Wednesday, January 17, 2018

Pure Nerd Fun: The Grasshopper Problem

illustration of grasshopper.
[image: awesomedude.com]
It’s a sunny afternoon in July and a grasshopper lands on your lawn. The lawn has an area of a square meter. The grasshopper lands at a random place and then jumps 30 centimeters. Which shape must the lawn have so that the grasshopper is most likely to land on the lawn again after jumping?

I know, sounds like one of these contrived but irrelevant math problems that no one cares about unless you can get famous solving it. But the answer to this question is more interesting than it seems. And it’s more about physics than it is about math or grasshoppers.

It turns out the optimal shape of the lawn greatly depends on how far the grasshopper jumps compared to the square root of the area. In my opening example this ratio would have been 0.3, in which case the optimal lawn-shape looks like an inkblot

From Figure 3 of arXiv:1705.07621



No, it’s not round! I learned this from a paper by Olga Goulko and Adrian Kent, which was published in the Proceedings of the Royal Society (arXiv version here). You can of course rotate the lawn around its center without changing the probability of the grasshopper landing on it again. So, the space of all solutions has the symmetry of a disk. But the individual solutions don’t – the symmetry is broken.

You might know Adrian Kent from his work on quantum foundations, so how come his sudden interest in landscaping? The reason is that problems similar to this appear in certain types of Bell-inequalities. These inequalities, which are commonly employed to identify truly quantum behavior, often end up being combinatorial problems on the unit sphere. I can just imagine the authors sitting in front of this inequality, thinking, damn, there must be a way to calculate this.

As so often, the problem isn’t mathematically difficult to state but dang hard to solve. Indeed, they haven’t been able to derive a solution. In their paper, the authors offer estimates and bounds, but no full solution. Instead what they did (you will love this) is to map the problem back to a physical system. This physical system they configure so that it will settle on the optimal solution (ie optimal lawn-shape) at zero temperature. Then they simulate this system on the computer.

Concretely, the simulate the lawn of fixed area by randomly scattering squares over a template space that is much larger than the lawn. They allow a certain interaction between the little pieces of lawn, and then they calculate the probability for the pieces to move, depending on whether or not such a move will improve the grasshopper’s chance to stay on the green. The lawn is allowed to temporarily go into a less optimal configuration so that it will not get stuck in a local minimum. In the computer simulation, the temperature is then gradually decreased, which means that the lawn freezes and thereby approaches its most optimal configuration.

In the video below you see examples for different values of d, which is the above mentioned ratio between the distance the grasshopper jumps and the square root of the lawn-area:





For very small d, the optimal lawn is almost a disc (not shown in the video). For increasingly larger d, it becomes a cogwheel, where the number of cogs depends on d. If d increases above approximately 0.56 (the inverse square root of π), the lawn starts falling apart into disconnected pieces. There is a transition range in which the lawn doesn’t seem to settle on any particular shape. Beyond 0.65, there comes a shape which they refer to as a “three-bladed fan”, and after that come stripes of varying lengths.

This is summarized in the figure below, where the red line is the probability of the grasshopper to stay on the lawn for the optimal shape:
Figure 12 of arXiv:1705.07621

The authors did a number of checks to make sure the results aren’t numerical artifacts. For example, they checked that the lawn’s shape doesn’t depend on using a square grid for the simulation. But, no, a hexagonal grid gives the same results. They told me by email they are looking into the question whether the limited resolution might hide that the lawn shapes are actually fractal, but there doesn’t seem to be any indication for that.

I find this a super-cute example for how much surprises seemingly dull and simple math problems can harbor!

As a bonus, you can get a brief explanation of the paper from the authors themselves in this brief video.

Tuesday, January 16, 2018

Book Review: “The Dialogues” by Clifford Johnson

Clifford Johnson is a veteran of the science blogosphere, a long-term survivor, around already when I began blogging and one of the few still at it today. He is professor at the Department of Physics and Astronomy at the University of Southern California (in LA).

I had the pleasure of meeting Clifford in 2007. Who’d have thought back then that 10 years later we would both be in the midst of publishing a popular science book?

Clifford’s book was published by MIT Press just two months ago. It’s titled The Dialogues: Conversations about the Nature of the Universe and it’s not just a book, it’s a graphic novel! Yes, that’s right. Clifford doesn’t only write, he also draws.

His book is a collection of short stories which are mostly physics-themed, but also touch on overarching questions like how does science work or what’s the purpose of basic research to begin with. I would characterize these stories as conversation starters. They are supposed to make you wonder.

But just because it contains a lot of pictures doesn’t mean The Dialogues is a shallow book. In contrast, a huge amount of physics is packed into it, from electrodynamics to the multiverse, the cosmological constant, a theory of everything and to gravitational waves. The reader also finds references for further reading in case they wish to learn more.

I found the drawings were put to good use and often add to the explanation. The Dialogues is also, I must add, a big book. With more than 200 illustrated pages, it seems to me that offering it for less than $30 is a real bargain!

I would recommend this book to everyone who has an interest in the foundations of physics. Even if you don’t read it, it will still look good on your coffee table ;)




Win a copy!

I bought the book when it appeared, but later received a free review copy. Now I have two and I am giving one away for free!

The book will go to the first person who submits a comment to this blogpost (not elsewhere) listing 10 songs that use physics-themed phrases in the lyrics (not just in the title). Overly general words (such as “moon” or “light”) or words that are non-physics terms which just happen to have a technical meaning (such as “force” or “power”) don’t count.

The time-stamp of your comment will decide who was first, so please do not send your list to me per email. Also, please only make a submission if you are willing to provide me with a mailing address.

Good luck!

Update:
The book is gone.

Wednesday, January 10, 2018

Superfluid dark matter gets seriously into business

very dark fluid
Most matter in the universe isn’t like the stuff we are made of. Instead, it’s a thinly distributed, cold, medium which rarely interacts both with itself and with other kinds of matter. It also doesn’t emit light, which is why physicists refer to it as “dark matter.”

A recently proposed idea, according to which dark matter may be superfluid, has now become more concrete, thanks to a new paper by Justin Khoury and collaborators.

Astrophysicists invented dark matter because a whole bunch of observations of the cosmos do not fit with Einstein’s theory of general relativity.

According to general relativity, matter curves space-time and, in return, the curvature dictates the motion of matter. Problem is, if you calculate the response of space-time to all the matter we know, then the observed motions doesn’t fit the prediction from the calculation.

This problem exists for galactic rotation curves, velocity distributions in galaxy clusters, for the properties of the cosmic microwave background, for galactic structure formation, gravitational lensing, and probably some more that I’ve forgotten or never heard about in the first place.

But dark matter is only one way to explain the observation. We measure the amount of matter and we observe its motion, but the two pieces of information don’t match up with the equations of general relativity. One way to fix this mismatch is to invent dark matter. The other way to fix this is to change the equations. This second option has become known as “modified gravity.”

There are many types of modified gravity and most of them work badly. That’s because it’s easy to break general relativity and produce a mess that’s badly inconsistent with the high-precision tests of gravity that we have done within our solar system.

However, it has been known since the 1980s that some types of modified gravity explain observations that dark matter does not explain. For example, the effects of dark matter in galaxies become relevant not at a certain distance from the galactic center, but below a certain acceleration. Even more perplexing, this threshold of acceleration is related to the cosmological constant. Both of these features are difficult to account for with dark matter. Astrophysicists have also established a relation between the brightness of certain galaxies and the velocities of their outermost stars. Named “Baryonic Tully Fisher Relation” after its discoverers, it is also difficult to explain with dark matter.

On the other hand, modified gravity works badly in other cases, notably in the early universe where dark matter is necessary to get the cosmic microwave background right, and to set up structure formation so that the result agrees with what we see.

For a long time I have been rather agnostic about this, because I am more interested in the structure of fundamental laws than in the laws themselves. Dark matter works by adding particles to the standard model of particle physics. Modified gravity works by adding fields to general relativity. But particles are fields and fields are particles. And in both cases, the structure of the laws remains the same. Sure, it would be great to settle just exactly what it is, but so what if there’s one more particle or field.

It was a detour that got me interested in this: Fluid analogies for gravity, a topic I have worked on for a few years now. Turns out that certain kinds of fluids can mimic curved space-time, so that perturbations (say, density fluctuations) in the fluid travel just like they would travel under the influence of gravity.

The fluids under consideration here are usually superfluid condensates with an (almost) vanishing viscosity. The funny thing is now that if you look at the mathematical description of some of these fluids, they look just like the extra fields you need for modified gravity! So maybe, then, modified gravity is really a type of matter in the end?

I learned about this amazing link three years ago from a paper by Lasha Berezhiani and Justin Khoury. They have a type of dark matter which can condense (like vapor on glass, if you want a visual aid) if a gravitational potential is deep enough. This condensation happens within galaxies, but not in interstellar space because the potential isn’t deep enough. The effect that we assign to dark matter, then, comes partly from the gravitational pull of the fluid and partly from the actual interaction with the fluid.

If the dark matter is superfluid, it has long range correlations that give rise to the observed regularities like the Tully-Fisher relation and the trends in rotation curves. In galaxy clusters, on the other hand, the average density of (normal) matter is much lower and most of the dark matter is not in the superfluid phase. It then behaves just like normal dark matter.

The main reason I find this idea convincing is that it explains why some observations are easier to account for with dark matter and others with modified gravity: It’s because dark matter has phase transitions! It behaves differently at different temperatures and densities.

In solar systems, for example, the density of (normal) matter is strongly peaked and the gradient of the gravitational field near a sun is much larger than in a galaxy on the average. In this case, the coherence in the dark matter fluid is destroyed, which is why we do not observe effects of modified gravity in our solar system. And in the early universe, the temperature is too high and dark matter just behaves like a normal fluid.

In 2015, the idea with the superfluid dark matter was still lacking details. But two months ago, Khoury and his collaborators came out with a new paper that fills in some of the missing pieces.

Their new calculations take into account that in general the dark matter will be a mixture of superfluid and normal fluid, and both phases will make a contribution to the gravitational pull. Just what the composition is depends on the gravitational potential (caused by all types of matter) and the equation of state of the superfluid. In the new paper, the authors parameterize the general effects and then constrain the parameters so that they fit observations.

Yes, there are new parameters, but not many. They claim that the model can account for all the achievements of normal particle dark matter, plus the benefits of modified gravity on top.

And while this approach very much looks like modified gravity in the superfluid phase, it is immune to the constraint from the measurement of gravitational waves with an optical counterpart. That is because both gravitational waves and photons couple the same way to the additional stuff and hence should arrive at the same time – as observed.

It seems to me, however, that in the superfluid model one would in general get a different dark matter density if one reconstructs it from gravitational lensing than if one reconstructs it from kinetic measurements. That is because the additional interaction with the superfluid is felt only by the baryons. Indeed, this discrepancy could be used to test whether the idea is correct.

Khoury et al don’t discuss the possible origin of the fluid, but I like the interpretation put forward by Erik Verlinde. According to Verlinde, the extra-fields which give rise to the effects of dark matter are really low-energy relics of the quantum behavior of space-time. I will admit that this link is presently somewhat loose, but I am hopeful that it will become tighter in the next years. If so, this would mean that dark matter might be the key to unlocking the – still secret – quantum nature of gravity.

I consider this one of the most interesting developments in the foundations of physics I have seen in my lifetime. Superfluid dark matter is without doubt a pretty cool idea.

Tuesday, January 09, 2018

Me, elsewhere

Beginning 2018, I will no longer write for Ethan Siegel’s Forbes collection “Starts With a Bang.” Instead, I will write a semi-regular column for Quanta Magazine, the first of which -- about asymptotically safe gravity -- appeared yesterday.

In contrast to Forbes, Quanta Magazine keeps the copyright, which means that the articles I write for them will not be mirrored on this blog. You actually have to go over to their site to read them. But if you are interested in the foundations of physics, take my word that subscribing to Quanta Magazine is well worth your time, not so much because of me, but because their staff writers have so-far done an awesome job to cover relevant topics without succumbing to hype.

I also wrote a review of Jim Baggott’s book “Origins: The Scientific Story of Creation” which appeared in the January issue of Physics World. I much enjoyed Baggott’s writing and promptly bought another one of his books. Physics World  doesn’t want me to repost the review in text, but you can read the PDF here.

Finally, I wrote a contribution to the proceedings of a philosophy workshop I attended last year. In this paper, I summarize my misgivings with arguments from finetuning. You can now find it on the arXiv.

If you want to stay up to date on my writing, follow me on Twitter or on Facebook.

Wednesday, January 03, 2018

Sometimes I believe in string theory. Then I wake up.

They talk about me.
Grumpy Rainbow Unicorn.
[Image Source.]

And I can’t blame them. Because nothing else is happening on this planet. There’s just me and my attempt to convince physicists that beauty isn’t truth.

Yes, I know it’s not much of an insight that pretty ideas aren’t always correct. That’s why I objected when my editor suggested I title my book “Why Beauty isn’t Truth.” Because, duh, it’s been said before and if I wanted to be stale I could have written about how we’re all made of stardust, aah-choir, chimes, fade and cut.

Nature has no obligation to be pretty, that much is sure. But the truth seems hard to swallow. “Certainly she doesn’t mean that,” they say. Or “She doesn’t know what she’s doing.” Then they explain things to me. Because surely I didn’t mean to say that much of what goes on in the foundations of physics these days is a waste of time, did I? And even if, could I please not do this publicly, because some people have to earn a living from it.

They are “good friends,” you see? Good friends who want me to believe what they believe. Because believing has bettered their lives.

And certainly I can be fixed! It’s just that I haven’t yet seen the elegance of string theory and supersymmetry. Don’t I know that elegance is a sign of all successful theories? It must be that I haven’t understood how beauty has been such a great guide for physicists in the past. Think of Einstein and Dirac and, erm, there must have been others, right? Or maybe it’s that I haven’t yet grasped that pretty, natural theories are so much better. Except possibly for the cosmological constant, which isn’t pretty. And the Higgs-mass. And, oh yeah, the axion. Almost forgot about that, sorry.

But it’s not that I don’t think unified symmetry is a beautiful idea. It’s a shame, really, that we have these three different symmetries in particle physics. It would be so much nicer if we could merge them to one large symmetry. Too bad that the first theories of unification led to the prediction of proton decay and were ruled out. But there are a lot other beautiful unification ideas left to work on. Not all is lost!

And it’s not that I don’t think supersymmetry is elegant. It combines two different types of particles and how cool is that? It has candidates for dark matter. It alleviates the problem with the cosmological constant. And it aids gauge coupling unification. Or at least it did until LHC data interfered with our plans to prettify the laws of nature. Dang.

And it’s not that I don’t see why string theory is appealing. I once set out to become a string theorist. I do not kid you. I ate my way through textbooks and it was all totally amazing, how much you get out from the rather simple idea that particles shouldn’t be points but strings. Look how much consistency dictates you to construct the theory. And note how neatly it fits with all that we already know.

But then I got distracted by a disturbing question: Do we actually have evidence that elegance is a good guide to the laws of nature?

The brief answer is no, we have no evidence. The long answer is in my book and, yes, I will mention the-damned-book until everyone is sick of it. The summary is: Beautiful ideas sometimes work, sometimes they don’t. It’s just that many physicists prefer to recall the beautiful ideas which did work.

And not only is there no historical evidence that beauty and elegance are good guides to find correct theories, there isn’t even a theory for why that should be so. There’s no reason to think that our sense of beauty has any relevance for discovering new fundamental laws of nature.

Sure, if you ask those who believe in string theory and supersymmetry and in grand unification, they will say that of course they know there is no reason to believe a beautiful theory is more likely to be correct. They still work on them anyway. Because what better could they do with their lives? Or with their grants, respectively. And if you work on it, you better believe in it.

I consent, not all math is equally beautiful and not all math is equally elegant. I yet have to find anyone, for example, who thinks Loop Quantum Gravity is more beautiful than string theory. And isn’t it interesting that we share this sense of what is and isn’t beautiful? Shouldn’t it mean something that so many theoretical physicists agree beautiful math is better? Shouldn’t it mean something that so many people believe in the existence of an omniscient god?

But science isn’t about belief, it’s about facts, so here are the facts: This trust in beauty as a guide, it’s not working. There’s no evidence for grand unification. There’s no evidence for supersymmetry, no evidence for axions, no evidence for moduli, for WIMPs, or for dozens of other particles that were invented to prettify theories which work just fine without them. After decades of search, there’s no evidence for any of these.

It’s not working. I know it hurts. But now please wake up.

Let me assure you I usually mean what I say and know what I do. Could I be wrong? Of course. Maybe tomorrow we’ll discover supersymmetry. Not all is lost.