and one day before its second and final performance. The show itself is a fifty minute story about two travelers guided by the spirit of knowledge who, looking like a cross between a flapper and Amelia Earhart, immerses them into the innards of various quantum systems. It is a Dante’s Inferno type of affair except with much more whimsy and a lot more science (and almost no damnation and suffering). This somewhat classic storytelling structure makes it into a great vehicle for explaining the quantum aspects of our world in a way that is not prohibitively formal or difficult but still represents the relevant ideas in a scientifically accurate way. Given that it’s production has also been driven largely by undergraduates and volunteers, working on it has also been an incredible experience in collaborative work and I remain impressed by movement pieces, projections and original music students have contributed to it.
In my role as the Production Manager for the show I learned a lot about what really happens behind the scenes of out-of-the-box, inter-disciplinary art projects and mostly it involves countless emails and extremely patient conversations with everyone from administrators and secretaries to students chasing deadlines. I have physically carried lights and chairs, I helped craft a large prop that ended up being unsuitable for the actual show, I worked out the logistics of officially borrowing and transporting a table from a university classroom to a hotel ballroom and, since I do happen to be a physicist, I consulted on how to depict concepts such as the Fermi sea and Cooper pairing in oil paints and light projection. And I bought a truly outrageous amount of batteries to power little hand-held lights used by actors depicting photons. The whole process has been quite a challenge for my organizational and ‘soft’ or ‘people’ skills and though it has been quite stressful I think it has made me feel more prepared for the next instance when someone randomly puts me in charge of an event. However, I am, not unexpectedly, starting to feel myself crave the usual routine of long research days interrupted by talks and seminars and finished by trips to the gym rather than late night hang-outs at impromptu performance venues. It has been very special working on this show but I think it’s quickly dawning on me that my regular work is pretty special as well.
Finally, I have also been spending a fair amount of time on the teaching course I am enrolled in for the remainder of the semester and I am somewhat disappointed to acknowledge that my original impression of it – the one of being largely unimpressed – has not changed much in the past two weeks. Although being able to familiarize myself with research into teaching has definitely been informative and helpful the structure of the class feels stiff, the textbooks we are referencing discuss undergraduates in ways that often sound overly condescending and the faculty, though friendly, do not always seem to be taking their own advice. At the same time, having to reference long lists of in-class activities to complete assignments and spending a day creating a graphical and written syllabus for a course I have assisted in teaching in the past has given me what feels like a more realistic taste of what teaching might be like for young faculty. This sort of insight is very valuable, and I am hoping to use this course, all of its flaws not withstanding, to identify specific practices that could be useful to me and my students in the future and that I likely would not be exposed to through the rather uniform mode of teaching required from teaching assistants in physics.
LISTENING: In thinking about either clinging to memories or aiming to recognize patterns in one’s own life, I was reminded of this recent episode of Invisibilia in which, in addition to an exploration of one particular human story, a set of researchers have to come to grips with not even big data always being able to save us from randomness of life. I was similarly reminded of the past few episodes of Dear Franklin Jones in which the narrative has moved close enough to present day that the host has to confront just who he had been when he decided to devote himself to his parents’ cult leader. On a less factual yet equally powerful note, this episode of Welcome to Night Vale (first podcast I had ever listened to over five years ago!) in which Cecil realizes how muddy his own past and memories are remains to be one of my favorites – even outside of the elder gods infused friendly desert town community that is Night Vale, our teenage years and memories can seem like they belong to someone foreign, disconnected and potentially slightly monstrous.
One the music side of things, a friend from Croatia recently tagged me on Facebook to share ten albums that have over time been my favorites and that I consider to have been very influential on me even if I don’t revisit them often. In no particular order, here are the albums that I shared:
Iron Maiden, Powerslave
Cult of Luna, Salvation
Johnny Cash, American IV: The Man Comes Around
Ash Borer, Bloodlands
Fuzz, Fuzz
Ramones, It’s Alive
Boris, Pink
Vektor, Outer Isolation
Janelle Monae, Archandroid
Opeth, Blackwater Park
Many of these likely deserve their own letter and the more I think about the list the more I am torn between claiming that it represents me well and that it is fairly incomplete so I am deferring this discussion for another day but stand firmly by all ten of my choices.
WATCHING: Given that I did get a break and a chance to spend time with my boyfriend since my last letter, it may not come as a surprise that I have consumed a pretty serious amount of media in that time. In the week that my boyfriend was visiting we finished Netflix’s Altered Carbon and made our way through a good chunk of their internationally produced show Dark. Last week I also finally saw Blade Runner 2049 in the local art theater during a rare evening of freedom from chasing down theater props or crunching through equations. Oddly enough, all three of these pieces deal with memories and a sense of self as constructed around them.
Altered Carbon was a fun watch with some interesting details sprinkled in alongside a heavy dose of polished cyberpunk visuals, but I was overall not taken by it as much as I had hoped I would be. The show raises some interesting questions about whether being human is contingent on being able to die and what would happen to any of us as coherent personalities if we could hop between bodies, but the treatment of those issues is inconsistent and often embedded in the fairly shaky main plot. It is a shame that the main villain is so not compelling and with such poorly developed motivations, which really takes away from the rest of the story, because the actors give some great performances and I could see this being a great show if just paced somewhat differently and treated with a bit more respect and less manic need to show us neon lights and futuristic guns. My boyfriend remarked that compared to the old cyberpunk adage ‘hi tech, lo life’ it seemed almost too polished and not gritty enough and I am inclined to agree. While I cannot speak about the books it was based on, the visual identity of this show seemed too reminiscent of other people’s visions of the dystopian future but cleaned up and turned to eleven.
While we did not have a chance to finish Dark, it was definitely my favorite out of the two Netflix offerings we had sampled. I heard it described as ‘art house German Stranger Things’ on the Watch and that is likely as compelling of a pitch as anything I can write and gives a good overview of the elements that make this show work. It is a show about time and time travel and whether people can change or have to stay locked in personality loops. It is also a show about nuclear power, the 80s, immaculately styled German teenage angst and dead birds falling out of the sky. I am looking forward to finishing it.
It is hard to write anything overly coherent about Blade Runner 2049 without taking up a tremendous amount of space (for that there is always this very good episode of Supercontext) but I was generally quite impressed with it. Partly, this is a consequence of my having fairly low expectations, as I do for most sequels, but mostly I do believe that this was objectively one of the better films I had seen in a while. Denis Villenueve managed to recreate the gloominess and the absolutely fantastic architecture of the original film without seeming repetitive or uncreative, the cast excelled and while some of the plot points around them were executed in ways that may not be the most socially aware, the main themes of the film left me with the feeling that I need to digest them, or chew on them, further. The film does have a major problem with representation in the sense that in a world filled with Japanese and Korean symbols, where everyone eats noodles, there is not a single Asian person in sight and the only people of color we see are either criminals or afforded less than a minute on the screen. Even if one were to argue that the future will be that much more segregated than the present, the director has done a poor job conveying that. The issues of gender that the movie touches upon are less clear-cut and I am willing to believe that misogyny was consciously woven into this world and plays a role in the story that is in part about autonomy, labor and being more of a product than a person (as, for instance, Facebook users who’s data got scrapped might feel like already, about 31 years early). The last part of the film does seem to be in need of some polish and the same can be said of its antagonists who do not get to develop as much as I would have wanted, but despite being absolutely gorgeous this film does offer a plethora of true ‘lo life’ moments and I am glad I had found the time to immerse myself in that particular vision of future bleakness.
EATING: As a testament to how busy I have been recently, all of the meals I have considered sharing in this letter have been either very quick or mostly improvised – like rice noodles with a spicy miso sauce that seems to morph every time I make it or a breakfast bowl of savory oatmeal, beans and whatever else is lurking in my fridge and fits a vaguely Mexican theme. Ultimately, I settled on a baked shawarma tofu (adapted from this recipe) I had made a few times this year and that can serve as great component to more meals than just noodles. I am inclined to not give a spiel on tofu here since there are many resources out there on the web explaining how to make it taste good and why it is not terrible for your health (and the environment) as long as it is made with organically grown soy. Other than having developed a taste for it over the years of not eating meat, I am really not that much of an expert anyway. A typical block of tofu makes for four or five servings so this is a good meal prep option and since it relies heavily on savory spices it is versatile enough for salads, sandwiches or grain bowls, all of which I very much enjoy.
You will need:
1 block tofu
2 tablespoons olive oil
1 teaspoon salt
2 teaspoons ground coriander
1 teaspoon turmeric
1 teaspoon garlic powder
Press the tofu: cut the block horizontally in half then wrap in a paper towel and put on a cutting board or a large plate. Place something heavy on top of the tofu, some books on top of another cutting board or a skillet, and let sit for at least 20 minutes. After 20 minutes the paper towel will be drenched and you should see some liquid on the plate as well. Discard it and cut the tofu into bite-sized pieces (or any other desired shape)
Make the marinade: in a small bowl mix the rest of the ingredients until a liquid yet thick paste forms.
Mix: place the tofu into a small container or a Ziploc bag and pour in the marinade. Shake until all pieces are evenly covered
Bake: bake the tofu on a sheet pan lined with parchment paper, evenly spaced and brushed with any leftover marinade, for 20-25 minutes (depending on the size of your pieces of tofu) at 400F. Check and flip the tofu pieces after about 12 minutes. When they are done, they should be springy but not too hard and golden in color with a nice spice crust.
Serve: serve with rice, steamed veggies and cashew cream, as a protein in sandwiches, or toss on top of a cucumber and tomato salad topped with parsley and lemon juice. I mostly opted to eat it with this curry roasted cabbage (extremely easy yet mind-blowingly good), some paprika roasted eggplant, cucumber slices and homemade hummus.
Tips: The marinade will work on basically any vegetable in addition to the tofu so to turn this recipe into a sheet pan dinner double all of the ingredients except for the tofu and add a couple of red potatoes or half a head of cauliflower to the sheet pan after mixing them with the oil and the spices in the same way as above. Serve with a side salad and hummus or just top with lemon juice, chopped parsley and a bit of tahini. Alternatively, add zucchini and cherry tomatoes and serve over rice or pasta.
If I were to use this preparation of tofu in a sandwich, I would cut it into thin slabs and bake it for no longer than 20 minutes. As I am slightly obsessed with homemade hummus at the moment my sandwich would likely still consist of hummus, tofu, cucumbers and maybe a tomato on some variety of seedy bread or possibly in a whole wheat wrap.