Visit the fantastic El Yunque Rainforest

I haven't written anything for the past few days because my fiancee and I were vacationing in Puerto Rico. If you haven't been to the island, I highly recommend that you visit. The people are generally nice and the natural scenery is incredible.

The highlight of our trip was our visit to El Yunque National Forest, the only tropical rainforest in the United States. We spent two days there; the first hiking the Rio Sabana trail on the closed and remote south side of the forest, and the second following the sites along Highway 191, the main drag through the forest to the north. Both sides offered distinct and interesting vegetation.

The most interesting thing I learned about (thanks in part to my fiancee, who has some training in rainforest ecology) is one process by which rainforests manage their species numbers. Some species of trees, such as the yagrumo, which feature large, mitten-shaped leaves, serve as pioneer species, which quickly grow and establish themselves in an area where the forest canopy has broken or the local soil content is unfavorable to other species. These trees live for a short amount of time (~40 years for the yagrumo!), and then are followed by more permanent trees in a process known as secondary succession. In the tabonuco forest, these are the tabonuco trees. Pioneer species tend to be hardy, fix nutrients into the soil, and provide more nutrients once they die and are decomposed.

As I hiked through the forest, I was also in awe of the natural energy balance involved with the rainforest. One particular process in this balance occurs with all plants: the conversion of sunlight into glucose and eventually ATP. Since the yagrumo trees grow very quickly in a short amount of time in poor nutrient-conditioned soils, I imagine that they are relatively efficient converters of light energy into chemical energy. Should we ever wish to harness photosynthesis as a controllable energy source, I believe we should look to the pioneer species and their biochemistry as a first step.

Will light from two independent lasers give a beat note?

In my line of research I usually assume that light from two independent lasers will not produce an interference pattern when combined. This is because the light from one is not coherent with the other. For this reason I was surprised when a colleague of mine who works with frequency combs told me that light from two independent lasers will produce a beat note when combined interfered. This means, for a short time, there will be an interference pattern, though it may change too rapidly for our eyes to see. One can understand this by assuming an ever-shrinking line width for each laser until each one is perfectly monochromatic. [It also helps that the center wavelengths be slightly different.]

Of course, in retrospect, I realize now the error in my thinking. Supposedly "general" rules when applied to the topic of coherence are almost always wrong because one must specify the relevant timescales involved to determine whether light is coherent or not. These time scales include the integration time of the detector, the width of a wave packet, the period of the carrier wave, etc. Because there are so many different parameters, optical coherence problems do not lend themselves to an easy generalization, at least when one is first learning the topic.

And even when one is experienced with it, he or she will likely continuously be surprised like I was this morning.

Energies and fields in statistical mechanics

I've recently encountered a failure of understanding in my attempts at a statistical mechanics treatment for systems that are best described by fields rather than energies.

The problem lies with constructing the partition function of a system whose degrees of freedom are field quantities, rather than energetic quantities. In the classical Boltzmann formulation, the partition function is a sum over exponentials whose arguments are functions of the energies of the corresponding microstates:

The partition function ultimately goes into the calculation of how likely it is that a system will be in a given state subject to constraints and may determine important system parameters such as total energy and entropy. 

Importantly, in the equation above, the energy Es of each microstate is a sum of the energies of each degree of freedom in the system. For example, the energy associated with a microstate of a container of gas is the sum of all the energies of the gas molecules in that microstate. However, if the degrees of freedom of the system are best described by field quantities, then the energy of a given microstate becomes nontrivial. Fields are represented by two numbers (real and imaginary parts) and the energy carried by the total field is not simply the sum of the energies corresponding to each degree of freedom taken alone.

This result is commonly known in optics. Irradiance (which intuitively may thought of as how much energy a beam carries) is the time-averaged square of the electric field. Two beams that are coherent and allowed to interfere with one another will produce a different irradiance than two incoherent beams. As another example, electrical engineers know that the power delivered to a device is a product of current and voltage, which are also field quantities. The power thus becomes a function of the phase lag between the two.

The conclusion is that the energy of a microstate now depends on the sum of field quantities, which depends on quantities like their relative phase and state of coherence. This makes the Es appearing in the equation above much more difficult to calculate.

I have just discovered the topic of statistical field theory, which apparently deals with the statistical mechanics of fields, but it looks a bit difficult and may require a large amount of time to grasp its concepts.

Are there better indicators for causality than correlations?

In a recent post concerning the use of analytics to do science, I hypothesized that the notion of cause-and-effect is an ill-suited tool for describing the effects of input parameters on complex systems. In other words, cause-and-effect are ideas associated with models and many complex systems do not readily admit description by models. Now, to give fair warning, I have had no formal training in complexity or dynamical systems analysis, so most of what I write on these topics is an exploration of the relevant concepts to further my understanding--a sort of self-teaching, if you will.

I was therefore pleased to read the current thesis by Mark Buchanan in Nature Physics about work from dynamical systems theory concerning ways of determining causation in a complex system. He references an article from this year's Science journal (Science 338, 469-500; 2012) that contains an example of a model for the interaction of two species. The model consists of two equations for the population of each species that contains coupling parameters linking the two populations. Despite the fact that the population of each species affects the other, the populations are uncorrelated in the long run because the model goes through times of correlation, anti-correlation, and no correlation.

This is an example of the maxim "causation does not imply correlation." (Of course, many of us with scientific training have been chided endlessly about the maxim's well-known converse.) On the face of it, this example seems to support my idea about causation.

However, the main focus of Buchanan's article is about finding descriptors for causation beyond correlations. As he states:

Correlation alone isn't informative, nor is the lack of it. Might there be more subtle patterns, beyond correlations, that do really signify causal influence?

He mentions two major works, one old and one new, that address this question. The old one, introduced by Clive Granger in 1969, states that two variables are causally-linked if including one in a predictive scheme improves prediction of the other. The new work addresses problems with this idea and is somewhat technical, but it is capable of solving the problem of two populations mentioned above and one outstanding problem in ecology concerning the population of two species of fish.

I think now that my earlier conclusion about causality was wrong. I thought that a correlation needed to exist for there to be a causal link between two system parameters. Though I was paying heed to "correlation does not imply causation," I was ignorant of its converse, "causation does not imply correlation." Thus, causation can be an important concept for complex systems; we may only have to find better indicators than correlations.

Hyper-ballistic transport of waves

In this month's Nature Physics there is a paper entitled "Hyper-transport of light and stochastic acceleration by evolving disorder" by Levi, et al. The work is an experimental and numerical study of the propagation of light in a disordered medium that has been carefully constructed to serve as a model for the transport of a 2D quantum wavepacket in a spatio-temporal random potential, i.e. a potential energy landscape that changes randomly in space and time. The authors demonstrate that a beam's spot-size and angular spectrum spreads faster as it propagates through this particular medium than it would if the beam propagated in free space or in a random distribution of parallel waveguides (the Anderson localized regime).

The crux of their demonstration is provided by the comparison of their measured transport regime to the two aforementioned regimes: ballistic and localized transport. Ballistic transport is characterized by a beam spot size that grows with propagation distance and a constant angular spectrum with many longitudinal plane wave components. Localized transport is characterized by a beam spot size that does not grow in size with propagation and takes place in a disordered medium with refractive index fluctuations that possess an angular spectrum of plane waves with all the same longitudinal components. These characteristics are illustrated in Figure 2 of the article.

In contrast, hyper-transport is defined by a spot size that grows faster with propagation than in the ballistic case and by an angular spectrum of the beam (not of the disorder!) that widens with propagation as well (see Figure 3C).

The authors do not provide a comparison with the case of diffusive propagation of the waves. I think that this may be because diffusive transport (increasing spot size and constant but uniform angular spectrum over all propagation directions) is a limiting case of the transport regime that they studied. In other words, they looked at the transient process of the waves becoming diffusive, but not the limiting case. I think that similar work has already been done in the area of beam propagation through atmospheric turbulence, though I can't provide any references.

To be fair, the authors do state that:

"Strictly within the domain of optics, the results described below are intuitive. However, this direct analogy to transport in quantum systems makes our findings relevant for very many wave systems containing disorder."

I would have liked to have seen a comparison to or discussion about the diffusive regime since it would reveal that any multiply scattering medium displays this hyper-transport for short propagation distances.

Finally, I like their technique for controlling the disorder's correlation distance in the z-direction. This seems to be a very good tool for studying transport in disordered systems.

Our latest paper on optically-controlled active media

My colleagues and I just published a paper in Nature Photonics entitled "Superdiffusion in Optically Controlled Active Media." You may access the paper via this link to Nature Photonics or read about it in UCF Today.

I should point out that the claims made in the article in UCF Today is a bit over-reaching. What we have done is demonstrated that the coupling between light and particles in suspension models nonequilibrium processes that share some characteristics with similar processes inside cells. This is because the colloidal particles exchange energy randomly with their thermal bath (the water) and with the laser's radiation.

Importantly, the nature of the light-matter coupling is random due to multiple scattering by the particles, which establishes a three dimensional speckle inside the suspension.  This speckle exerts random forces on the particles which adds an additional component to their motion, besides that of Brownian motion. The resulting effect is that the particles move superdiffusively for times that are shorter than the decorrelation time of the speckle (which was about 1 millisecond).

Besides serving as a model system, I think it may be interesting to explore its ability to control some types of reaction kinetics. If reactants in solutions are driven apart from one another before they have a chance to react, then this would present a mechanical way of slowing the reaction of the bulk solution.

If you can access the article, then I hope you enjoy it!

Important equations and numbers to memorize

Many students in science and engineering today claim that the need to memorize certain numbers or equations is outdated with the advent of the internet and the ease with which we may find information. As an undergraduate, I often heard my engineering colleagues complain when they were not offered an equation sheet for a test because real-world engineers don't have to look up "well-known" equations when solving problems.

However, I contend that there are advantages to memorizing a few important equations and numbers. These reasons include the following:

1. You may work faster since you will not need to constantly reference those equations.
2. You may mentally check your work to ensure that the numbers are correct.
3. You may mentally approximate the solutions to certain calculations if you don't have access to reference material. I remember reading once how Feynman would challenge mathematicians by approximating powers of numbers in his head by knowing the natural logarithms of 2 and 10.
4. You may impress the ladies with your abilities (not guaranteed).

So what are some equations and constants I've found useful throughout the years? Here is my list:

1. The binomial expansion
2. The Taylor expansion
3. The power series representation of e 
4. The speed of light in vacuum and the impedance of free space (about 377 Ohms)
5. The refractive index of common glass (~1.5) and water (~1.33)
6. Boltzmann's constant and the thermal energy at 300 K (4.14 Joules) and thermal voltage at 300 K 0.026 eV)

Other important things to memorize, depending on your field, include:

1. Stirling's approximation
2. The Krebs cycle (a biologist once told me that many ideas in molecular biology come back to this)

Consider these as tools in your toolbox. They can make your job a lot easier.

Addendum: Not memorizing the basics is a mistake to avoid in physics according to this Back Page of APS News.

Working around distractions

I've been trying a method for enhancing my productivity during the day. Why am I doing this? Well, I want to maintain a healthy work-life balance; if I get more done at work, I have more time to spend with my fiancee, workout, cook, read, etc. This is incredibly difficult in academia, where there is no limit to the work that can be done. However, I value my personal life as well as my professional one, so a balance must be found.

My approach is simple and is inspired by the No Meat Athlete and Zen Habits blogs. I first make a list of things that need to be done. It can be large or small; it doesn't matter. I then work on one thing on the list and only open programs and browser tabs that directly relate to this task. That's the most important part. Multitasking, I've decided, is the bane of productivity.

If I see an interesting paper, I use Instapaper to save it for later. If I have too many e-mails or unread items, I go through and delete the ones that were once relevant but no longer are. The goal is to reduce the noise and possibilities to be distracted while working. This also includes choosing not to listen to music if I am writing anything or doing something that requires more than passive attention.

Of course, I also account for when I can not avoid being distracted. I allow time to be interrupted (and I will be numerous times) during the day by my colleagues and adviser. I also set time limits that are proportional to the magnitude of the task. If I have lab work, I give myself an hour minimum and two hour maximum; conversely, computer work is usually delegated 15 minutes to an hour. By setting upper time limits, I allow myself to be satisfied with unfinished work, knowing that when I pick it up again I will have a fresher mind. And by setting lower time limits, I ensure that I have enough time to accomplish something meaningful. Once an item on the list is done, I remove it. If it's not done, it stays.

This may be a bit too much micromanaging for some, but with the amount of things that are required of one in academia, I think it's necessary.

I believe that Henri Poincaré had habits similar to these. His goal, in part, was to maximize his creativity. I think this was so because he believed that a creative mind must be allowed to wander around different tasks. In this way, he recruited his subconscious to work on problems while he consciously worked on others.

CREOL Seminar - Becoming a Faculty Member

Today Dr. Mercedeh Khajavikhan gave a brief seminar to the CREOL students about her experiences with finding a job in academia. Though I'd already heard a lot of the information she presented, some points stuck out.

The following are a few notes I made during the talk. Keep in mind that they are her ideas that I've paraphrased (so if I misinterpreted them, then I'm sorry :)

• Considering that a post-doc position is a time to build your CV and publish, a post-doc offering research problems that can easily turn into publications within a year or two would be highly desirable.
• Apply to schools where there is not much overlap between your research and that of the existing faculty. This will minimize competition for valuable resources and money within your own department.
• Related to the point above, consider emphasizing how your strengths can help other faculty while applying. This will make you appear more valuable to the committee considering you for a job.
• Letters of reference are invaluable in the job search. (I need to work on building relationships with more faculty members!)

Astronomical crowdsourcing

I just read a short piece in this month's issue of Nature Physics about a number of astronomical discoveries made this past month. One exciting discovery was of a Neptune-sized exoplanet found in a complicated orbit around four stars about 5000 light years away. I find this absolutely amazing, and even more so considering that it was found by a crowdsourcing website called PlanetHunters.org that recruits public users in identifying transits from a large database of astronomical data.

This is one example of data-centric science, but it is slightly different from the way I've presented it in past posts. In particular, this is an observation that was not driven by some unanswered question. Rather, people were simply looking at data to find planets.

I think that this is really cool, but I do wonder why somebody hasn't written some code to do the data analysis.

The Effective Temperature as a Description for Non-equilibrium Systems

Cugliandolo wrote a survey last year of recent work involving the effective temperature. This is a macroscopic quantity that characterizes a system driven out of equilibrium. The review states that the effective temperature was initially used as an intuitive description of glassy and slowly relaxing systems, but only recently have theoreticians placed it on firmer ground by linking it to the fluctuation-dissipation theorem (FDT).

In practice, the effective temperature is the negative inverse slope of a system's dc susceptibility (a.k.a. its time-integrated impulse response) vs. the time-correlation function of some observable (a.k.a. the description of its thermal fluctuations). Importantly, a departure from the straight line joining the points (1,0) and (0, 1/temperature) on a properly normalized plot may signify a system that is not at equilibrium with its bath. In the paper, Cugliandolo assumes a canonical ensemble, or a system coupled to an equilibriated thermal bath. Also, because it is based on the FDT, this treatment is only valid for extremely small perturbations to the system such that an impulse response is an appropriate description.

Most recent work has been focused on determining whether the effective temperature meets our intuitive requirements for a temperature, like being measurable by a thermometer, and whether it is an appropriate thermodynamic description, i.e. it is a single number that summarizes the state of a large ensemble of random system parts. It seems that very slow relaxations, either forced or natural, must be present for this quantity to be useful.

What is nonequilibrium thermodynamics?

An important topic in thermodynamics and statistical mechanics is the description of systems that are not in equilibrium. It is important because most systems are not in thermodynamic equilibrium and routinely exchange energy and matter with their surroundings. Somewhat surprisingly, the equilibrium thermodynamics of pioneers such as Boltzmann, Gibbs, and Carnot has sufficed for many years, in part, I think, because of its success at guiding the design of heat engines and describing chemical reactions. A theoretical description of nonequilibrium systems, though, still remains a challenge and active area of research.

So what is a nonequilibrium thermodynamic system? I am seeking an intuitive answer, not the unenlightened, yet all-too-common statement "a system that is not in equilibrium."

Unfortunately I cannot find the answer in any one place. I've read several research articles, particularly on active matter, which provide about zero insight to the question. This is probably because journal articles typically assume some familiarity with a topic. Wikipedia's page on nonequilibrium thermodynamics, which I linked to above, seems to provide a good answer in the form of a long description. However, I usually run into problems when trying to identify what about a particular system drives it out of equilibrium or why classical thermodynamics fails to describe the system. For example, on the Wikipedia page noted above under basic concepts, a system between two thermostats at different temperatures is described as a nonequilibrium system, even though basic heat engines from (equilibrium) thermodynamics texts are modeled in this way.

I suspect that a general definition of a nonequilibrium system is elusive because we usually must appeal to specialized statements about the system at hand, such as what spatial and temporal scales we are interested in, and whether the system is in a steady state.

My intuitive understandings about nonequilibrium steady states are given below:

1. Thermal, pressure, or particle density gradients are present, resulting in fluxes.
2. The behavior of the system changes with spatial and temporal scales.
3. There are many ways to drive a system out of equilibrium, so general descriptions must include the nature of the driving processes.
4. Macroscopic properties are not easily defined. Microscopic properties, however, appear easier to describe.

Addendum
The link about nonequilibrium steady states contains one important quality about these systems: work must continuously be performed on a system to maintain its state.

Finding similar ideas in other fields - ecology and thermodynamics of complex systems

I very much like to find connections between ideas in different fields. The Wikipedia page on Ecology contains the following quote:

System behaviors must first be arrayed into levels of organization. Behaviors corresponding to higher levels occur at slow rates. Conversely, lower organizational levels exhibit rapid rates. For example, individual tree leaves respond rapidly to momentary changes in light intensity, CO₂ concentration, and the like. The growth of the tree responds more slowly and integrates these short-term changes.
O'Neill, et al., A Hierarchical Concept of Ecosystems, Princeton University Press, p. 253 (1986).

This is not much unlike the microscopic and macroscopic descriptions of a material system. Small-scale fluctuations in the microscopic arrangement of a material often don't significantly impact the macroscopic qualities of this system [1]. Rather, many small-scale changes must occur in concert to cause a large-scale change. For example, the momenta and positions of various gas molecules in a container at equilibrium with its surroundings may change about some average, but the pressure of the gas on the container's walls will not. If the average molecular speed changes, though, pressure will likely change as well.

In addition, if the interaction between the microscopic constituents is nonlinear, it becomes much more likely that small changes in the microscopic description will result in macroscopic changes as well.

Notes
[1] In fact, most fluctuations are dissipated by the system. If the system is in equilibrium with its environment, then energy conservation, among other things, implies that a fluctuation away from equilibrium of a local volume inside a system will eventually be "smoothed out" by the system. This is an important property of complex systems known as the fluctuation-dissipation theorem.

How does bias impact the physical sciences?

One of my favorite scientists, Ben Goldacre, recently posted a Ted Talk that he gave last year on how biases affect the methodology of food and drug trials and the reporting of results. One solution he proposes is an increase in transparency of reporting on science. Presumably, people are not willing to check all the facts because it is perceived as a tedious and unwelcome job.

It's worth watching and thinking about how similar biases affect the physical sciences. Some important biases to identify in our own work include

1. Publication bias - The increased likelihood of publishing positive results over negative results.
2. Experimenter's bias - The bias to perform an experiment in such a way that one is more likely to achieve an expected result.

Most information I've found on scientific bias comes from medicine and social sciences. Is this because they are more susceptible to bias, because the implications for error are larger, or because the physical sciences have not adequately addressed these issues?

Open-Access Explained by PhD Comics - and more Data-centric science

My girlfriend sent me this video link from PhD comics last week about open-access publishing. The video is narrated by Nick Shockey and Jonathon Eisen, brother to one of the co-creators of the Public Library of Science (commonly known to academics as PLoS).

One of the arguments presented by the two is that research should be free to re-use. I think this means that scientists should be able to use all of the knowledge and material presented in a journal paper to advance their own work. The narrators mention that the full content of papers should be searchable to easily find connections between works and to facilitate locating relevant papers.

This is another hint towards the shifting focus to a data-centric approach to science, only here the arguments for it are coming from the open-access movement.

I also want to add that I support this movement. I can't access a paper that I've written because our campus doesn't have access to the journal. And when I graduate, I will not have access to any of them unless I or my employer has the appropriate subscriptions. Ridiculous.

Data-centric science - Correlation and causation

Yesterday I defined a model as a description that involves a cause-and-effect relationship between phenomena. In contrast, a data-centric approach to science looks only for correlations between data sets to answer scientific problems. This approach relies on very large data sets to come to accurate conclusions.

After thinking about yesterday's post I realized that there is a relationship in my arguments to the common admonishment "correlation does not imply causation." This fallacy is most often made when complex systems made of many interconnected parts are involved, such as in human health. Statements like "taking vitamin C tablets will cause me to not get sick" and "eating vegetables prevents me from getting cancer" are statements about cause-and-effect. As we have been taught again and again, though, taking vitamin C tablets may only decrease the chances that I get sick.

So here is the dichotomy that I was looking for: model-based science is useful in simple systems for which I may make cause-and-effect statements. Data-centric science is more useful for complex, coupled systems for which causality is a poor descriptor.

This is certainly a new way of thinking. Depending on the complexity of what we are observing, we should either employ or abandon causality as a means of interpretation.

Data-centric science - What is a model?

Chris Anderson's The End of Theory: The Data Deluge Makes the Scientific Method Obsolete suggests that models may no longer be necessary to solving scientific problems due to the large amount of data now contained in databases across the globe. Rather, looking for correlations between events may be enough to solve these problems.

I'm going to assume that the article's title is an overstatement; not all scientific problems may be solved with a data-centric approach. Some are very well suited to this method, however. To discern between these types of problems, I think it's necessary to first address the question "what is a model?" After this is answered, I hope to address why models may sometimes be circumvented.

Wikipedia's site on the disambiguation of the word model is quite long. It can mean many things within a scientific context. However, several words continuously appear on this page and its links: description, simulation, representation, framework. More informative (albeit complicated) is the explanation found at the Stanford Encyclopedia of Philosophy. The central question to this post is addressed on this site in Section 2: Ontology. A model may be a physical or fictional object, a description, an equation, or a number of other things.

Based on this information I think it's reasonable to state that a model is an attempt at replicating the behavior of some phenomenon, whether physically or as a result of an application of logical rules. I think further that a model establishes cause-and-effect relationships to do this. For example, Newton's theory of gravity contained the idea that something (gravity) caused the apple to fall. As another example, energy input from the ocean causes (among other things) hurricanes in weather models.

Models satisfy some human desire for causality. I read once (though I don't remember where) that people use reasoning as a coping mechanism for emotionally difficult situations, such as when a loved one dies. Somehow, finding a reason or a cause for things provides us some degree of comfort.

The data-centric approach to scientific problem solving obviates the establishment of a cause-and-effect relationship. Insurance companies don't need to know why married, twenty-something men get in fewer car wrecks than their single companions in order to charge them less. Instead, they only need to know whether this is true.

But other than to make ourselves feel good, why would we need to find a cause-and-effect relationship in the first place? I think that this could be because the ability to make correct predictions is an important part of any model. We make predictions when we're unable to carry out an experiment easily or when we don't have enough data already to answer a question. It is my suspicion that cause-and-effect relationships are central to a model's ability of prediction, though I'm not sure how right now.

So, in summary, a model is a physical or mental construct meant to replicate the behavior of some phenomenon or system. I believe that the main difference between a model-based approach to science and a data-centric approach is that a model-based approach creates a causal chain of events that describe an observation. I don't necessarily see this chain ever ending. Once we determine a cause, we might wonder what caused the cause. And what caused the cause that caused the cause? At some point, data-centric science responds with "Enough! Just give me plenty of data and I will tell you if two events are correlated." That's all we can really hope for, anyway.

Notes: The never-ending chain of causes sounds very familiar to Pirsig's never-ending chain of hypotheses in Zen and the Art of Motorcycle Maintenance. Is there a connection?

Also, I remember E. T. Jaynes arguing in Probability Theory: The Logic of Science that we can't really know an event will occur with 100% probability. This seems to suggest that cause-and-effect relationships do not really exist. Otherwise, we would always know the outcome of some cause. And if they don't really exist but are actually good approximations, then models really are what we've been told since middle-school science: imperfect and intrinsically human attempts at describing the world.

Note, October 25, 2012: I wrote this post late last night after having had a beer with dinner, so my mind wasn't as clear as when I normally write these posts. I realized this morning that the reason for building cause-and-effect relationships is that we can control a phenomenon if we know its proper cause. Many things are correlated, but a fewer number of things is linked by a causal relationship. Therefore, accurate models provide us the ability to control the outcome of an experiment, not just predict it. I don't believe that correlative analytics necessarily allow us to do this.

Data-centric science - My initial thoughts

"The scientific method is built around testable hypotheses. These models, for the most part, are systems visualized in the minds of scientists. The models are then tested, and experiments confirm or falsify theoretical models of how the world works. This is the way science has worked for hundreds of years... But faced with massive data, this approach to science — hypothesize, model, test — is becoming obsolete."

This quote is from a 2008 article in Wired Magazine called "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete" by Chris Anderson. In this article, Anderson addresses our ability to solve scientific problems by looking for correlations in data without the need to form models. This ability has been enabled by the huge amount of searchable data that the internet has generated over the past two decades, which has led us into the so-called Petabyte Age.

This approach, sometimes referred to as analytics, has been successfully employed to translate between written languages, sequence genomes, match advertising outlets to customers, and provide better healthcare to people. Now, Anderson argues, it may be applied to problems across the full range of sciences. This is a welcome evolution, partly because many fields now possess too many theories and lack the experiments to validate or deny their predictions. Take particle physics or molecular biology, for example. There are arguably more theories and models now about these systems  than ever before, and many of them can not be verified. A data-centric approach could solve this problem.

This is a very interesting idea and I've been thinking about it for a few weeks now. I think that, to make any sense of it, I need to address several issues and assumptions. Questions to consider include:

• What is a model? When is it useful and when is it not?
• Are only certain fields of science able to benefit from a data-centric approach?
• What is the human component to research? How would it change if this approach was implemented?
• What has already been done to solve scientific problems with data-driven solutions?
• What are the philosophical implications to changing our idea of science? The scientific method has existed in some form or another for almost 2000 years (I'm referring all the way back to Aristotle, even if his ideas contained flaws). A significant change to the scientific method, especially given its importance to modern society, could have major sociological consequences.

I'll consider these questions in future posts.

Come see my talk at FiO

I'm giving a talk on optically controlled active media today at Frontiers in Optics. The talk number is FTh3D.7 and it's in Highland E.

The talk is about our work concerning the optical forces on colloidal particles in 3D, space and time dependent speckle. I'm pushing it as a model system for testing ideas from nonequilibrium thermodynamics, but I think that our method of treating the field-particle coupling is equally interesting.

Come check it out.

Tommorow at Frontiers in Optics

Right now I'm in Rochester, New York for this year's Frontiers in Optics conference hosted by the OSA. So far I've attended the plenary talks, which dealt with quantum optics and thermodynamics, 2D IR spectroscopy, retinal imaging, and, of course, the Higgs boson. In addition, I visited the Omega laser facility, which was incredibly fascinating. If you're in the area and you have any interest in incredibly powerful lasers or inertial confinement nuclear fusion, then I recommend making a visit.

Tomorrow I plan to visit some talks and work at the CREOL exhibition booth from 1:00 PM to 3:00 PM. Additionally, I'm giving a talk for a group mate who couldn't make it to the conference on Wednesday morning at 11:30 AM and my own talk on optically-controlled active media on Thursday at 3:00 PM. Briefly, the talk deals with solutions pumped by light to expand their free energy so that they may do carry out additional work. I hope the project will eventually be applied to controlling reaction kinetics in cells.

If  you're there, let me know and we can talk over coffee or a beer!

Better charts and graphics

I was recently asked to supply some annotated figures to a journal in a vector format, rather than the bitmap format that I had submitted them in. Unfortunately, I did not originally save the graphs in this format and had to hurriedly redo them with a program I was unfamiliar with. As you might expect, this wasn't a pleasant experience.

This event has lead me to offer the following tip: when you're making charts, figures, and other graphics that you intend to publish, be sure you save them in a production-quality format at the step where they are generated. It might mean a bit more time as you make them, but it saves a lot of hassle in the long run.

It's also worthwhile to explore the tools that are available to you at your institution for making figures. I am most comfortable with visualizing and analyzing data in Matlab, but I feel like too much effort is needed to make the plots look nice. Origin is a common alternative, but I find that it is not so intuitive to use and that there is a paucity of tutorials available. I also sometimes use free packages like SciPy along with Inkscape and GIMP to prepare for when I may find myself in a situation where I don't have access to (expensive) commercial options. Ultimately, it seems like the best approach is to use many different visualization tools depending on your purpose for the plot.

Here is an old but useful post on making production-quality graphs in Matlab. Has this process become simpler?

Addendum: I just found this on the Matlab File Exchange: plot2svg.m. I haven't yet tried it, but it should convert your Matlab plots to the .svg format, which is readable by Inkscape and a W3C standard.

A tip for providing useful feedback on writing

A common complaint I hear about feedback on someone's writing is that it focuses too much on stylistic changes (grammar, word choice, word order) and lacks good suggestions about improving the content or the message of the work. This is likely because it's much easier to comment on style than on content.

To prevent myself from doing this while giving feedback, I have begun following this general rule: if I can't provide a good, clear reason for why I am suggesting a change, then the change is unnecessary.

The only problem with this rule is that sometimes a paragraph or sentence really can be too difficult for a reader to understand because of its vocabulary. In this case, a stylistic change that simplifies the message is warranted.

The Reynolds number: A case-study about a simplified concept

Lately I've been re-reading some older papers on random walks in colloids to prepare for my dissertation proposal. One such paper from a 2007 PRL issue is entitled "Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk." The first sentence reads

"The directed propulsion of small scale objects in water is problematic because of the combination of low Reynolds number and Brownian motion on these length scales."

This struck me as particularly interesting because I had never considered the Reynolds number as useful for describing particle transport within a fluid; rather I always imagined it as a number that somehow answered the yes-or-no question "is this bulk fluid turbulent when flowing?" This conceptualization of the Reynolds number comes from my fluid and thermal systems class in the sophomore engineering curriculum at Rose-Hulman. Since I am so much wiser now than I was then, I decided to re-examine this number and its importance.

After consulting Wikipedia, I now understand the Reynolds number as a dimensionless ratio of the magnitude of the inertial forces transmitted by the fluid to the viscous forces. In other words, higher Reynolds numbers means that the fluid molecules will all move in more-or-less the same direction for longer periods of time and over larger domains. If something perturbs a region of the fluid at very large Reynolds numbers, the perturbation is transmitted by a large-scale, correlated motion of the molecules. This is because the dissipation of the fluid does not suffice to damp this motion.

Going further, this picture explains why scientists who study complexity and emergence love turbulence. If a system can not quickly dampen a fluctuation (in this case a small pocket of correlated motion within the fluid), it grows chaotically into a large-scale turbulence. Often, ordered patterns of fluid motion emerge from this chaos, like in Benard cells. So the Reynolds number leads to much more than a simple answer to a yes-or-no question; the physics behind it describes the chaos and complexity of turbulence itself.

This introspective exercise also demonstrates the short-comings of over-simplifying a concept when teaching it to others. I was blind to the connections between Reynolds number, the microscopic behavior of a turbulent fluid, chaos, and emergence for so long* because I only considered the yes-or-not question above, not the concept behind the number.

*It's been 8 years since I took that class. Yikes, I'm getting old.

Purposefully structured chapters, paragraphs, and sentences

With my dissertation proposal approaching, I've found myself writing quite a bit lately. One skill that I've happened to discover during this time that has greatly improved my writing is to insist that every structure within the document has an explicit purpose.

I envision the document as a hierarchy of structural units that contain information. At each level of the hierarchy, one message should be espoused. For example, each chapter is assigned one particular idea that I want the reader to know. The ideas communicated by chapters are usually very general. Moving down the hierarchy are sections, paragraphs, and sentences. With each level, the complexity of the ideas often increases, but each unit at a particular level still contains only one message. Each message must also contribute in some way to the units sitting above it in the hierarchy.

This approach has two benefits. The first is that it makes my writing more concise by lessening unwanted redundancy. I have no doubt that redundancy can be an effective way to communicate a message, but sometimes writing the same thing at too many locations within the document makes it difficult to extract the new content from the already established ideas. Instead, I use summary paragraphs and examples to better establish ideas.

The second benefit of enforcing a purpose to every unit in the hierarchy is the large-scale structure it lends to the document. This structure makes it easier to revise and establishes a good flow for the reader. While writing, I usually add a comment at the start of every paragraph that states what the reader should know after reading that paragraph. If I find that I start writing sentences that do not support this message, I delete these sentences or move them to a new paragraph.

I think that this approach is necessary for a document as long as a dissertation. Some structure must absolutely be imposed. Otherwise, I think that it would be impossible to make the entire thing effective at communicating its message.

Scientist personality types

Today is Friday, so as you might expect my mind is on the weekend and I don't have the desire to write too in-depth on anything. However, I wanted to make some quick notes about a train of thought I've had recently concerning the types of people who pursue science.

We often stereotype scientists as introverted individuals who tinker away in labs or on computers and who are awkward in social settings. This stereotype has the unintended consequence as portraying all scientists as the same type of person. My experience in graduate school has taught me differently. Scientists engender many different personalities, from brash and confident individuals to shy introverts to downright strange people. As we might expect, each personality type benefits science in a different way.

What are the common personality types in science and what is their impact? My question is motivated partly by the thesis in Five Minds for the Future by Howard Gardner. I have not read it, though I feel I that I have an idea of what his thesis entails.

I also wonder how a personality type might be better matched to a career field before they enter a career path. I suppose this is the work of many guidance counselors and career service offices across the country, but I feel that information provided by these services about some career fields, in particular scientific careers, is simply lacking.

Online career path guidance for science PhD's

I saw a link to myIDP this morning from the Condensed Concepts blog. myIDP (IDP is short for individual development plan) is a free service for assisting individuals who have science PhD's with finding careers that fit their skills and interests. The service presents and discusses a list of possible career routes, including many that lie outside of academia.

I am quickly approaching graduation, so I took a look at the site and filled out a few of the questionnaires. The three primary questionnaires for evaluating my career options considered my interests, skills sets, and values. I was pleased to see that many skills that I consider myself to be strong in (writing, communicating, and presenting, for example) were included in the survey. I was also pleased that "traditional" graduate student and scientific skills (performing experiments, processing data, etc.) filled only a small fraction of the possible skill sets that might be considered important.

myIDP goes further than simply displaying a percentage for how well a career field correlates with my responses to the questionnaires. It listed several questions that were based on my responses that I am to ask myself concerning these fields and my values. The questions relate to my highest rated values and serve as a guide while establishing my goals. For example, I responded that a good work location was very important to me. The questions I need to consider are whether there are geographic clusters for a specific field and whether these  areas would fulfill my lifestyle requirements.

The inclusion of many different career possibilities in myIDP has been very reassuring. I became concerned about two years ago that my original choice of career path, a faculty research and teaching position, did not correlate well with my values. I also realized that my institution and advisor are poorly equipped to place students into non-traditional science jobs. All of this led to some anxiety on my part because I was beginning to suspect that I had taken a wrong career path. Fortunately, I've learned that there are many options for me and that I shouldn't necessarily compromise on my interests and values just to maintain a sort of status quo for scientific careers.

How does subdiffusion arise in cells and why is it important?

In this month's Physics Today there is an interesting article entitled "Strange kinetics of single molecules in living cells" that discusses recent interpretations of single-molecule tracking experiments. In these experiments, fluorescent molecules or microbead probes are attached to some organelle or other piece of cellular material in a live cell and then tracked using video tracking microscopy. The paths taken by the molecules or beads are then analyzed and their motion is interpreted through random walk models. The goal is to learn something about the intracellular environment from the complicated motions of these probes.

The diffusion of these probes is almost always subdiffusive, which means that their mean squared displacements (the second moment of their position vs. time) grows slower than linearly with time, or

where r(t) is the position of the probe at time t and 0 < a < 1. The brackets denote an ensemble average, or a second moment calculated over a large number of probe trajectories. In these experiments, there does not exist a large enough number of probes for sufficient averaging, so instead the time average of a few probes is calculated. However, this produces wildly different results from particle to particle. The reason is that ergodicity may not apply to cellular transport. Ergodicity is a well-known property from statistical mechanics of systems whose ensemble averages are equivalent to time averages as time tends to infinity.

This article presented two possible models for why the probes behave in this way. One model, the continuous time random walk (CTRW), is nonergodic and subdiffusive for heavy-tailed probability distributions of particle waiting times. The other interpretation is that the cellular environment is spatially inhomogeneous so that "the environment sampled by the molecule during its motion through the cell differs from one trajectory to another."

If my understanding of their reasoning is correct, then I don't think that these two possibilities are logically equivalent. The random environment of the cell is a real, physical thing. The CTRW model is just that: a model. I feel that presenting these as two possibilities to explain the motion of the probes is like saying the earth revolves around the sun because either there is an attractive gravitational force between the two or the orbit is roughly elliptical with the sun at one foci. The first is a statement about the physics of the phenomenon and the other is a mathematical model. Perhaps the random cellular environment is the cause for the CTRW model to be valid. This line of reasoning I can accept.

The article concludes with very interesting remarks on why subdiffusion of proteins and biomolecules should occur at all. Subdiffusion is a way to make certain reactions more efficient by preventing the reactants from diffusing too far apart from one another. Considering normal diffusion (a = 1 in the expression above) as the most efficient manner of passive transport for cellular materials, subdiffusion may be understood as an evolutionary compromise between fast transport and efficient use of cargo in a crowded environment. Cells should not be viewed as "small, well-mixed reaction flasks," since their order actually enables crucial cellular processes.

Other notes:

1. Advances in improving the experiments' temporal resolution and finding smaller and brighter light emitters are the primary challenges to optics from single-molecule tracking experiments.
2. Fractional Brownian motion (first developed by B. B. Mandelbrot) is another random walk model that leads to subdiffusion but does not break ergodicity. It may model single particles in many-body systems, such as a monomeric unit in a polymer chain.
3. A fundamental question in cell biology concerns how the chromosomes are packed inside the nucleus. Are they separated by unseen walls or does their connectedness and limited volume keep them effectively disentangled.
4. While reading this article the following thoughts came to mind: superdiffusive transport, like transport of vacuoles by molecular motors, is a characteristic of nonequilibrium systems. Subdiffusion does not require a nonequilibrium system since the cells' physical constrains are the likely limiting factor to transport. Does this make subdiffusion and superdiffusion fundamentally different things?

Notes from the Chaos Cookbook, Chapters 2 and 3

The remaining portion of chapter 2 in the Chaos Cookbook involves properties of the logistic equation logistic map, which was first used by biologists and ecologists to model population growth. This function displays chaotic behavior as its parameter k is varied between 2.7 and 4, roughly.

There are regions of relative stability in the logistic equation's map's bifurcation plot of the possible output values vs. k, followed subsequently by a period doubling cascade. There is also self-similarity in the plots (I've added one of my own below).

A strange attractor for the logistic equationmap is the set of all possible output values for given k. It is strange because the values do not appear in a predictable or meaningful order.

It's been proved that a period 3 system (three possible output values) indicates the onset of chaos.

Feigenbaum's constant is roughly 4.669. It applies generally to many chaotic systems.

Chapter 3 concerns differential equations, their phase space plots and features, and numerical methods for solving them. I am largely familiar with all of this information, so I only loosely read this chapter.

Of special note in this chapter was a reminder to myself that phase space trajectories for a system of differential equations may not overlap.

Also of note was that, for these systems, a strange attractor is a phase space trajectory that does not settle into some periodic limit cycle and often shoots around to different parts of phase space.

Python code for some of these projects may be found at my personal website: http://quadrule.nfshost.com

Thoughts on P. W. Anderson's "More is Different"

Philip Anderson wrote a well-known article for Science in 1972 entitled "More is Different" whose goal was to refute the "constructionist hypothesis," i.e. the idea that all phenomena can be explained by a small set of fundamental laws. Presumably, these were the laws that govern elementary particle interactions. The constructionist hypothesis states that everything, from cellular biophysics to human thought processes, can be understood in terms of these laws so long as one is sufficiently clever in applying them. This hypothesis also leads many scientists to consider other fields as applied subsets of the fundamentals, such as biology existing as a form of applied chemistry, which would be just applied many-body physics and so-on down the line until particle physics is reached again.

Anderson claimed that, contrary to the constructionist hypothesis, new and "fundamental" science is performed at each level of the logical hierarchy of scientific fields and that this is because of the emergence of unexpected behavior at each level of complexity. His primary arguments lay with many-body physics and the idea of broken symmetry. As a system becomes more complex (that is, it takes on more components or the interactions between components become more intricate), it seeks to minimize the interaction energy between its components, which leads to a reduction in the symmetries of the components and an entirely different behavior of the system.

One example of emergent behavior in many-body physics is a crystal lattice, whereby translational and rotational symmetry is reduced by the ordered arrangement of atoms. Instead of a continuous translation or rotation, space must be shifted by an integer amount before the lattice looks the same again, and so these symmetries are reduced. The behavior that emerges from this is rigidity. If certain regions of the crystal experience a force, then the entire crystal moves as a result.

Another example—which demonstrates the unpredictability of emergent behavior—from many-body physics is the ammonia molecule. The nitrogen atom in ammonia undergoes inversion at a rate of roughly 30 billion times per second, which means that the nitrogen atom flips between its location above and below the plane containing the hydrogen atoms. Quantum mechanically, the stationary state of the molecule is a superposition of the two states representing the location of the nitrogen atom. This stationary state is symmetrical and represents what is actually measurable about the molecule. However, the understanding of inversion as a superposition of two unsymmetrical and unmeasurable states required intellectual machinery that was independent of the fundamental rules of atoms. Anderson's argument here suggests that human intuition led to the understanding of inversion, not the laws of physics, which at the fundamental level deal with symmetries and their consequences.

On a minor level, Anderson notes that scale and complexity are what lead to faults with the constructionist hypothesis. He also cautions that the nature of emergence at one level of complexity may not be the same at other levels.

My only question from this article is exactly what does fundamental mean? He seems to assume that fundamental science is always good science, so with his arguments chemists, biologists, and even psychologists can use the word to describe their work and win back their prestige from the particle physicists. However, it also might suggest that any scientific work is fundamental, thereby reducing the word's value and meaning.

Notes from the Chaos Cookbook, Chapter 15

I've skipped ahead to this short chapter in the Chaos Cookbook since I wanted to incorporate some of its ideas into my dissertation proposal. This chapter is entitled "An overview of complexity" and provides a brief and limited definition of what complexity is and several examples to broaden this definition.

Complexity is the study of emergent behavior from systems operating on the verge between stability and chaos. However, chaos is considered a subset of complexity. Complex systems also involve interactions between their individual components. The behavior that emerges from these interactions is often unexpected since the rules of the components don't necessarily predict this behavior.

Examples of complex systems in this book include traffic, autocatalytic systems, sand piles, and economies.

The bunching of cars and subsequent spreading out on highways is an emergent phenomenon that can depend on factors such as driver reaction times, car speeds, and the distances that drivers feel comfortable with when following other cars. I think that the variability in these individual factors leads to the random bunching of cars on the road.

The angle of repose of a sand pile is the angle that the pile makes with the horizontal plane that the pile is on. This angle emerges as the pile grows and may depend on how the pile is formed (dumping, pouring, etc.). Any changes to this angle caused by the addition of more sand leads to small avalanches that "correct" the perturbation so that the angle of repose is maintained. This is known as a self-organized critical state.

Not all sets of system behaviors can give rise to complex behavior.

Economies represent adaptive systems. In these systems, each agent adapts their behavior to the rules of the system to maximize their profits/utility. There is not one best strategy for this; rather, each agent must adapt their strategy according to what the whole system is doing to succeed.

Notes from the Chaos Cookbook, Chapter 2, pp. 37-42

Today I have more notes and insights from the Chaos Cookbook, Chapter 2. I'm trying to understand what characteristics are common to all chaotic systems and what it means for these systems to be chaotic. I'm also trying to find what links the ideas chaos, fractals, emergence, and complexity together. Chapter 2 deals with a specific type of chaotic entity: an iterated function.

"For a function to exhibit chaotic behavior, the function used has to be nonlinear." However, a nonlinear equation does not necessarily display chaotic behavior (example: y = x**2).

An iterated function is one whose value depends on the prior iteration. This is a form of mathematical feedback. Feedback is common in all chaotic systems.

There are two important parameters that describe an iterated function: the initial value and the number of iterations. Chaotic functions may be very sensitive to the initial value, meaning that very small changes to it will produce very drastic differences in the outcome of the iteration process.

Despite the sensitivity of the values of an iterated function to the initial value used, each graph may display common features.

The sequence of numbers obtained by iterating over a function is known as an orbit. Orbits may be stable, unstable, or chaotic.

Harnessing Light 2

The US National Academy of Sciences has released the second iteration of Harnessing Light, which analyzes and recommends action for maintaining or increasing US competitiveness in global photonics markets.

I just watched the OSA webinar of the roundtable discussion on this document that occurred today at Stanford. Of interest was the committee's strong recommendation to increase US manufacturing capabilities in both optics and other areas that utilize optics for their manufacturing processes. They also addressed the stigma of manufacturing being a blue-collar field and made note that manufacturing engineers address very challenging and technical problems. One member also mentioned that the personal satisfaction from manufacturing jobs is often very great because of the tangible reward of seeing a product that one has designed come to market.

Tom Baer said that industry is better-suited for multidisciplinary research because the historical barriers across fields do not exist there.

Also of interest was the notion that the US has been a good innovator for ideas and technologies but has increasingly lost its ability to capitalize on these ideas to other countries.

Notes from The Chaos Cookbook, Chapter 1

I'm reading through a used copy of Joe Pritchard's "The Chaos Cookbook" that I recently obtained from Amazon. This post contains my notes from Chapter 1.

On page 26 he notes that Newton's laws of motion cannot predict the general result of three solid balls colliding simultaneously. At first I thought that this was not true since I remember solving problems like this in general physics. However, after further thought I realized that we can predict which direction a ball will be traveling after the collision only if we already know what happened to the other balls. In other words, we cannot predict the outcome of the collision entirely; we must know something about what happened.

The above example may be considered as a subsequent loss of information that occurs following the collision of all three balls.

Historically speaking, nonlinear systems subjected to analysis were either 1) treated only to first order where higher order effects were negligible (e.g. a simple pendulum displaced slightly from its resting position), or 2) too difficult to analyze thoroughly.

A system may appear periodic in a reduced-dimensional phase space (see Fig. 1.3). A corkscrew trajectory along one axis will appear circular when looking down this axis.

Period doubling leads to splitting in a system's power spectrum. As the number of period doublings occur, does the spectrum fill with peaks and appear as white noise (uniform power spectrum)? Does this form a link to diffusive processes? (The Facebook link to white noise contains some block diagrams. Perhaps these can be shown to be equivalent to some chaotic systems.)

Chaos can emerge both in iterated function systems and from sets of differential equations.

Chaotic systems may eventually settle into seemingly stable states and vice versa.

Sequestration: yes or no?

Members of scientific organizations, like myself, are being urged to sign petitions asking that congress resume talks on methods to avoid resorting to sequestration. In case you haven't heard, we've run out of fancy terms to describe globally important ideas. As a result, sequestration is no longer a group of techniques for removing carbon from the atmosphere but now refers to budget cuts to U.S. spending that give no regard to which government programs receive the cuts. More precisely, funds that exceed the budget set forth by congress are sequestered by the treasury and not made available to congress for appropriations. This is generally viewed as a bad thing, and through this post I'm trying to determine why it is bad and whether or not the arguments make sense.

The argument of we scientists and engineers appears to be this: science and engineering help the U.S. remain competitive in an increasingly global technology market while contributing more to economic growth than other government-funded programs. Therefore, science and technology should receive less of the burden imposed by the budget cuts. Sequestration is seen as unfair and harmful since other organizations that contribute less to the economy receive more-or-less the same cuts to their funding. I have not seen arguments from other federally-funded sectors that may be hurt by sequestration, but I can imagine that similar appeals are being made.

Two possible requests are made, so far as I can tell, in the petition above. The first is that we are asking that sequestration not occur. The second is that so long as budget cuts are made, fewer cuts should be applied towards science and technology than other sectors. If this is indeed a zero sum game, then that would seem to indicate that other organizations will suffer more cuts than they would under sequestration. Perhaps some government-sponsored organizations are viewing sequestration as very appealing in this light.

I have two concerns about the value of the arguments put forth by APS and the many other scientific professional organizations. The first is that science and engineering is a very big field that employs many people. If every bit of it is contributing equally to the growth of the economy, then I agree that sequestration is unfair. But we also know very well that there are too many people, at least in academia, and that this is driving down the quality of scientific research in favor of publishing to gain a competitive edge. I suspect that only a certain percentage of science and engineering is actually of value to our economy, so perhaps it would be more appropriate to suffer sequestration and let the ineffective parts choke from the lack of funding. Harsh, I know, but this is the reality of cutting a budget.

My second concern is based on an intuition I have about complex systems, particularly the stock market. Over the long run (20 years or more), index mutual funds will always outperform actively traded funds, and they do this by matching the mean growth of the market rather than reacting to the daily or monthly fluctuations of stock prices. In other words, reacting to fluctuations and trying to understand the market with a reductionist approach will eventually fail to account for every factor that affects prices, and as a rule of complex systems, even minor factors have big consequences. Additionally, you often pay more in fees for the large amount of trading that is performed. My intuition is that this analogy applies, at least loosely, to federal spending. It may simply be impossible to effect a detailed budget plan that helps the economy in the long run. Any attempt to do so might work, but it's cost in man-hours and other resources may lead to other problems. Across the board cuts imposed by sequestration might be the best approach we have to reducing spending within the government.

Finally, let's keep in mind that the purpose of a government is to secure the welfare of its people. The simplest question we can ask is this: which of the two options—sequestration or a detailed budget reduction plan—will ultimately make people happier? I haven't been convinced yet which one is best for U.S. citizens.

I'm going to end this post for now in favor of other work I need to do, but I think that there are many other subtleties here that are worth considering. I welcome any and all comments, thoughts, and questions as we try to sort this out before January.

Recycling made easier

I'm moving soon, and since I attempt to live as a minimalist that means that I've been getting rid of things I don't use anymore. Here is a list of resources that I've found useful for recycling old stuff:

1. Goodwill The status-quo for reuse and recycling. I probably drop stuff off here once every couple of months. Despite their reputation for reselling used clothing (which never fits me), they also accept and sell kitchenware, furniture, books, electronics and appliances, and shoes, among other things. Plus, you know that people will be able to directly use the items that you're donating. I worry though that these people simply trash such things once they're done with them.
2. Dell Reconnect Dell has partnered with Goodwill to recycle computers and some other electronic equipment. All you have to do is drop your old computer off at a participating Goodwill (of which there are many near me). Dell will determine whether a used computer is good enough to resell at Goodwill or simply recycle it for raw materials.
3. Best Buy Best Buy will recycle a whole slue of electronics and electronics accessories, including batteries, cell phones, computers, computer hardware, ink cartridges, speakers, and appliances. Just take the items to the service desk if there isn't already a bin for them near the front door.
4. Home Depot Home Depot will recycle burnt-out compact fluorescent light bulbs (CFL's). I'm surprised that this campaign hasn't received more publicity since CFL's contain mercury.
5. Target Target recycles plastic bags, cell phones, ink cartridges, and MP3 players. Look for the bins near the front of the store.
6. Publix Florida's most popular grocery store. Most Publix have bins near the front door for recycling plastic bags and foam cartons. This makes me feel a bit better about Publix putting what seems like half of their entire produce stock on foam trays. Tsk, tsk, Publix, you are so wasteful...

Satisfying the energy needs of the UK with renewables

I love real numbers. Here's an interesting Ted Talk on the amount of land and other resources required to fuel the UK with alternative energy sources.

http://www.ted.com/talks/david_mackay_a_reality_check_on_renewables.html

Write for ideas

"Writing doesn't just communicate ideas; it generates them. If you're bad at writing and don't like to do it, you'll miss out on most of the ideas writing would have generated." - Paul Graham

I really could not agree more with this sentiment. In fact, it's why I started the blog. Don't underestimate the power of writing for developing new ideas. It's good for finding flaws in your ideas, too.

Measurements and theory - the required number of data points

I hosted group meeting today. The topic was on Brownian motion, but during it I came to an interesting realization about measurements of displacement, velocity, and acceleration.

Consider the measurement of a displacement of an object, say a car, from some fixed point, let's say its owner's house. A measurement of the car's distance from home consists of taking a yardstick and determining how many yardsticks away from the house the car lies at a moment in time. This is one measurement. If we wish to determine the car's (average) velocity, we measure the distance at another moment in time, subtract this from the previous measurement to obtain its displacement, and divide by the time interval between measurements.

This means that if we wish to determine the velocity of the car, we necessarily must make two measurements of its distance from home. If the car is accelerating, then this may be determined from a minimum of three distance measurements. The two (average) velocities determined by the three points where the distance measurements occurred are subtracted and divided by the time difference between the first and last measurement, resulting in the car's acceleration.

One could argue that the car has an instantaneous velocity and acceleration, so to talk about the number of measurements required to determine one of the car's dynamical quantities is nonsense. For a car this is more or less true, practically speaking. However, the fact is that at very small length and time scales, we can not measure instantaneous quantities. Rather, they are sampled at specific time intervals. For example, a digital camera works by sampling the light that falls on each pixel of its CCD array. So, any measurement is in reality a collection of discrete points of data. Quantum mechanics also tells us that in the microscopic world, measurements are discrete and that the continuum results from combining a large number of discrete measurements.

Back to my main point. In kinematics and calculus we learn that velocity is the first order derivative of displacement and that acceleration is the second order derivative. The previous argument suggests that if we wish to determine some quantity that is a derivative of the quantity that we actually measure, then we need to make N+1 measurements, where N is the order of the derivative representing the quantity that we're after.

So there you have it. You can't determine an acceleration from one or two measurements of displacement. If I had a device that measured velocity, however, then the acceleration could be determined from two velocity measurements, since acceleration is the first derivative of velocity. All of these arguments aren't too surprising when you consider that these quantities are essentially differences, and differences require more than one thing.

As an aside, fractal motion becomes a bit more enlightening in this context, but I'll save that for a later discussion.

The importance of knowing your random number generator

I recently gained access to the Stokes computing cluster at UCF and ran some long-time simulations that I had written in Matlab. These were not in parallel, but rather utilized each node to run a separate simulation. Each simulation took a few days to run.

I was a bit irritated to find the output from each simulation was exactly the same, despite having run on separate nodes. The reason, I discovered, is that Matlab will generate the same string of random numbers every time you start a new instance.

I tried this experiment with Matlab R2011a. After opening a fresh instance of Matlab, I entered and received the following in the Command Window:

>> randn(1,5)

ans =

    0.5377    1.8339   -2.2588    0.8622    0.3188

I then closed Matlab and reopened it. Entering the same command gives the same output as before:

>> randn(1,5)

ans =

    0.5377    1.8339   -2.2588    0.8622    0.3188

Oh, boy. I performed the same procedure on a different model of computer in our lab and received the same output. I also tried this with Python 2.7 and Numpy's random.rand() method. The output is different each time I start an iPython session, so the seed must be automatically shuffled at the start.

Lesson learned. If you're working with random numbers in Matlab (at least with R2011a), be sure to randomize the seed with the command rng shuffle. Now to go redo a few week's worth of work...

Matlab's userpath and how to use it

I've been having some trouble lately on my computer dealing with MATLAB's search path (specifically MATLAB R2011 A). I try to change the user path from the GUI via File -> Set Path..., which is the same as entering 'pathtool' in the Command Window, but my settings are never saved when I restart MATLAB. I suspect that the problem involves the permissions which are managed by Windows 7, which typically prevents me from changing files on my hard drive from within MATLAB.

I managed to change the user path by instead using the command sequence

userpath('clear')

userpath('c:\mypath\')

where of course the second command points to my desired user path.

I discovered that the string containing of all the paths is contained in matlabroot/toolbox/local/pathdef.m, where matlabroot is the parent folder that contains the MATLAB program files. For example, my root folder is C:\Program Files\MATLAB\R2011a\. At the end of this file, there is the line

p = [userpath,p];

which appends the string in userpath to the front of the string of paths to all of the toolboxes (note that pathdef.m returns the string p).  Also of note is that pathdef.m is called by MATLAB's initialization m-file, matlabrc.m, which is located in the same folder and executed at MATLAB's startup.

Finally, there exists in the same directory an m-file called startupsav.m. It is a template which, when renamed to startup.m, contains any lines of code that one wishes to run at startup. I believe that these lines are executed once the Matlab GUI opens and well after matlabrc.m is executed.

I'm still not sure where userpath is defined.

Addendum: You may place your pathdef.m or startup.m files in the userpath directory, and I believe that these take precedence over those in the matlabroot/toolbox/local/pathdef.m file.

In my startup.m, I've added paths that contain files that I do not wish to be associated with MATLAB's toolboxes. The commands in this file look like

addpath some_directory\MatlabMPI\src\ -END

where the -END flag places this path at the end of the string returned by the path command.

The purpose of numerics, part 2

In a post that I wrote nearly two years ago, I wondered about the usefulness of numerics and simulations for science. After all, a simulation necessarily tests the outcomes of a model, and a model is only an approximation for the real world. Because of this, I placed a higher emphasis on understanding experimental data than on numerics.

At the time of that writing I was irritated by a trend I had found in papers to present numerical results that exactly matched an experimental outcome. This mode of thinking, I believe, consisted of

1. performing an experiment;
2. developing a model that explained the experiment;
3. using computer software to show that the model can exactly reproduce the experiment and was therefore good.

Unfortunately the only value I can see in doing this is confirming that the model reproduces the experimental data. What is perhaps more interesting to pursue numerically is to explore the effect of varying the model's parameters on the observable quantities, which is especially true for simulations of multi-body interactions, like the Ising model. For cases such as this, either it is not feasible to explore the full parameter space of this model experimentally or the outcome of such models is too difficult to predict analytically.

Essentially I've restated what I said in that earlier post, but I needed to do it. It's easy, at least for me, to dive head first into developing a computer simulation because it's great fun, but if I don't slow down and think about why I'm doing the simulation, I risk wasting my time on a menial task that adds little to the body of scientific knowledge.

What is a good description for entropy?

"Insight into Entropy," by Daniel F. Styer, is a nice paper that appeared in the American Journal of Physics in 2000. In the paper, he argues for a qualitative explanation of entropy that involves two ideas: disorder and freedom.

Entropy as disorder is a common analogy given to students who are learning about thermodynamics, but Styer provides several arguments for why this qualitative description fails to adequately explain the idea. One such argument involves a glass of shredded and broken ice. Despite the fact that the ice has been shattered into many pieces, the entropy of the bowl of ice is less than that of an identical bowl filled with water. The water may seem to be more ordered because it is homogeneous, but it does not possess a lower entropy.

Styer's idea of entropy as freedom attempts to explain how systems can possess multiple classes of states (commonly known as macrostates) and how entropy limits the microscopic details of each class. In the game of poker, the probability of getting a royal flush is identical to any other five-card selection without replacement. However, the number of configurations that form a royal flush is extremely small, so the entropy of the class of hands forming a royal flush is low. This very low entropy class of poker hands restricts the possible configurations of the microstate—the description of what five cards are in one's hand—and completes the analogy with freedom. High entropy macrostates have greater freedom in choosing their microstate by having a larger number of microstates to choose from; low entropy macrostates (royal flushes, for example) have less freedom.

Styer does propose retaining the "entropy as disorder" description by suggesting that both the freedom and disorder analogies be presented simultaneously to negate any emotions commonly associated with either word. His example of such an analogy goes as "For macrostates of high entropy, the system has the freedom to choose one of a large number of microstates, and the bulk of such microstates are microscopically disordered."

Finally, on a different train of though: teaching ideas by analogy apparently must be done with sensitivity to the common emotions associated with a word. I've never considered this idea before, but will surely be mindful of it in the future.

Thoughts on Trends in PhD's in Physics

There's an interesting article on The Back Page of APS News (Vol. 21, No. 4) written by Dr. Geoff Potvin from Clemson University. The article discusses the current plight of physics PhD's—trends in graduation rate, factors for success, and inherent biases against women and minorities. I'm not offering a full analysis here, just jotting down the first things that came to my head as I read the article.

Dr. Potvin states that the growth in the number of PhD's in physics is stagnant and that this is a problem for the US as it tries to remain scientifically competitive in an increasingly global community. However, other STEM fields have grown at much faster rates. Claiming that the stagnant growth in PhD's awarded in physics is bad for US technological competitiveness seems to be a weak argument for enhancing physics graduate education. After all, the number of independent STEM fields has grown enormously (for example I am working on my degree in optics), so any argument along these lines should look at the total trend in all STEM fields.

Dr. Potvin's research has shown that students' motivations for attending graduate school often determine their level of success as measured by publication rates and funding. All too often, graduate advisors assume that their students inherently possess the interest and motivation to perform research. However, the interests and goals of the advisor and student often differ. Dr. Potvin suggests that proper attention paid to graduate students' motivations would enhance their productivity.

There is an inverse relationship between doctoral completion time and future salary for men; higher pay goes to those who took less time to complete their PhD. Unfortunately, completion time is almost entirely uncorrelated with factors that students can control, and is instead determined by things such as the riskiness of research topics or becoming involved in multi-group projects. The pay for women PhD's is uncorrelated to almost all factors in their graduate education and is lower than their male counterparts, on average.

Faculty mentors tend to replicate their graduate school experiences. In my opinion, this keeps them out of touch with their students since they do not adapt to changing attitudes towards work, social life, and career choices for their students. As managers of new work-force members, it would behoove the greater community to adapt their advising style towards these new attitudes.