Dr. McKenzie defines emergence

Dr. Ross McKenzie has provided a definition of emergence on his blog, Condensed Concepts. This definition nicely summarizes and simplifies emergence by laying out the properties that describe emergent phenomena.

I've always struggled with defining emergence because it seems that a definition by example is the best way to do it, but this is often difficult to do simply. Trying to define emergence in one sentence, rather than the list that Dr. McKenzie provided, seems nearly impossible. There's always some form of discussion that's required.

What other kinds of models or theories require expanded definitions involving enumerations of their qualities?

A satisfying definition of emergence (at least, for me)

Building a bit off of yesterday's blog topic concerning biology, mathematics, and complexity, I wanted to note a satisfying and simple explanation of emergent phenomena in P. W. Anderson's "Physics: The Opening to Complexity."

To paraphrase, emergent phenomena are not logical consequences of underlying physical laws. In other words, one can't deduce the behavior of, say, monarch butterflies from the laws of quantum electrodynamics. However, emergent behavior cannot contradict the physical laws upon which they are built.

'Living crystals' reported in Science

Living Crystals of Light-Activated Colloidal Surfers is a recent publication in Science. It presents a study of the dynamics of interacting particles that are propelled by a light-catalyzed reaction between hematite (located on the surface of the colloidal particles) and hydrogen peroxide. These particles experience a nonequilibrium driving force from the reaction, repulsive forces between one another due in part to SDS surfactant present in the solvent, and attractive phoretic forces towards other particles. They observe that when the system is illuminated with blue light and the hydrogen peroxide reaction is catalyzed, the particles form crystalline arrangements that dynamically grow, shrink, merge and split. This is a form of self-organization and is fueled by the energy delivered to the system in the form of light.

Importantly, the attractive pair forces and and driving forces are not present when the light is off, which demonstrates that the formation of the crystals occurs under nonequilibrium conditions.

This rather elegant work demonstrates how complex behavior in systems can emerge from interactions between the parts of the system.

A PopSci article summarizes the work, though I think it focuses too much on the properties of life that the crystals satisfy.

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.

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.

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.

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.