Exploring Chaos Theory Constants: Beta, Prandtl, and Rayleigh

[osilmag]

One of the constants in chaos theory is symbolically labeled as beta, however I haven't found an official definition. The other constants Prandtl and Rayleigh deal with viscosity and diffusivity so they must be appropriate for the specific situation. Is beta simply a constant that can be changed to generate different results or is there more to it?

 

[Mark44]

One of the constants in chaos theory is symbolically labeled as beta, however I haven't found an official definition. The other constants Prandtl and Rayleigh deal with viscosity and diffusivity so they must be appropriate for the specific situation. Is beta simply a constant that can be changed to generate different results or is there more to it?

Chaos theory covers a broad sweep of topics, so I'm not sure what you're asking about. One possibility is bifurcation diagrams in a logistic map, one example of which is shown here: https://en.wikipedia.org/wiki/Bifurcation_diagram
In the section titled "Logistic Map" the equation is $x_{n + 1} = rx_n(1 - x_n)$. Here r is the parameter, which might be the same as the $\beta$ you mentioned. Some values of r yield predictable behavior, but others lead to chaotic bifurcations.

To the best of my knowledge, there is no special definition for such constants, other then the possible values they can take on.

 

[S.G. Janssens]

One of the constants in chaos theory is symbolically labeled as beta, however I haven't found an official definition. The other constants Prandtl and Rayleigh deal with viscosity and diffusivity so they must be appropriate for the specific situation. Is beta simply a constant that can be changed to generate different results or is there more to it?

These constants suggest you have in mind the well-known Lorenz system, not chaos theory in general.
Sparrow (and references to it) is (are) a good place to start. Or is that what you were already reading?

In his original article, Lorenz obtained the system as a truncation approximation. There $\beta$ is called $b$.
In other contexts modeled by the same equation, it may of course have a different interpretation.

 

[Auto-Didact]

One of the constants in chaos theory is symbolically labeled as beta, however I haven't found an official definition. The other constants Prandtl and Rayleigh deal with viscosity and diffusivity so they must be appropriate for the specific situation. Is beta simply a constant that can be changed to generate different results or is there more to it?

As @Krylov mentioned, you are probably referring to the Lorenz equations. I learned them as:
$\dot x= \sigma (y-x)$
$\dot y= rx - y - xz$
$\dot z= xy - bz$

where
$\sigma$ is the Prandtl number,
$r$ is the Rayleigh number,
$b$ is the unnamed parameter $\beta$ to which you are referring.

There is an experimental setup called the Malkus-Lorenz waterwheel, which is a mechanical analogy of the Lorenz equations, actually a special case thereof with $b=1$. The rotation of this waterwheel is actually also analogous to convection rolls, which occur in convection when $r$ passes a critical treshold; here $b$ is related to the aspect ratio (role wavelength to boundary layer depth) of these convection rolls, which explains why $b$ often appears as a fraction.

To the best of my knowledge, there is no special definition for such constants, other then the possible values they can take on.

Actually that stance seems to be a rampant misconception widely occurring, both within and outside of this field; contrary to popular belief there are definitely special definitions for such numbers. These pure numbers always arise from combinations of several dimensional parameters which are subsequently non-dimensionalised using specific products and ratios; the resulting pure number is called a dimensionless group in the dimensional analysis literature.

Physics - in particular fluid mechanics - is absolutely rife with such dimensionless groups, for example the Reynolds number, Mach number, all coefficients (drag, pressure, lift, etc) from basic physics, the fine structure constant from QED and so on. Moreover, the above procedure of non-dimensionalisation should sound eerily familiar to anyone familiar with constructing natural units in physics, seeing that is exactly how they are constructed using only physical constants. Even stronger, from the perspective of dimensional analysis, I would posit that all important quantities and formulae in physics are always distinguishable into two types: dimensional groups (regular physics quantities) and dimensionless groups.

The reason that the specialness of these dimensionless pure numbers doesn't seem to be so well known in the wider scientific community is that (this aspect of) nonlinear dynamics (NLD) and/or chaos theory is often marginalised and glossed over in the treatment of the subject. Instead, the focus tends to be predominantly on application, i.e. numerical plugging and chugging on a computer model in order to just quickly find solutions, without any further regard for developing a deeper mathematical understanding of the theory or gain physical intuition. This is worth elaborating on a bit more, for it seems to have large implications for both the present state and future of physics.

This difference in focus might be explained by the fact that NLD courses tend to be more often taught by applied mathematicians than by physicists, and so also attract more applied math majors than physics majors; pure mathematicians tend to already turn up their nose far before ever coming close to the subject. This literally seems to be a case of differing philosophies of mathematics between physicists and (applied) mathematicians stalling progress in both areas, a phenomenon occurring not just among scientists, but even occurring within single scientists who (often unconsciously) carefully separate out their physics knowledge from their mathematics knowledge, as Freeman Dyson masterfully illustrated in his paper 'Missed Opportunities'.

Lets get back to the heart of the topic at hand, namely non-dimensionalisation. Of course, it goes without saying that reformulating an equation containing several parameters into a simpler equation using dimensionless groups can make a solution easier to obtain, but doing this might change qualitative properties of the equation in some non-obvious way. Doing this however also often tends to result in some particular singular limiting case of the original nonlinear equation, which then requires perturbation theory to find an approximate solution. In the strong nonlinear limit however perturbation theory eventually ends up breaking down, requiring more potent techniques such as multiple scale analysis.

Moreover, reinterpreting the input of numerical values in computer models as actually carefully varying geometric and physical properties of a system, i.e. of the parameters within these dimensionless groups enables the use of physical intuition and more importantly, a direct route to experimental verification. Varying particular dimensionless groups past critical values while keeping others constant causes (period doubling) bifurcations in the system; this not only explains but also unifies wholly separated and vastly different branches of physics through the use of renormalisation group theory. It doesn't take an enormous stretch of the imagination to see how the above discussion directly relates very intimately to the mathematical structure of QFT and also to its many troubles - troubles which happen to remain completely unresolved in string theory.

In any case, it seems clear to me that non-dimensionalisation, taken to heart, has enormous implications for the interrelations of theories in physics, i.e. the ontological status of physical analogies and therefore for the entire mathematical structure of physics as a whole. Doing this carefully, i.e. without merely a focus on application and solutions in mind, but actually trying to gain mathematical insight and also develop physical intuition, forms a natural mathematics based guide forward in theoretical physics, a path upon which fractal geometry (FG) and NLD have demonstrably key roles.

The history of mathematics and physics have shown us how these two mathematical fields have in a short timespan already pretty much respectively infiltrated and subsumed all areas of not just physics, but almost all fields of science. This strongly suggests that from a theoretical physics point of view, any contemporary accepted physical theory currently regarded as fundamental, yet not deeply or fundamentally compatible with either FG, NLD or both should be regarded with the necessary amount of suspicion, since such a theory is probably merely provisional (I'm looking at you, QM). I end by paraphrasing Feynman, "Physics isn't linear, dammit, and if you want to make a fundamental theory of physics, you'd better make it nonlinear, and by golly it's a wonderful problem, because it doesn't look so easy."

 

[osilmag]

Thanks for the write-up Auto-Didact. You make great points about dimensionless numbers being used for plugging and chugging, instead of understanding the deeper physical connection or gaining a mathematical insight.

Yes, the Lorenz equations were what I was referring to. I had casually modeled a couple things and wanted to find more about "b." In the Malkus - Lorenz example you described, it is a ratio of two properties that describe the system. As I searched a little more, I found from another source "b" to be defined as a geometric factor.

 

[Jimster41]

@Auto-Didact. Thanks for link here.

I am grasping for a better understanding of what is meant by non-dimensionalization... got to read up.

 

[Auto-Didact]

Non-dimensionalisation is literally what the term implies: to make a dimensional number (e.g. a length expressed in meters) dimensionless (e.g. length as just a number).

All parameters in the equations of NDSs are always such non-dimensionalised numbers called a dimensionless (or nondimensional) group. Here's a simple example using a popular dimensionless group, the Reynolds number $\mathrm {Re}$ from fluid dynamics:
$$\mathrm {Re} = \frac {\rho vL} {\eta}$$The units for the density $\rho$ are kilograms divided by cubic metres, the units for speed $v$ are meters divided by seconds, for the length $L$ simply meters, and the units for the dynamic viscosity $\eta$ are kilograms divided by meters times seconds.

Using dimensional analysis we can therefore rewrite this equation in the form of the units of the quantities:
$$\mathrm {Re} = \mathrm {\frac {[\frac {kg} {m^3}][\frac {m} {s}][m]} {[\frac {kg} {m\cdot s}]}} = \mathrm {\frac {[kg][m^3][s ]} {[kg][m^3][s ]}} = 1$$By algebraic simplification we are now left with a pure number, meaning $\mathrm {Re}$ is a pure number characterizing all these physical quantities in that exact algebraic combination, in other words a dimensionless group: varying all the physical quantities changes this pure number.

 

[Jimster41]

Non-dimensionalisation is literally what the term implies: to make a dimensional number (e.g. a length expressed in meters) dimensionless (e.g. length as just a number).

All parameters in the equations of NDSs are always such non-dimensionalised numbers called a dimensionless (or nondimensional) group. Here's a simple example using a popular dimensionless group, the Reynolds number $\mathrm {Re}$ from fluid dynamics:
$$\mathrm {Re} = \frac {\rho vL} {\eta}$$The units for the density $\rho$ are kilograms divided by cubic metres, the units for speed $v$ are meters divided by seconds, for the length $L$ simply meters, and the units for the dynamic viscosity $\eta$ are kilograms divided by meters times seconds.

Using dimensional analysis we can therefore rewrite this equation in the form of the units of the quantities:
$$\mathrm {Re} = \mathrm {\frac {[\frac {kg} {m^3}][\frac {m} {s}][m]} {[\frac {kg} {m\cdot s}]}} = \mathrm {\frac {[kg][m^3][s ]} {[kg][m^3][s ]}} = 1$$By algebraic simplification we are now left with a pure number, meaning $\mathrm {Re}$ is a pure number characterizing all these physical quantities in that exact algebraic combination, in other words a dimensionless group: varying all the physical quantities changes this pure number.

oh, that! Well, that kind of thing is familiar. I thought you were talking about something more exotic. I got to ponder your statements about how such values serve to connect theoretical frameworks (that are different but similar).

 

[Auto-Didact]

oh, that! Well, that kind of thing is familiar. I thought you were talking about something more exotic. I got to ponder your statements about how such values serve to connect theoretical frameworks (that are different but similar).

The connection is geometric: a small triangle with some particular angles and lengths is just a scaled down version of a bigger triangle with the same angles but with the lengths of the smalles triangle now all multiplied by a scaling factor. This same geometric proportion argument directly applies to dimensionless groups.

The difference between different physical systems which are described by the same dimensionless group just means that the constituent materials from which the system is built up (as well as their properties such as size, speed, etc) is different; this is analogous to different kinds of triangles (right, isosceles, equilateral, etc).

 

[Jimster41]

The connection is geometric: a small triangle with some particular angles and lengths is just a scaled down version of a bigger triangle with the same angles but with the lengths of the smalles triangle now all multiplied by a scaling factor. This same geometric proportion argument directly applies to dimensionless groups.

The difference between different physical systems which are described by the same dimensionless group just means that the constituent materials from which the system is built up (as well as their properties such as size, speed, etc) is different; this is analogous to different kinds of triangles (right, isosceles, equilateral, etc).

I do kind of get what you are saying and I do sort of buy it. you might call them the eigenvalues of a set of formal systems (though what the eigenvector is I'm not sure)

All of which reminds me I am only half way through Hofstadter's awesome tome...
Gödel, Escher, Bach: An Eternal Golden Braid

 

[Auto-Didact]

Viewing dimensionless groups as eigenvalues works for scaling up and down systems; like in the geometric case with triangles, squares and so on in the plane, all vectors in the plane are then eigenvectors, making the entire concept somewhat redundant.

Far more interesting, at least in my opinion, is if the geometric analogy can tell us why the dimensionless groups when varied past a certain number suddenly leads to qualitative changes in the object in question, e.g. as in the case of a large $\mathrm {Re}$ leading to turbulence.

In other words, is there a way of scaling up triangles which suddenly leads to the scaled up triangle suddenly changing its properties? The answer is yes. For illustrative purposes, just imagine drawing a triangle on the floor, measuring its internal angles and then summing them up, which of course is a constant.

Then imagine scaling up the triangle on the floor in size; rinse and repeat, with each time measuring and summing of angles. If you would actually carry this out, you would keep on getting the same answer; past some large size however the answers would start to change, can you see why?

 

[Jimster41]

Viewing dimensionless groups as eigenvalues works for scaling up and down systems; like in the geometric case with triangles, squares and so on in the plane, all vectors in the plane are then eigenvectors, making the entire concept somewhat redundant.

Far more interesting, at least in my opinion, is if the geometric analogy can tell us why the dimensionless groups when varied past a certain number suddenly leads to qualitative changes in the object in question, e.g. as in the case of a large $\mathrm {Re}$ leading to turbulence.

In other words, is there a way of scaling up triangles which suddenly leads to the scaled up triangle suddenly changing its properties? The answer is yes. For illustrative purposes, just imagine drawing a triangle on the floor, measuring its internal angles and then summing them up, which of course is a constant.

Then imagine scaling up the triangle on the floor in size; rinse and repeat, with each time measuring and summing of angles. If you would actually carry this out, you would keep on getting the same answer; past some large size however the answers would start to change, can you see why?

curvature of the earth? and if you did it in outer space - the curvature of that...? But that wouldn't be due to the size of the triangle per-se' only the embedding of it's plane in a - not plane. If that's the implication you are alluding to, the curious effect running these parameters has - i.e. they may hold clues about the geometry we flatlanders can't access, well them I am with you...

I need to learn more about the dimensionless-group. All my questions regarding it are naive. Got any recommendations for a primer?

 

[Jimster41]

yeah, the network stuff is great. This wiki was helpful in making the connection to physics.
it's also a nice connection (at the intuitive level, for me anyway) to Nowak's stuff on evolutionary dynamics.

https://en.wikipedia.org/wiki/Bose–...etwork_theory)#Connection_with_network_theory

 

[Klystron]

Viewing dimensionless groups as eigenvalues works for scaling up and down systems; like in the geometric case with triangles, squares and so on in the plane, all vectors in the plane are then eigenvectors, making the entire concept somewhat redundant.

Far more interesting, at least in my opinion, is if the geometric analogy can tell us why the dimensionless groups when varied past a certain number suddenly leads to qualitative changes in the object in question, e.g. as in the case of a large $\mathrm {Re}$ leading to turbulence. ...[snip]...

[edited for brevity]

Does this statement agree with the example?
"A large change in Reynolds number indicates transition between laminar and turbulent flow" in fluids.

For instance, "transition between subsonic and supersonic air flow is measured by change in Mach number" avoids the invalid statement "Mach number causes a sonic transition". Perhaps I consider dimension-less numbers as passive indicators, measurements of conditions; not causes?

 

[Jimster41]

Does this statement agree with the example?
"A large change in Reynolds number indicates transition between laminar and turbulent flow" in fluids.

For instance, "transition between subsonic and supersonic air flow is measured by change in Mach number" avoids the invalid statement "Mach number causes a sonic transition". Perhaps I consider dimension-less numbers as passive indicators, measurements of conditions; not causes?

I find that clear. The difference between correlation and cause is always tricky. If correlation is strong I think I would argue they are passive indicators of something active (causal) as yet unidentified. You could also say as I think Hofstadter, Smolin and Unger, Dawkins and other do, that no formal system no matter how accurate can be equated, ultimately, with cause... there is always the lurking problem of infinite regression... i.e. what causes that? But they, the formal systems and "passive" or worse yet "contrived" indicators are all we have.

 

[Jimster41]

To try and flesh out @Auto-Didact's point a bit, after reading more on just what dimensionless numbers are... and assuming I understood the triangle example which was cool... dimensionless terms are odd because there is no obvious intrinsic degree of freedom in them to describe the way they map a single domain to multiple (or what appear to be very different) ranges. There must be something but it's only visible in their movement across the domain - critical points. @Auto-Didact's point, I think, is that there might be, must be, some ability to step up a level (get meta about them) to try and understand, perhaps, the way different non-linear systems manifest these kind of scary "tuning" knobs - maybe based on aspects of their structure. I don't know I'm trying to peer into the fog a bit... but it is intriguing. The reason I mentioned using Ai to characterize such weird geometrical systems is that - it will probably take a much wider, deeper, more robust perception to explore such a jungle of wild maps.

I get a lot of stuff on material science in my various feeds. It strikes me as a situation where exactly what he's alluding to is happening inch by inch. It's in pursuit of practical knowledge (like high temperature superconductors) and it's all done firmly in the context of a stable and trusted formal system (physical chemistry) but it strikes me they are really cataloging in a semi-brute force (computer simulation aided way) a vast and wild zoo of non-linear dynamics. And at least to my eye they are discovering suprising bizarre stuff on a daily basis.

 

[Auto-Didact]

curvature of the earth? and if you did it in outer space - the curvature of that...?

Exactly. If you are unfamiliar with this, this is how Gauss figured out how the intrinsic curvutare of a space can be measured; suffice to say the curvature of spacetime in GR is based on this.

Moreover, there is actually an experiment being planned, called (E)LISA, which could directly measure the curvature of spacetime in this manner, by making an enormous triangle using lasers and directly measuring and summing the angles.

But that wouldn't be due to the size of the triangle per-se' only the embedding of it's plane in a - not plane.

Indefinitely expanding the size of a triangle on a flat surface, such as the floor in your apartment, to arbitrary large sizes means you are working with a tangent space to the surface of the Earth; this tangent space is clearly not physically real in any meaningful sense.

On the other hand, the meaning of the state space and the parameter space are absolutely clear; there is no reason whatsoever to assume that these spaces and the objects residing within them do not contain similar mathematical richness as physical space contains.

If that's the implication you are alluding to, the curious effect running these parameters has - i.e. they may hold clues about the geometry we flatlanders can't access, well them I am with you...

I was just making the point that clearly the geometry, not only of the formal system and its ambient space, but also the geometry of the state space, the parameter space and so on, is absolutely paramount.

There actually often is a curved surface in a higher dimensional generalization of parameter space, neither of which have own proper name as far as I know; in any case, cusps of this curved surface are the subject of catastrophe theory.

I need to learn more about the dimensionless-group. All my questions regarding it are naive. Got any recommendations for a primer?

Lin and Segel, 1988

Does this statement agree with the example?
"A large change in Reynolds number indicates transition between laminar and turbulent flow" in fluids.

For instance, "transition between subsonic and supersonic air flow is measured by change in Mach number" avoids the invalid statement "Mach number causes a sonic transition".

You are correct formally, but this is just because I'm not careful in my phrasing; I would say "some particular increase in Mach number means sonic transition".

But this is just arguing semantics in my opinion: practically speaking, given some tube length and some fluid, the cause of turbulence in the flow of this system is a high velocity to viscosity ratio; this is completely equivalent to saying "a large Reynolds number".

Perhaps I consider dimension-less numbers as passive indicators, measurements of conditions; not causes?

Yes, and this is problematic because these dimensionless indicators are anything but passive (or to use my terminology 'just tools'): they contain all active indicators and so capture everything relevant about the behavior of the system in a unique manner.

 

[Auto-Didact]

yeah, the network stuff is great. This wiki was helpful in making the connection to physics.
it's also a nice connection (at the intuitive level, for me anyway) to Nowak's stuff on evolutionary dynamics.

https://en.wikipedia.org/wiki/Bose–...etwork_theory)#Connection_with_network_theory

Very nice, I recently did some complex network research applied to economics; I was very surprised to see that some of the pictures on that page are indistinguishable from my own, implying Bose-Einstein condensation taking place in the economic system I was researching!

To try and flesh out @Auto-Didact's point a bit, after reading more on just what dimensionless numbers are... and assuming I understood the triangle example which was cool... dimensionless terms are odd because there is no obvious intrinsic degree of freedom in them to describe the way they map a single domain to multiple (or what appear to be very different) ranges. There must be something but it's only visible in their movement across the domain - critical points. @Auto-Didact's point, I think, is that there might be, must be, some ability to step up a level (get meta about them) to try and understand, perhaps, the way different non-linear systems manifest these kind of scary "tuning" knobs - maybe based on aspects of their structure. I don't know I'm trying to peer into the fog a bit... but it is intriguing.

Yes. I should mention that nothing what I am saying here is controversial in any sense; there have been multiple books and textbooks written about what I am describing here, starting around the 80s. This is essentially the subject matter of the highly interconnected fields of nonlinear differential equations, network science, statistical mechanics, chaos theory, bifurcation theory, catastrophe theory, topology, dynamical systems, complexity science and so on; it is currently a highly rapidly evolving field of science but as we can see it doesn't have a single clear moniker apart from 'nonlinear dynamics' which seems to be too vague for most people.

The only thing that is peculiar is that most theoretical physicists seem to be unaware of these developments, while condensed matter theorists seem to think that they invented these subjects. In actuality however, it is clearly the engineers, applied mathematicians and other scientists who have been doing most of the heavy work w.r.t. these fields for the past two centuries by studying fluid and continuum mechanics in depth as well as other fields of classical physics and science more broadly.

The reason I mentioned using Ai to characterize such weird geometrical systems is that - it will probably take a much wider, deeper, more robust perception to explore such a jungle of wild maps.

AI isn't even necessary until things become extremely complicated; everything I have described so far and much more was done without AI.

I get a lot of stuff on material science in my various feeds. It strikes me as a situation where exactly what he's alluding to is happening inch by inch. It's in pursuit of practical knowledge (like high temperature superconductors) and it's all done firmly in the context of a stable and trusted formal system (physical chemistry) but it strikes me they are really cataloging in a semi-brute force (computer simulation aided way) a vast and wild zoo of non-linear dynamics. And at least to my eye they are discovering suprising bizarre stuff on a daily basis.

Indeed, among many others, Strogatz in his books and textbook clearly describes this. I actually cannot even think of a single research route in any science, business or practice which cannot benefit directly from utilizing the tools and fruits of nonlinear dynamics as part of their methodology.

Utilizing these advanced mathematical methods for doing research, can be far more productive but is also far more complicated than utilizing simple statistics - especially for non-physicist scientists - practically making interdisciplinary research with mathematicians (or perhaps AI) mandatory; this can be seen as the only 'drawback'.