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.