Ryan Westafer

An article relayed by the ASEE highlights something researchers have known for years – the boundaries between fields are coming down:

MIT Promotes Convergence as Model for 21st Century Research

At first I thought this might imply that people with interdisciplinary knowledge, like me :-), will be necessary to facilitate the teaming of super-specialists.  That is, people acting analogously to catalysts, matching networks, surfactants, etc.  However, I realize that it’s much bigger; it’s global, and it’s universal (essentially by diffusion).  The global availability of information in this era is allowing the diffusion of disparate knowledge to everyone.

Consider the papers published by researchers at CERN… I saw at least one with 50+ authors.

While I agree with recognizing all contributors, it is increasingly apparent to me that a prevalent and perhaps necessary way to “succeed,” “stand out,” or have an “advantage,” will be to withhold information – to create or maintain barriers — to be exclusive.  Trade secrets, what-have-you.  Why patent something?  You have to bother with litigation – the patent is written as broadly as possible, but its protection requires “pockets” as deep as possible ($$$).  Maybe that’s a new metric for IP success: (Claimed Scope)*Capital.  Measuring the scope could be challenging.

As fields intersect, this is why I love open software and designs.  There’s inherent scalability and mutual respect in sharing.  I wonder, though, what the world will be like in 20+ years.  The culture of community innovation may stay, but what will be the scarce resource, the differentiating factor?

A basic but interesting fact:

At the point on a dispersion diagram, where and , the usual phase velocity is undefined; that is, .

So, what do we do when this occurs? We use L’Hospital’s rule. This yields . So, in the large wavelength (low momentum) and low frequency (large (rest) mass) limit, the phase velocity is mathematically identical to the group velocity – regardless the dispersion relation.  In an interesting whiteboard discussion with Saeed Mohammadi, this realization seemed to be helpful.

When , the “original” or “root” modes (e.g. “acoustic” rather than “optical” phonons) have infinite temporal duration and spatial extent. This seems like a point of modal equilibrium. It would be interesting to check whether operating points on a dispersion curve transition toward when transient phenomena are considered.  This point represents the “smoothest” real-(phase)space solution, which should be the proper direction for gradient descent over total energy for a general physical PDE.

I’ll go ahead and sling this out… a realization is that the group velocity, (effective)mass, etc. lose significance when spatiotemporal “texture” disappears.

Read more…

In a group presentation I gave this week, I showed the math (now improved with a few corrections) for characteristics of a wave equation in general media (i.e. over a periodic basis).  It is well known that Fourier techniques can be applied in partial differential equations to represent distributions of concentration, heat, electric charge, etc.  This is especially appropriate for crystalline materials with well-defined periodic structure, but it is extensible to general media.

The first step was to reduce Newton’s 2nd Law from tensor form to a 1-D form, i.e. the propagation of longitudinal waves along x.  (A nice feature of acoustical treatments in general, is that its formulation permits waves of general polarization, including longitudinal.  Electromagnetic waves do not, except in the presence of media which can impart phase front tilt.)

I’ll eventually give the math here (and hopefully it will pass muster in my thesis), but it turns out that spatial periodicity of the material stiffness gives rise to a complex (mathematical sense) wave equation having orthogonal components.  It represents a coupled heat equation and wave equation, where the dissipation term is converse to our usual expectations; it is (without coefficients):

Now there are several details and other terms which appear, but this snippet captures a form which many might find surprising.  After visualizing the spatially periodic characteristics of the general PDE, I believe this has improved my understanding of dispersion diagrams: why they are generally complex (again, mathematical sense), and how we design devices using “meta” materials, and PxCs.

This investigation of periodicity-induced modal diffusion may only be novel to me, but I just haven’t seen this phenomenon described in simple terms before.  See the references for some particularly impenetrable notation describing dispersion and mode coupling … but no explicit mention of inherent diffusion equation(s).

Read more…

Last year, NPR’s Science Friday marked SETI’s 25th birthday. This sparked a few thoughts…

What if we’ve already got evidence of “extraterrestrial intelligence” in every signal we observe?

How would contemporary science explain that?  This idea stemmed from consideration of the signal characteristics I would expect to receive from a super-intelligent being or civilization:

• Coding: pseudo-noise. An optimal universal signal might indeed have characteristics of noise. It might be so spread-spectrum that it would be felt at all scales: from the shockwaves of supernovae to the invisible rainfall of high energy cosmic rays.  It would have very large uncertainty products; large negentropy.
• Content: survival. What sort of signal would “someone” exert the effort to craft and produce? I’d argue that it would likely be a necessary one. A signal conveying recipe(s) for life might be a good start.
• Radiation: omnidirectional. If the universe expands, the signal ought to have evolved with the expansion of the universe… i.e. it ought to have been present all along, imparted to and diffused among all the initial components, rather than being emitted from an already distant location to our present-day Earth.

Are we all extraterrestrials?

OK, that’s a weird statement; but surely I am not here on this planet of my own plan & design.  Explaining, would some scientists agree that are we all created by an unknowable universal energy (information) source?  The biomolecular chemistry which enables terrestrial life at the cellular level is powered by the most natural and universal behaviors of stochastic motion, diffusion, and equilibrium.  We are fundamentally driven by signals of indeterminate origin and assumed to be “noise.”  From such stochastic processes we can obtain “order” in the limit of distributions and averages.

Contemporary scientific theories suggest that all the structure we perceive originated in a singular event of pure (scalar) power comprising everything: time, space, etc.  This creation singularity, maybe mapped through the point (as in crystal dispersion diagrams), represents the ultimate “degeneracy” (mathematical sense…) point, from which structure condenses (emerges).  It’s amazing that, for all the triumphs of science, it seems even the most gifted of scientists cannot trace information through the singularity and must accept this limitation on a sort of faith.  Moreover, it seems that life is universal… it’s a beautiful solution interwoven throughout (and dependent upon) the entire universe; a sort of inherent gift balancing the “messy,” yet essential, random/thermal component.

One might ask why we don’t see evidence of life elsewhere… perhaps that is because life is fractal, and our very measurements have their bases within that fractal, making observable things in some sense self-similar.  We know measurements to be inner products, and thus, projections of the world onto the world around us.  This also relates to mutual information in communications theory: a source and receiver should speak the same language, having that information in common.  But, I suspect, thanks to an interplay of heat and waves, that the fractal has a “forward” direction in which we find new variations on old observations.

And I can’t help but note: this whole “extraterrestrial” creator thing seems like a secular perspective of God – that which cannot be seen but is omnipresent and ultimately responsible for all phenomena.  We as humans perceive only things our amazing but limited senses permit.  Yet we do find new perspectives – rearrangements of state – and this might be likened to looking through a kaleidoscope.

And of course I have comments on the expert commentary during the Science Friday broadcast:

• We’re looking for a “radar signal?” A “series of pulses?” We expect the signal to be polarized? We expect the signal to be modulated?I suspect an intelligent and dense signal would in fact vary all of these parameters… looking pretty random and confusing to us.
• A “focused laser beam” would be detectable “thousands of light years away” … if only one could sufficiently reduce the random angular variance in the transmitter…  and create a large enough effective aperture… and… and…

Group theoretic expressions of optical singularities in photonic crystals
by Wheeldon, Jeffrey F., Ph.D., University of Ottawa (Canada), 2009, 268 pages; AAT NR51824
(PDF costs 37.00 USD for members of academic institutions.)

I contacted Jeffrey Wheeldon, and he seemed interested that his work could be relevant in that way. However, I’m not sure anything will come of the connection. Regardless, it so happens that Maxwell actually published a “molecular vortex model” which was later refined in his “On the Physical Lines of Force,” so the father of E/M has already opened the door.

This resonates with several viewpoints I sometimes think about: the wave structure of matter (atoms from waves), the conveyance of angular momentum (photons) among singularities (hearts of atoms), and others.

So I pose a question (thesis topic anyone?): “Would a 3-D extension of Wheeldon’s 2-D treatment of singularities in linear photonic crystals correspond to a structure of matter?” The scope would require much refinement.

While the claim may be a stretch for anyone, perhaps even Dr. Wheeldon, the applicability of group theory to the conditions on optical vortices in in photonic crystals just screams “standard model” to me. We shall see…