The rules of the scientific game

While listening to NPR recently I encountered the book "The Structure of Scientific Revolutions" by Thomas S. Kuhn. This book, which was published in the seventies, is an explanation of how normal science emerges from times of discord between scientists (the so-called revolutionary periods). It also places the practice of science onto a firm sociological foundation, addressing how scientists think and what drives them to pursue their art.

Kuhn's influential work is credited with introducing the word paradigm into science. A paradigm is a set of beliefs which provides some basis for scientists to do their work. For example, in optics, I always assume the validity of Maxwell's equations to explore more esoteric optical phenomena. A paradigm is not just a scientific theory, though. It is a mode of thought that dictates what theories, experiments, methodologies, and instruments constitute valid science.

A particularly enlightening section of the book has been in Chapter 4, which explains why scientists do their work. Kuhn likens science to puzzle-solving, which is an often-used analogy for attracting students to scientific fields. A scientific puzzle consists in rendering coherent observations and theories from some class of phenomena. A scientific puzzle also consists in extending the theory to a broader range of phenomena, like the particle-physicist's search for a Theory of Everything.

What's necessary for this puzzle-solving to advance, Kuhn argues, is a set of rules for ensuring that a solution to a scientific problem can be found. A paradigm thus becomes a useful tool in filtering out what class of observations can even be addressed with the theory. If observations can not be adequately accounted for, then a theory must either be modified or thrown out altogether.

These ideas are important for any scientist wishing to make his or her own scientific revolution. Moving into uncharted waters is very difficult because one must first upend the existing paradigm, which most-likely exists for good reason. But the most difficult part is addressing those problems which lie outside the scope of the paradigm itself. In some sense, a scientist addressing such observations needs to start from scratch. I credit anyone who is explicitly working in these areas of science because they are likely to be shunned by their peers and encountering problems never before rationalized.

I highly recommend this book to anyone working in science, since its discussions effectively remove us from the constraints of our own tools for understanding scientific problems and lays them bare before us, without the hubris that is sometimes associated with trying to rationalize our own mode of thought.

A beautiful experiment on a nonequilibrium thermodynamic system

I just finished reading an impressive article from 2003 entitled "Observing Brownian motion in vibration-fluidized granular matter". In the article's beginning, the authors established a simple question: can linear response theory describe a nonequilibrium thermodynamic system? This question is important because systems that are not in thermodynamic equilibrium are both difficult to analyze and serve as appropriate models for most natural phenomena. However, very powerful mathematical tools exist for systems that are in equilibrium, so it would be very convenient if their mathematical formalism could be extended to nonequilibrium cases.

In particular, the authors explore whether a torsion oscillator driven by a "heat bath" of randomly vibrating glass beads can be described by the fluctuation-dissipation theorem (FDT). The FDT is arguably the hallmark of linear response theory and describes the return to equilibrium of a many-body system subjected to a small perturbation. (A small perturbation means that the response is linearly proportional to the perturbation.)

A canonical example where the FDT finds use is in describing the motion of ions in a fluid between two plates of a capacitor after a voltage difference has been applied to the plates. Prior to the voltage being applied, the motion of the ions is erratic and Brownian. A long time after the constant voltage is applied, they move with an average velocity that is proportional to the electric field between the plates, the proportionality constant being called the mobility. The FDT describes the very short times immediately after the field is applied. It also links the noise (the random movement of the ions in equilibrium) to the mobility of the ions.

Returning to the article, the authors find that the motion of the oscillator is described by the FDT so long as an "effective" temperature is adopted. Effective temperatures are very appealing as analytical tools for describing nonequilibrium systems because they are very, very simple modifications to the FDT. Simply replace T with T_eff and you're done.

As Cugliandolo points out, to be a good thermodynamic descriptor, an effective temperature should be measurable by a thermometer. I'm not sure what the thermometer is in this system, but I suspect that it's the torsion oscillator itself. Furthermore, she stresses that not all nonequilibrium phenomena are describable by effective temperatures. It seems that one requires coupling between fast processes and slower observables, among other requirements. The beauty of the Nature article is that the authors not only confirmed this point (which seems to currently be an area of contention), but did so convincingly by measuring the relevant quantities directly and under a number of different conditions.

I'm not sure whether the effective temperature is a universal property of nonequilibrium systems; I'm inclined to say it is not. Hopefully more experiments like this one will be done that may further elucidate the current maze of theoretical papers on the topic.

Mathematics has a new friend: biology

In 2004 Joel E. Cohen wrote an article in PLoS Biology entitled "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better." This short article, which has been on my "To Read" list for a long time, is a brief history and assessment of the contributions that each field has and continues to make in the other.

As the title suggests, half of Cohen's claim is that the problems in modern biology will fuel the development of new mathematics. These mathematics should address the fundamental questions posed by biologists, such as "How does it work?" and "What is it for?" There are six such questions, and they are further divided according to the many, many orders of magnitude of space and time that are spanned by biological processes: from molecular biology to global ecosystems and from ultrafast photosynthesis to evolution. Scale-dependent systems and emergent phenomena are the primary themes in modern biological problems.

To illustrate his idea of mathematics, Cohen paints a picture of a tetrahedron with the topics of data structures, theories and models, algorithms, and computers and software located at the four vertexes. Each topic affects the others and has certain strengths for addressing the problems of biology. No weaknesses in the current state of mathematics are mentioned per-se, but any real weakness is likely the notion that, for some problems, the appropriate mathematics simply don't yet exist.

For Cohen, mathematics is decidedly applied mathematics. I doubt he has much to say about topics with no direct relevance for biological applications.

The article is divided into past, present, and future. Cohen first goes into a brief review of the historical interplay between math and biology, starting with what I think is an excellent example: William Harvey's discovery of the circulation of the blood. Just enough background is given to appreciate how novel and unexpected this discovery was. Notably, empiricism, aided by calculations, was in its infancy during Harvey's time. Cohen then pays homage to several other co-developments of math and biology, some of which are nicely summarized in the article's Table 1.

For present matters, Cohen notes that issues of emergence and complexity should lead to great discoveries in mathematics. What is notable here is that emergence in biological systems at one particular level of organization is driven by events at both lower and higher levels. For example, both the genes of an organism and evolution determine many aspects of species. Cohen also provides an example of recent research that marries ideas from statistics, hierarchical clustering, and cancer cell biology. This example is a bit difficult to follow, but I think it is a good analogy of the interplay he is discussing. (To be fair, I was reading the article in an airplane flying through some turbulence, so it was difficult to give this section my full attention.)

The article finishes with a future outlook for his thesis and very briefly presents some ethical problems and opportunities for the continued correspondence between the two fields. I didn't find this section terribly insightful.

This article and those similar to it can't help but make me feel like the Age of Physics is near an end. The problems that occupy most practical people's minds today seem to be concerned with complexity. Physics, which is concerned with constructing models based on the simplest possible assumptions, is by its very nature a difficult tool for understanding phenomena that emerge from the entangled interactions of many heterogeneous parts. Biology just happens to be one field that can push forward our understanding of complex systems. Computation, information science, and neuroscience are other fields that will help further mathematics.

Physics will always be important, but the domain of natural phenomena in which it finds itself useful is lessening as the Information Age comes into full swing.

Understanding the correlations between model parameters of speckle

Today I read "Structural correlations in Gaussian random wave fields," an old PRE by Freund and Shvartsman. The authors analytically found the existence of correlations between the amplitude and phase gradients in random electromagnetic fields commonly known as speckle. Notably, while the amplitude and phase are not correlated at certain points, the amplitude is correlated to the gradient of the phase. Higher amplitudes usually are found with smaller phase gradients and vice-versa.

What's not clear to me is if this treatment works for vector fields or only scalar fields. Notably, I'm not sure what phase means for a random vector field.

Perhaps the authors make the assumption that the components of the vector are independent and thus a scalar treatment is sufficient, but I'm not sure that this is so.

Notes on "Biological Physics," Part II

I finished the review of "Biological Physics" today, which included the sections on bioenergetics, forces, and single-molecule experiments. I skipped the section on reaction theory because I am not familiar with the topic and it didn't interest me as much as the others.

There are two primary topics in bioenergetics at the biomolecular level: charge transport and light transduction. Charge transport refers to the process by which isolated charges travel amongst different sites in a complex molecule. This process is inherently quantum mechanical, since electrons and holes may actually tunnel into different sites in the molecule, depending on the molecule's conformation.

Light transduction refers to the conversion of energy in a photon to chemical or electronic energy. A paragraph is dedicated to human vision and the photo-induced isomerization that is central to its operation, but the rest of this sub-section is devoted to photosynthesis.

During photosynthesis, "antenna systems" in the chlorophyll molecules capture light energy, which is transferred to other parts of the plant cell along excited molecular states, much like in Foerster resonance energy transfer. The transfer is so fast that the quantum mechanical coherence of the excited states likely plays a role. It seems that most of the work done up to the point in time when the article was written has been performed by theorists.

The various forces in the cell are typically "effective" forces models typically neglect the fundamental electromagnetic nature of the primary forces in the cell. At the protein level, enzymes may actually pull apart the covalent bonds in "violent" events. It's also been hypothesized that mechanical vibrations in the form of solitons can propagate along the covalently-bonded protein backbone, but this is strongly debated.

The transmission of forces through a heterogeneous medium, like the cell membrane is also a topic of study.

Finally, single-molecule studies are gaining prominence as experimental techniques become more refined, but "the challenge of studying individual protein molecules is still very much in its infancy... The key is to use extreme dilution so that only a single biomolecule is in the reaction volume."

Much single-molecule work has been done on DNA because it is simple and readily obtained. Spring-like forces in DNA are both enthalpic, which means they depend on the energy change due to deformation of electronic orbitals, and entropic, which means the DNA resists changing its shape due to interaction with its thermal environment.

In the conclusion, the authors anticipate that problems relating to the brain lie ahead as major areas of work in biological physics.

It would seem that the experimental study of proteins remains a major challenge to biological physics, but also is perhaps the most worthwhile to pursue. Photosynthesis, the effects of a protein's environment on its folding and charge transport, disordered protein behavior, and the forces between parts of proteins are not very well-understood. If there are new discoveries to be made, then I think they lie in protein dynamics.

Notes on "Biological Physics," Part I

There is an article from 1999 in Reviews of Modern Physics entitled "Biological Physics." This review summarizes research during the twentieth century where "physics has influenced biology and where investigations on biological systems have led to new physical insights." The exchange of ideas between the two fields has not been of equal magnitude, the authors note. Many tools from physics have found their way into the biological sciences, though some biological systems have led to new physics, usually in the form of providing experimental testbeds for new physical theories. The article is primarily concerned with molecular biological physics.

The seven primary sections of the review are

1. The structures of biological systems
2. Complexity in biology
3. Dynamics, mostly within proteins
4. Reaction theory, where biology has provided testbeds for new physical theories
5. Bioenergetics
6. Forces
7. Single-molecule experiments.

Most of the interesting ideas I've found so far in the article are associated with the complexity and dynamics of biomolecules. Particularly, there is an idea known as the principle of minimal frustration. From the Wikipedia article,

"This principle says that nature has chosen amino acid sequences so that the folded state of the protein is very stable. In addition, the undesired interactions between amino acids along the folding pathway are reduced making the acquisition of the folded state a very fast process. Even though nature has reduced the level of frustration in proteins, some degree of it remains up to now as can be observed in the presence of local minima in the energy landscape of proteins."

This idea came from a theory of energy landscapes for proteins that was developed by Bryngelson and Wolynes. In language that I'm more familiar with, the potential energy of the molecules has some fractal-like structure, because from Section III in the article the authors state that

"The kinetic observations suggest that the energy landscape might have a hierarchical structure, arranged in a number of tiers, with different tiers having widely separated average barrier heights."

It seems like structural determination of proteins and other biomolecules has become something akin to bookkeeping. The tools exist and are refined to find static structures, like neutron scattering and NMR. Additionally, the energy landscape theory for protein folding seems to be mature at this point as well. So what open-ended questions still exist in biological physics? After reading up to section V, I've compiled the following grand problems in biological physics as I've interpreted them from this paper only:

1. "A synthesis that connects structure, energy landscape, dynamics, and function has not yet been achieved." This seems to suggest that there is some degree of incoherence between these individual fields of study, so ideas that link them together are required.
2. Biochemists can now synthesize their own proteins, but can they do this in a useful manner, for, say, molecular and microscopic engineering purposes?
3. Sensing and characterizing phase transitions, especially in glassy systems, could lead to better experimental investigations into protein folding.
4. "Understanding protein folding can improve the capability of predicting protein structure from sequence." Apparently there's a lot of DNA sequence information, but predicting what proteins come from it is nontrivial.

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.

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.

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.

Understanding the generalized Stokes-Einstein equation

Mason and Weitz published a paper in 1995 about a technique for extracting bulk material parameters from dynamic light scattering measurements on complex fluids. That is, they established a mathematical relationship between the fluctuations of scattered light intensity from a colloidal suspension and the shear moduli of the complex fluid as a whole.

One primary assumption in this derivation is the equivalence of the frequency-dependent viscosity to a so-called memory function:

 

where η(s) is the Laplace frequency-dependent viscosity and ς(s) is the memory function. As a special case example, ς(s) is a delta-function at s=0 for purely viscous fluids since they do not store energy (i.e. they do not possess any elasticity). Substituting this into the well-known Stokes-Einstein equation leads to a relation between the colloidal particles' mean-squared-displacement (measured by dynamic light scattering) and the complex shear modulus of the fluid, G*(ω) (after conversion to the Fourier frequency domain):

The authors note in the end of the paper that it's unknown why light scattering techniques should produce the shear modulus of the fluid since they measure elements along the diagonal of the system's linear response tensor, whereas the shear moduli are contained in the off-diagonal elements.

They also note (with explanations I don't quite understand) that "...the light scattering may not provide a quantitatively exact measure of the elastic moduli; nevertheless, as our results show, the overall trends are correctly captured, and the agreement is very good." (emphasis mine)

Question everything

As you may know, my major field is optics, which concerns the study and application of light. Throughout my studies I've been constantly amazed that Maxwell's electromagnetic theory of light, which has been around since the late 1800's, still contains features that have not been settled or have been simply overlooked by scientists. One such artifact is the dissimilarity between Minkowski's and Abraham's descriptions of the momentum carried by an electromagnetic wave.

In a 2010 PRA Rapid Communication, Chaumet et al. expand on earlier work by Hinds and Barnett that examines the force on a dipole in a time-varying (i.e. pulsed) plane wave. This force is written completely as

 

where Pj is the dipole moment, E is the electric field, B is the magnetic induction, and ε is the Levi-Civita tensor. The first term in the sum relates to both the radiation pressure and the gradient force. The second term, according to Hinds and Barnett, is usually absent in laser trapping and cooling texts because it is proportional to the time-derivative of the Poynting vector, which is zero in common cooling setups. This term is responsible for repulsion of systems such as a two-level atom from the leading edge of the wave when the first term alone predicts an attraction.

Works like this make me wary of blindly using formulas when performing calculations since it reminds me that a theory may not be complete or its assumptions explicit when presented to a niche audience.

More notes from Jaynes

The introduction to Probability Theory: The Logic of Science has been useful for explaining what various statistical procedures are used for when making an inference about data.

Maximum entropy is a technique used to establish probabilities for outcomes from data given no prior information or assumptions. It is essentially an algorithm that comes to a conclusion without any bias from the experimenter. Bayesian techniques, on the other hand, require some prior information, and this will affect the conclusion.

Typically, when performing acts of inference, one begins with maximum entropy if very little is known except what's given in the data. Once more is known, one may turn to Bayesian analysis.

Bayesian analysis requires five things: a model, sample space, hypothesis space, prior probabilities, and sampling distributions.

There is much work to be done in developing techniques when even little is known about the raw data; this could lead to steps that can assist in studies where even maximum entropy may fail to give adequate results.

A physical standard for time

I've now completed Chapter 2 of Cook's "The Observational Foundations of Physics," my current lunchtime reading. In this chapter, Cook describes a thought experiment in which a beam of caesium atoms is polarized by a strong magnetic field, then enters a region where a strong RF field is applied. Following the RF region, the beam passes through another magnetic field such that atoms whose magnetic dipole moments are not flipped by the RF field are deflected into a beam block. Those atoms that do undergo an electronic transition that is accompanied by a flip of the magnetic dipole moment reach a detector that reports the intensity of the beam. A feedback mechanism adjusts the frequency of the RF field so that the beam intensity at the detector is maximized; in this way, a quantum standard of time is established through the frequency of the RF field that maximizes the atomic beam intensity (you may note the similarity to the Stern-Gerlach appartus).

Cook then proceeds to argue for his thesis, namely that the experiments and observations that are available to us dictate the form of our physical theories. He starts first with the theory. The time-evolution of the caesium atoms is described by the Schroedinger equation. This equation contains a first order time derivative which is a consequence of the wavefunction containing all information about the system at any one point in time. If only one initial condition is required to establish the wavefunction, then it must be first order in time (this is in contrast to the wave equation which is second order in time and whose solution requires an initial condition on the wavefunction and its derivative).

Cook next mathematically defines the operations of the experiment described above, postulating that the two magnetic states of the atoms are described by stationary states of a wavefunction. Using only mathematical arguments derived from the nature of the experiment, he obtains the form of the equations governing the time evolution of the system; the wavefunction is affected by a first order time derivative. This suggests that how we perform experiments determines the form of our theories. The time standard need not be quantum in nature as he repeats the argument for a classical, mechanical oscillator. Again he stresses that once the time standard is set, it is meaningless to ask whether or not its period remains invariant with time, since the standard defines time itself. It is recognized that differences between the same apparatus for establishing the standard exist when the apparatus are spatially separated due to the geometry of spacetime.

These are some of the thoughts I had while reading this chapter:

1. Many times physical theories are developed first and then experiments follow that verify their predictions. Does this fact weaken Cook's argument that experiments shape our theories? If the purpose of theory is to predict experimental outcomes, then why argue for the reverse? Which came first, the chicken or the egg?
2. Cook was careful to explain that his arguments are based on a physical world that is independent of a subjective observer. Still, I wonder how the human perception of time can be reconciled with these arguments. As stated earlier, it makes no sense to ask whether or not the time standard is invariant within the context of observation and theory. But a human can perceive large changes in the period of a slow mechanical oscillator. What is it that acts as an internal time standard for a subjective observer and can it be related to the physical standard?
3. Cook only obtains the form of the equations of motion for the systems he describes. On the other hand, the theories give meaning to his unspecified parameters, such as energy and the unit of electronic charge. What determines how these mental concepts are developed? Energy is a relatively easy concept to understand. Was this why the fathers of thermodynamics used it as a core concept in physics as opposed to some other mental construct?

Notes from "The Observational Foundations of Physics"

Section 1.3, Measurements and Standards, is a continuation of the setup for the arguments for Cook's thesis on how measurement affects the logical structure of physics. First, Cook states that the equations of physics are simply relationships between physical states or quantities. These relationships are congruent to the relationships between observations. I am a bit unclear as to what congruent means here, but aside from that the setup so far seems fairly obvious.

He continues onto a more lengthy discussion of the role that standards play in measurement. Every measurement consists of comparing some quantity to a standard quantity. When measuring the length of an object, for example, one simply compares the object's length to the length of a ruler (the standard). Our system of standards plays a significant role in shaping the nature of physical theories.

What was very surprising is that, traditionally, standards for all physical measurements can be derived from four independent standards: length, mass, time, and current. These standards have since been replaced by other physical constants and quantities, but the number of independent standards has remained the same. For example, length is measured as a ratio between the speed of light in free space to a unit of time, which is derived from a standard of frequency from a certain atomic process.

The standard of voltage comes from the standard of frequency and the Josephson effect with help from another fundamental constant, the ratio of Planck's constant to the unit of electronic charge. Mass currently (as of the book's publishing) escapes a relation to the standard of frequency, but it's conceivable that it could be related to energy, voltage, and current through the quantum Hall effect.

The shift from mechanical standards to electronic and quantum standards has greatly increased the precision with which we can measure physical quantities. It has also changed the nature of our physical theories, Cook claims.

Lunchtime reading: The Observational Foundations of Physics

I have started reading "The Observational Foundations of Physics" by Sir Allan Cook during my lunch breaks. The book's purpose, as Cook states in the first sentence of Section 1.1, "is to attempt to unravel some ways in which the practice of physics determines the form and content of physics and physical theory." In other words, Cook wishes to understand how the practices found in physics affect physical theories and the practices themselves. It is as if there existed a feedback loop such that performing experiments changed not simply the theory used to describe a phenomenon but the nature of theory itself.

Further in Section 1.1, he poses these questions that are central to his analysis:

1. "Why should physics be so effective, and what does that tell us about the world of physics and our ways of gaining knowledge of it?"
2. "Is there a real world that exists independently of whether I or anyone else is looking at it, or are all the ideas I have about a world external to me just the construction of my mind?"

He defers a thorough answer to the second question until the end of the book, but does offer that he believes that most physicists, while working at the bench or on a computer, act as if an external world existed independent of their attention.

Section 1.2 deals with observations and sets many of the premises of his arguments. Observation and experiment are decided to be equivalent. Observations also consist of two aspects: objective and subjective. Of the subjective aspect, only the communal nature of observation is of consequence to his arguments. Science is a social construct and scientists hold great influence over each other such that the act of observation is never truly independent of people other than the experimenter.

Cook goes to some length to explain that physics is empirical, "with observation primary and theory secondary," but he concedes that rarely can observation be performed without some theory underlying the act of observing. He gives the example of reading a voltage from a digital multimeter. The direct observation is of figures on a LCD readout, a consequence of numerous electronic circuits that respond to potential differences between two probes and relates to the potential energy difference of electrons between two points in a circuit. Of course, electrons are theoretical constructs. The theories underlying an observation can in some ways assure an experimenter that the results are telling us something of the real world and not subject to some extraneous errors or misinterpretations. For simplicity, an observation is defined as the operations that lead to a measurement and result in  "raw data." The data is considered "raw" regardless of the complexity of the measurement.

Finally, theories are models of observations, not a model of the real world itself. "I take a theory to be a mathematical realisation of an abstract system that has properties corresponding to those of a set of observations... It is in that sense that I take a theory to be a model of the world of observations, with the implication that there is a more fundamental correspondence than just giving the right answers..." Theory is an abstraction of the real world, not vice versa.