What systems behave like fluids?

Housekeeping post to define what a fluid is. Of course, we all know what a fluid is – it’s a state of matter between solid and gas, right? Yes and no. The purpose of getting to an abstract definition of “what a fluid is” is to allow us to apply the “macro-architectures” we see arise in normal fluids like waves, spiral waves, flow, turbulent flow, and even periodic Reynold’s instabilities. The every day patterns where we see these are ocean waves, hurricanes, thundercells, water spray patterns from a hose nozzle, and wake patterns trailing behind a boat (or the atmospheric wake patterns seen in this picture).

The purpose of my previous posts on flows is to explain that the essential qualities of a “substance” or “system” that make it behave like a “fluid” are actually really simple. So here’s the abstract definition of a fluid:

1. A fluid is composed of coherent objects (in a typical matter-fluid, these are atoms or molecules). These objects are interchangeable in some way – ie, one object is “identical” to all the others in one (or more) attribute-dimensions, but the objects are never 100% identical. For instance, one water molecule is functionally equivalent to all other water molecules in a way that is, actually, an abstract property – the property we describe as having an oxygen atom “bound” to two hydrogen atoms. “Wait,” you might say, “that’s not an abstract property! Those are always absolutely concrete properties, aren’t they?” Actually, no. First, there are many suitable isotopes of each of these atoms. And even those descriptions are not absolutely concrete because the exact quantum states of any particular atom vary, even among identical isotopes. In the real world, we don’t even know if indivisible particles – like electrons – are all 100% identical. It seems that they are identical, but we really don’t know. So the point I’m stressing here is that we shouldn’t be concerned if we find ourselves talking about a fluid that is composed of “abstract,” non-identical objects – that is, actually, the common state of all the “real-world” fluids we are already familiar with.

2. The coherent objects comprising a fluid must be able to affect – and be affected by – their adjacent objects. In water, this simply means that one water molecule can impact its neighbor and impart momentum in the process. The nature of the “impact” effect need not literally be “momentum transfer” for the fluid-system to behave like a fluid. “Momentum Transfer” is just one instance of the class of “affects” that may exist between objects in the fluid-system. To offer an abstract example, if Fred buys an apple from Sue, a kind of affect happened – a “transfer” – where Fred experiences a drop in his property, “cash in pocket,” while Sue experiences an identical – but opposite – increase in the same property – an elastic transfer (ie, the transfer is analogous to an elastic energy transfer event, ie, Conservation of [Energy][Stuff] applies.) Similarly, Fred and Sue experienced an elastic transfer in their “apples on hand” property. Their transfer can go in either direction, and they can experience the same type of exchange with any other person they come in contact with.

3. The objects in the fluid must have a “degree of freedom” that includes the capacity to completely separate from the objects that are their adjacent neighbors at any one moment (ie, translational movement). In water, this is simply the way one water molecule can move a meter to the left, leaving its former neighbors where they were. This is distinct from a “solid” like ice, which is defined as a state of matter where molecules can vibrate “in place” but not actually move to some new location relative to the other molecules, except in the sense that they become disassociated with the “solid” and take on some new “state” – such as gas (sublimation) or liquid (evaporation).

Some substances that behave like solids in the short-term are actually fluids in the long-term, such as glass. If you could observe a pane of glass over the course of 1,000 years, condensed into a few seconds, it would appear to flow. Likewise, the longer your perspective, the more no solid actually behaves like a “solid.” In the long run, even in the real world, nothing stays a molecular solid, because every rock is going to eventually fall into a star, meltdown, and/or fall into a black hole. Either result falls into a “fluid” category. This is not to try to claim that every system is a “fluid,” per se – but rather to illustrate that the sense of whether something fits the description of a “fluid” depends on how much the objects it is comprised of can experience a net displacement over the “observation period” that concerns you. This observation period is often some period of time, but doesn’t have to be. Time is just “one dimension” that can be used to “observe” a system’s properties. It is often extremely useful to analyze a system of objects in some dimension – or sets of dimensions – other than time. Systems can take on properties of abstract “flow” that are only “visible” when your “observation period” takes place on some spatial dimension, rather than time, or – more importantly – takes place on some abstract dimension of properties.

For instance (this is where the importance of “fluid” and “flow” really jumps out), imagine a 3-dimensional plot of the people in the United States in the year 2012, where the x-dimension is the person’s income, the y-dimension is the person’s net wealth, and the z-dimension is the person’s age. You’ll have a cube that has 300 million points distributed somewhere within it. Now imagine a series of these cubes, one for each nation on the planet in 2012, with each nation’s population charted the same way, but proportionally divided into 300 million points, too. Now imagine “flowing” through this series of cubes, putting them into whatever sequence minimizes the “jumpiness” of the resulting movie. The “movie” this would create is always going to be really jumpy, since there are such drastic differences between nations, but there are ways to smooth out the jumpiness and observe patterns of change. Notice that the “movie” you would see would not be one that occurs across time, but rather occurs across whatever your criteria was for sequencing the various cubes. The result is a “movie” that you observe over the time it takes to step through the sequence – perhaps 5 seconds (150 nations, 30 frames[nations] per second).

If you did smooth out this data, such as by extrapolating fictitious nations to fill in the gaps (which is functionally identical to simply blending the nation-images together using the simple graphics technique) you would see a smooth movie showing dots or blobs that “flow” – and the “real” concept of “time” would have nothing to do with the flow.

Sometimes time can be in the data, but the “flow” doesn’t technically occur over “real” time, but rather abstract time. Econometric impulse computations result in a graph that shows a curve that illustrates what, for instance, the average nation’s growth will be over the 6-year time span surrounding a 10% increase in debt-to-gdp, such as is shown here, from Arindrajit Dube’s critique of Reinhardt-Rogoff’s data:

This graph illustrates a flow that occurs over a kind of time, but what you’re actually looking at, from a “dimensional analysis” stand-point, can not technically be merely expressed as blah-per-year. The time here is an abstract dimension time that is too complicated to bother putting into words (or units), since it arises from a (not-all-that-complicated, but) “statistical” analysis of many real world points that really do have (simple) time as part of their dimensions. The true dimensions of the above graph would technically have to cite each nation and year and somehow note the weight that point imparts on the above chart. That may sound strange or even laughably incorrect, but that’s merely because we’ve gotten good at implicitly understanding what we mean when we look at statistical abstractions.

Even in averaging a small set of numbers, like {1 mph, 2 mph, 3 mph, 5 mph, 9 mph}, you can say that the average is “4” – but four of what? Most of the time we can “cheat” and collapse the units of all the items in the set because they are nominally identical – “mph.” However, that’s an abstraction. If this set is comprised of imaginary points that have idealistic precision, then they are also, implicitly, an abstraction – and (technically) the units should include that description (eg, “imaginary mph’s” – the wording is clumsy simply because we don’t have an English word that technically indicates the concept – but our brains do have a “mental-ese” “word”/symbol for the concept).

But in the real world, data points come from particular things in time and space, and have a distinct precision that necessarily varies over the time and space that the measurements are taken. Yes, the standard way of handling precision is to assert that the precision of each measurement from the same device is the same – but is it really? From a standpoint of quantum fluctuations, no. They could not possibly be exactly identical. But more importantly, the person reading the measurements varies in their attentiveness over time – even short time frames – and thus the exact estimation of precision associated with each data point is ultra–technically not identical. More over, the data points technically come from different “objects” in time and space (even if it is the same object at different times). (Ultra-)Technically, all of these properties should be included in the “units” of each data point.

But we’re not interested in all of those properties – we’re interested in JUST the abstracted property of “mph” with an abstractly truncated precision – so we toss out all the properties (units) we’re not interested in, and collapse all the remaining abstractly-truncated units into an identical moniker, and then treat the data points as functionally identical and interchangeable in properties, other than the magnitude – which in this case is the numerical quantity (such as “1” or “9”) – but could be any other property.  And that’s fine. But it is useful to be aware of what we’re really doing when we do it. Our final units (“4 mph”)  don’t even reflect that the magnitude of the average comes from the numerical magnitude of the items in the set – this is routinely assumed. But, again, that is, (ultra-)technically, a facet (or dimension) of the resulting units. After all, this misunderstanding is one aspect of why the “dimensional analysis” critique of the Cobb–Douglas production function is not a valid critique. The “dimensional-analysis” critique imagines that all units are somehow pristine, as in the “average” of my small set – but they aren’t.

Anyway, that’s all there is to abstract fluids – just those 3 properties – listed again in simple form:

1. comprised of similar “things”

2. the “things” interact with each other

3. the “things” move in relation to each other on any particular scale.

That’s it. And my point in all of these posts about “flow” and “fluids” is that the large-scale features that you can observe in any water or atmospheric flows can – and will – occur in any abstract fluid, too – whether the abstract fluid is the cosmological evolution of the known universe, or the macroeconomic picture of humans. This is not to say that macroeconomics literally consists entirely of predictable waves, as is asserted by many a crackpot. No. More often than not, the particular displacements you see on the surface of the ocean are essentially random and unpredictable. However, where there is a wave, it will flow. And where there is a circular pressure differential, a spiral phenomenon can evolve – and if it sits “on top of” a temperature gradient (like a hurricane), the spiral can be relatively self-sustaining as long as the spiral can draw “energy” from the temperature gradient. Similarly, some economic systems will “temporarily” fit the conditions for a vortex street.