Divergents and Directional Antennae

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★ Part Two: Measurement and Inference

So in the past two weeks or so, I dove in headfirst to a series of Plasma Physics lectures by V.K. Tripathi. I haven’t studied EMR/Maxwell or Thermodynamics or Netownian Dynamics. I even dropped out of my university physics class at Virginia Tech after about 2 weeks and haven’t tried to seriously study any physics. I had linear algebra, but hadn’t gotten to vector/tensor calculus.

I quickly found myself surrounded by math I hadn’t worked through before, like gradient, divergence, curl, Laplacian and operators in general. I’m working towards understanding Hamiltonian and Hermitian operators. I still don’t know all the details, but is very intuitive to me. Still, that was the greatest challenge. At a high level, I understood the content, but knew I couldn’t do much more without comprehending the low-level details of the math behind the physics. That’s one big difference between watching a TED Talk on fusion and actually taking a class on Plasma Physics.

Radiation of Electromagnetic Energy

I started thinking about directional antennae and reason about what kind of math would allow me to predict the shape of the energy distribution, first given idealized omnidirectional or directional antennae. And later, given the specific structure of a more realistic antenna, I hoped to identify some relationships that would allow computation of energy propagation and distribution.

Just a reminder that I’m working through this problem in my head. I realize that a lot of this is probably wrong and that I haven’t really done the math. But, at a high level, it all seems very simple to me. I’m starting to pick up some math/physics tools that would greatly augment my ability to reason about domains and problems which were formerly completely out of my grasp.

So uhh, keep in mind, this article isn’t about providing an accurate model of this problem. Instead, I’m walking through how I approached the problem.

I had just learned about surface integrals, which are used in the Laplacian operator. From intuition, it seems like the energy radiated by an antenna would be fairly constant over time. And for a medium in which the speed of light is constant, the energy radiated for one timeslice of length ∂t should be mostly equivalent to the energy radiated at some other slice. This assumes that ∂t is sufficiently larger than λ / c, where λ is the wavelength.

A Spherical Cow

This is much simpler given the spherical cow of a pointlike omnidirectional antennae, whose signal is one frequency composed of perfectly distributed energy, radiating in all directions. Given this scenario, and assuming there are no reflections, then calculating the energy distribution is dependent on a point’s distance from the source.

But once you start looking at directional antennae, things become more complicated. A simple model will just distort the distribution as an ellispoid-like object on one, two or three axes. That’s not right though, because the axes don’t need to be orthogonal to each other. And yet, that’s still not right because the EMR doesn’t even radiate from a single point. But assuming one goes down this path, this model is still fairly simple to use. You can model passing small volumes to a 3D parametric function E-dist(t,x1,x2,y1,x2,z1,x2) -> (E) that returns a scalar quantity of energy.

Power In = Power Out + Power Loss

Two very useful values that very useful to know is the rate of current/energy going into the antenna and how much energy is produced as EMR. You can correlate those values to the average rate of energy output, assuming some energy loss given the material specifics. This power P implies that for a timestep large enough such that the average rate of energy output holds, the sum of energy must equal the power output for that timeslice. This can be very roughly notated as: ∫E(r,t)∂t = P * ∂t, where we’re either summing over a thin shell using surface integral or laplacian or getting the difference in E(r,t+1) and E(r,t). But the point is that we may be able to use the total power in and efficiency of the antenna to estimate how much power should appear in a timeslice.

Antenna Mechanics

Spherical and multi-axis directional parametric - both of these models assume EMR from a pointlike object, but that’s impossible, isn’t it? So, if it’s not pointlike, then we can’t simply use a parametric function because that’s too much of an approximation. The mechanics of an antenna’s shape contributes to distorting that ellipsoid. So one could divide the antenna into increasingly small 3D pieces – Antenna Volume Element AVE for brevity. The space enclosing & surrounding the antenna is SVE.

Yet, that’s complicated, isn’t it? It’s going to require a tensor to map each antenna slice to it’s energy output. And it’s a bit more complex because for each SVE, to calculate it’s E-SVE, you need the sum of contributions for a particular timeslice. Because EMR travels at the speed of light, we’ll need to know that SVE’s distance to each AVE in the antenna. And therefore, the mapping for the tensor isn’t just space-to-space or [x,y,z] => [x,y,z] or [r1] => [r2], it’s space + time to space. Or [r1 | ∂t*c == dist(r1,r2)] => [r2] where ∂t here represents the difference in time for EMR traveling from r1 to reach r2, instead of the time difference for each slice or shell of the EMP radiating from the antenna.

So, given this somewhat complicated domain for the tensor, the energy contribution of each AVE can be summed to match the output of the antenna for each point in space surrounding it. We’ll call this pair of functions E-AVE() and E-SVE() – again, representing the energy contributions from each chunk of the antenna and the energy received into each chunk of space surrounding it. An algorithm to calculate these values can be set up in many different ways, therefore the arity or parameters of these functions could vary quite a bit. But we know they at least require some time t.

E-AVE can be calculated to return scalar, or a point-like spherical vector “field”, whose vectors’ magnitudes sum to that scalar, approximately. Technically, it might not be a vector field, since it only has vectors radiating outwards from that one point – and it could require a minimal volume/topology, in order to maintain some kind of continuity for the calculus. E-SVE could be calculated similarly as a scalar value of energy for that volume element, but for purposes of calculating values for regions on the surface integral of E-SVE for specific timeslices of t, it may help to retain that directional information. In other words, for each slice of the SVE, the EMR there can be calculated from components, but those components are actually continuous waves (.. particles) that can be traced backwords to each contributing AVE. I won’t be using this fact, and it gets significantly harder with electromagnetic forces that distort the path of EMR but it’s interesting that everything is kind of woven together.

So, given an E-AVE output with directional information, we should be able to calculate the E-SVE for each region in the surface integral. From here we can use the other vector calculus tools to get a bit more information about how the radiation dissipates as it moves further away from the antenna.

Surface & Volume Mapping

Another problem: in order to calculate a surface integral for the shell of radiation at a particular time point t, or to calculate the volume of a region of space within a timeslice at t, we can’t really specify that operation without either:

(1) Knowing a bit about the shape of that shell at t and its topology

(2) Or calculating it from the shell of EMR at t = 0 (the surface of the antenna)

Either way, you might have to account for changes in the topology, requiring your algorithm or math to remap the space. I think this implies a kind of discontinuity. E.G. you have a donut shaped antenna: at some point, as the EMR travels outward, that will change from a donut to a disk. Therefore, there will be a timestep that requires a discontinuity if you’re mapping the “surface” of the EMR radiating outwards over increasing timesteps. If you’re just mapping volume chunks and then integrating on those, it’s possible to maintain continuity. I wonder if that’s generally true: that you can just abuse dimensionality, do a bit of remapping and get around problems with continuity in topology. And if so, under what conditions will that hold and why?

This differentiability across a shape that could change topology is important when you want to structure higher order behavior on top of the distribution of EMR. E.G. if you want to design an algorithm that optimizes antenna design for phase coherence at specific distances, where you might want to have a receiver or something. I donno. I really don’t.

But, I think this differentiability across shapes that aren’t topologically continuous is important for differential manifolds, but I haven’t looked into it much yet. I just found Harvard Math 253’s compiled lecture notes and got that onto my Kindle. These look like good notes going from laplacian operator to manifolds to Reimann manifolds, which are used in relativity. I can’t really say I know what a manifold is. I just learned exactly what a vector field was, though they were in my high school textbooks. I thought they were boring because “vector” in the name.

Also, this paper from Toronto on fluid simulation using laplacian eigenfunctions looks cool. I think eigenfunctions allow you to vary operations & behavior based on state of local systems within a large complicated system. I’ve been trying to figure that out for a long time. How to specify various higher-order behaviors to emerge in a particle-based system. It’s probably important for physics based on statistical mechanics and quantum mechanics.

Minimum Distance to Surface

So, yet another “gotcha” and this one’s complicated. The entire antenna doesn’t produce signal. There are a lot of higher order effects here, including those from macro structures and micro structures.

One of the effects from macro structure is caused by the Skin Effect, which causes current to flow towards the outside of electronic structures and influences the design of antenna. This is fairly important in antenna design, but how do you mathematically model it? This is another topological issue where various topological fundamentals may result in drastically different models. It could be an opportunity for performance enhancement, since you might not have to calculate too far into the surface.

But basically, you need to map a coodinate system onto the surface and then unroll the shape at the midpoint into a 3D surface. where the the X and Y axis give you coordinates on the surface and the Z axis gives you depth into the antenna. Although, depending on the structure, you may only be able to penetrate some units into the design on the Z axis before running into mapping issues. This map will be used for calculating the current contribution of each AVE to the EMR output by checking the minimum distance to surface and checking the current flowing through its neighbors.

You will either need to hardcode in a simulation of the Skin Effect or model convolution onto each AVE and its neighbors. If you don’t do with a small enough AVE size, then you won’t see the current/potential looping in the wire that causes the Skin Effect. Convolution is incredibly expensive. More in a second. That distance will help you determine how much EMR that node is contributing, but you need to know how current flows between the nodes to determine the direction the EMR will be projected. So you need that convolution anyways.

If you’re modeling a more complex object, you have to specify how to unroll the object so that you don’t cover volumes twice. You may have to chain several mappings so that the X, Y and Z coordinates map proportionally and ideally one-to-one. This is a harder problem than mapping meshes to surfaces of 3D objects and probably requires algebraic geometry. Instead of keeping the object as one piece, you can break it into pieces using an indexed map, but then you may end up with continuity problems and you have to use another map to mark the surface that’s exposed.

Convolutional Behavior

Then, to tie it all together, you have to use another map to map the AVE as they are subdivided from the original object to the set of maps to calculate distance to surface. This is so that the convolution-based behavior can be modeled more easily, without needing fibers & sheaths and only God knows what kind of algebraic geometry objects to tie everything together …. I finally figured out how to read the wikipedia math articles and I found the good YouTube lectures … but I can’t really do this math. I’m probably making a lot of glaring mistakes here.

Convolutional behavior is found in machine learning and specifically in computer vision, where it’s used in the lower levels of neural networks to discover edge and color patterns. In computer vision, there are multiple layers, which connect into larger and larger pieces. These are used to identify higher-level and more complicated features. The lower levels often compare the difference in pixel value or contrast value in each pixel and 2 - 5 adjacent pixels.

A Convolution on a pixel and 2 adjacent pixels

[ ] [ ] [ ] [ ] [ ];
[ ] [ ] [ ] [ ] [ ];
[ ] [ ] [X] [ ] [ ];
[ ] [ ] [ ] [ ] [ ];
[ ] [ ] [ ] [ ] [ ];

Results in up to 25 operations per pixel at that layer.

But convolution in a 3D object would be similar, but in 3 dimensions. Except, instead of training layers for recognition of patterns – which actually might be useful – you’re passing these values into a function that updates the state of the center pixel based on the gradient of current flow and the divergence in adjacent pixels. Then it uses the ∂E value for that pixel to determine its EMR contribution, since AFAIK, EMR is based on fluctuations in current. Again, I haven’t taken an E & M class.

This is chained together in a 3D version of Conway’s Game of Life. Every timestep, the state of each pixel is updated based on the results of its convolutions. Sometimes this stuff can be GPU accelerated. Sometimes it can’t.

But the value for EMR produced for each AVE may more accurate if something like a plate model is used for each convolution. This means retaining several copies of each convolution, for each time slice. It’d be 4D instead of 3D. Plate models are from statistical modeling and are often used in AI for video games, where you want the computer to become adjusted to reacting to similar sequences of actions that may occur over similar but different timescales.

Absorption, Reflection, Dispersion and Anisotropy

These factors are much more complex to accomodate. They require modeling absorption and reflection data into the materials and then monte carlo sampling to cast rays. For video games, the rays are cast from lighting sources and bounce around the scene. Lighting is one of the first complicated aspects of graphics that you’ll run into. It’s computation intensive and you have one camera. If you’re calculating radiation reflection and absorption, each SVE is basically a camera. However, if there’s an unobstructed path, one can assume that the values for adjecent SVE nodes can be calculated by simply extending vectors, instead of recalculating everything from the sources.

orientation of energy radiation from AVE - orientation of energy radiating from AVE - E/M anisotropy of adjacent AVE affecting trajectory - alters dispersion and path - requires ray tracing (afaik) - also, i might be completely ignorant of E/M

Stateful Interactions

Some higher-level phenomena can emerge from the convolutions of AVE and their neighbors. These behaviors can cause the system or that part of the system to act in a completely different manner, unable to be sufficiently modeled by the basics. In other words, they are emergent phenomena. States of matter like solid, liquid, gas, and plasma are examples where the system takes on completely new behaviors that are not possible unless that part of the system satisfies specific conditions.

If I’m right with the wikipedia articles and papers I’ve been reading, then this is where the Hamiltonian operators become useful, which allow you specify complicated systems of behavior. And I think this is where eigenfunctions become useful. Again, you’re going to need to identify a set of conditions that can recognize where in the system these state rules begin to take effect. And then you’re going to need to be able to dynamically maintain a map of the part of the system that adheres to various states. And apply the functions that pertain to the active states in the right order.

As it approaches absolute zero, ice changes into many different forms. But just because a region of ice is at the temperature range for a particular form doesn’t mean it automatically assumes it. Sometimes, some level of kinetic energy must enter the system and then it has the energy it needs to reconfigure itself. Ice has another emergent phenomenon where it blocks out microwaves. Water vibrates and allows the microwaves pass through.

… This is why I have to thaw out my burrito’s because otherwise I end up with burrito slushies. Warm on the outside and a little cold in the middle. Just the way I like ‘em. Yummmm…

Further, there are many magnetic and EMR interactions that emerge from micro/macro structures in the antenna’s crystal lattice. From those interactions emerge stateful behaviors that would cause that. These are moreso micro-level interactions that can build up into higher level behaviors. I’m very much interested in this stuff.

That’s It

Moving on to the next article. Just wanted to explain some of the things I’ve learned lately. I feel like I’ve learned so much in the past two or three weeks, just from watching a few lectures and taking a few notes.

I wish I could dedicate myself to being a fulltime student on Coursera or maybe even a higher education institution. My issue with going back to school for four years officially is that – yes, it’s a bit too expensive, but that’s really not it – it’s that I’d have to settle on one thing for four years and really keep my blinders on. It’d be so hard for me to commit to school without getting distracted by some amazing technology, platform or startup idea.