Is there a single book or resource you would recommend for learning how group theory/symmetry can be used to develop theories and models?
I work in fluid dynamics, and I’ve mainly seen group theory/symmetry mentioned when forming simplifying coordinate transformations. Fluid dynamicists call these “dimensionless parameters” or “similarity variables”. I am certain other fields use different terminology.
See my response below to WhySpace on getting started with group theory through category theory. For any space-oriented field, I also recommend looking at the topological definition of a space. Also, for any calculus-heavy field, I recommend meditating on the Method of Lagrange Multipliers if you don’t already have a visual grasp of it.
I don’t know of any resource that tackles the problem of developing models via group theory. Developing models is a problem of stating and applying analogies, which is a problem in category theory. If you want to understand that better, you can look through the various classifications of functors since the notion of a functor translates pretty accurately to “analogy”.
I have no background in fluid dynamics, so please filter everything I say here through your own understanding, and please correct me if I’m wrong somewhere.
I don’t think there’s any inherent relationship between dimensionless parameters and group theory. The reason being that dimensionless quantities can refer to too many things (i.e., they’re not really dimensionless, and different dimensionlessnesses have different properties… or rather they may be dimensionless, but they’re not typeless). Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable. I suppose if you can rewrite an equation in terms of dimensionless quantities whose relationships are restricted to have certain properties, then you can treat them like other well-known objects, and you can throw way more math at them.
For example, suppose your “dimensionless” quantity is a scaling parameter such that scale * scale → scale (the product of two scaling operations is equivalent to a single scaling operation). By converting your values to scales, you’ve gained a new operation to work with due to not having to re-translate your quantities on each successive multiplication: element-wise exponentiation. I’d personally see that as a gateway to applying generating series (because who doesn’t love generating series?), but I guess a more mechanics-y application of that would be solving differential equations, which often require exponentiating things.
Any time you have a set of X quantities that can be applied to one another to get another of the X quantities, you have a group of some sort (with some exceptions). That’s what’s going on with the scaling example (x * x → x), and that’s what’s not going on with the !∘sqrt∘ln example. The scaling example just happens to be a particularly simple example of a group. You get less trivial examples when you have multiple “dimensionless” quantities that can interact with one another in standard ways. For example, if vector addition, scaling, and dot products are sensible, your vectors can form a Hilbert space, and you can use wonderful things like angles and vector calculus to meaningful effect.
I can probably give a better answer if I know more precisely what you’re referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Thanks for the detailed reply, sen. I don’t follow everything you said, but I’ll take a look at your recommendations and see after that.
I can probably give a better answer if I know more precisely what you’re referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Unfortunately, the subject is rather disjoint. Most fluid dynamicists would have no idea that group theory is relevant. My impression is that some mathematicians have interpreted what fluid dynamicists have done for a long time in terms of group theory, and extended their methods. Fluid dynamicists call the approach “dimensional analysis” if you reduce the number of input parameters or “similarity analysis” if you reduce the number independent variables of a differential equation (more on the latter later)
The goal generally is dimension reduction. For example, if you are to perform a simple factorial experiment with 3 variables and you want to sample 8 different values of each variable, you have 8^3 = 512 samples to make, and that’s not even considering doing multiple trials. But, if you can determine a coordinate transformation which reduces those 3 variables to 1, then you only have 8 samples to make.
The Buckingham Pi theorem allows you to determine how many dimensionless variables are needed to fully specify the problem if you start with dimensional quantities. (If everything is dimensionless to begin with, there’s no benefit from this technique, but other techniques might have benefit.)
For a long list of examples of the dimensionless quantities, see Wikipedia. The Reynolds number is the most well known of these. (Also, contrary to common understanding, the Reynolds number doesn’t really say anything about “how turbulent” a flow is, rather, it would be better thought of as a way to characterize instability of a flow. There are multiple ways to measure “how turbulent” a flow is.)
For a “similarity variable”, I’m not sure what the best place to point them out would be. Here’s one example, though: If you take the 1D unbounded heat equation and change coordinates to \eta = x / \sqrt{\alpha t} (\alpha is the thermal diffusivity), you’ll find the PDE is reduced to an ODE, and solution should be much easier now. The derivation of the reduction to an ODE is not on Wikipedia, but it is very straightforward.
Dimensional analysis is really only taught to engineers working on fluid mechanics and heat transfer. I am continually surprised by how few people are aware of it. It should be part of the undergraduate curriculum for any degree in physics. Statisticians, particularly those who work in experimental design, also should know it. Here’s an interesting video of a talk with an application of dimensional analysis to experimental design. As I recall, one of the questions asked after the talk related the approach to Lie groups.
For an engineering viewpoint, I’d recommend Langhaar’s book. This book does not discuss similarity variables, however. For something bridging the more mathematical and engineering viewpoints I have one recommendation. I haven’t looked at this book, but it’s one of the few I could find which discusses both the Buckingham Pi theorem and Lie groups. For something purely on the group theory side, see Olver’s book.
Anyhow, I asked about this because I get the impression from some physicists that there’s more to applications of group theory to building models than what I’ve seen.
Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable.
This is an important realization. The Buckingham Pi theorem doesn’t tell you which dimensionless variables are “valid” or “useful”, just the number of them needed to fully specify the problem. Whether or not a dimensionless number is “valid” or “useful” depends on what you are interested in.
Regarding the Buckingham Pi Theorem (BPT), I think I can double my recommendation that you try to understand the Method of Lagrange Multipliers (MLM) visually. I’ll try to explain in the following paragraph knowing that it won’t make much sense on first reading.
For the Method of Lagrange Multipliers, suppose you have some number of equations in n variables. Consider the n-dimensional space containing the set of all solutions to those equations. The set of solutions describes a k-dimensional manifold (meaning the surface of the manifold forms a k-dimensional space), where k depends on the number of independent equations you have. The set of all points perpendicular to this manifold (the null space, or the space of points that, projected onto the manifold, give the zero vector) can be described by an (n-k)-dimensional space. Any (n-k)-dimensional space can be generated (by vector scaling and vector addition) of (n-k) independent vectors. For the Buckingham Pi Theorem, replace each vector with a matrix/group, vector scaling with exponentiation, and vector addition with multiplication. Your Buckingham Pi exponents are Lagrange multipliers, and your Pi groups are Lagrange perpendicular vectors (the gradient/normal vectors of your constraints/dimensions).
I guess in that sense, I can see why people would make the jump to Lie groups. The Pi Groups / basis vectors form the generator of any other vector in that dimensionless space, and they’re obviously invertible. Honestly, I haven’t spent much time with Lie Groups and Lie Algebra, so I can’t tell you why they’re useful. If my earlier explanation of dimensionless quantities holds (which, after seeing the Buckingham Pi Theorem, I’m even more convinced that it does), then it has something to do with symmetry with respect to scale, The reason I say “scale” as opposed to any other x * x → x quantity is that the scale kind of dimensionlessness seems to pop up in a lot of dimensionless quantities specific to fluid dynamics, including Reynold’s Number.
Sorry, I know that didn’t make much sense. I’m pretty sure it will though once you go through the recommendations in my earlier reply.
Regarding Reynold’s Number, I suspect you’re not going to see the difference between the dimensional and the dimensionless quantities until you try solving that differential equation at the bottom of the page. Try it both with and without converting to dimensionless quantities, and make sure to keep track of the semantics of each term as you go through the process. Here’s one that’s worked out for the dimensionless case. If you try solving it for the non-dimensionless case, you should see the problem.
It’s getting really late. I’ll go through your comments on similarity variables in a later reply.
Thanks for the references and your comments. I’ve learned a lot from this discussion.
Could you guys cooperate or something and write an intro Discussion or Main post on this for landlubbers? Pretty please?
I have glanced at a very brief introductory article on dim.an. in regards to Reynold’s number when I wondered whether I could model dissemination of fern’s spores within a ribbon-shaped population, or just simply read about such model, but it all seemed like so much trouble. And even worse, I had a weird feeling like ‘oh this has to be so noisy, how do they even know how the errors are combined in these new parameters? Surely they don’t just sum.’
(Um, a datapoint from a non-mathy person, I think I’m not alone in this.)
Sure, I’d be interested in writing an article on dimensional analysis and scaling in general. I might have time over my winter break. It’s also worth noting that I posted on dimensional analysis before. Dimensional analysis is not as popular as principal components analysis, despite being much easier, and I think this is unfortunate.
I don’t know what a “ribbon-shaped population” is, but I imagine that fern spores are blown off by wind and then dispersed by a combination of wind and turbulence. Turbulent dispersion of particles is essentially an entire field by itself. I have some experience in it from modeling water droplet trajectories for fire suppression, so I might be able to help you more, assuming I understand your problem correctly. Feel free to send me a message on here if you’d like help.
And even worse, I had a weird feeling like ‘oh this has to be so noisy, how do they even know how the errors are combined in these new parameters? Surely they don’t just sum.’
Could you explain this a little more? I’m not exactly following.
Because dimensional homogeneity is a requirement for physical models, any series of independent dimensionless variables you construct should be “correct” in a strict sense, but they are not unique, and consequently you might not naively pick “useful” variables. If this doesn’t make sense, then I could explain in more detail or differently.
Yes, I remember that post. It was ‘almost interesting’ to me, because it is beyond my actual knowledge. So, if you could just maybe make it less scary, we landlubbers would love you to bits. If you’d like.
I agree about the wind and the turbulence, which is somewhat “dampered” by the prolonged period of spore dissemination and the possibility (I don’t know how real) of re-dissemination of the ones that “didn’t stick” the first time. The thing I am (was) most interested in—how fertilization occurs in the new organisms growing from the spores—is further complicated by the motility of sperm and the relatively big window of opportunity (probably several seasons)… so I am not sure if modeling the dissemination has any value, but still. This part is at least above-ground. It’s really an example of looking for your keys under a lamplight.
re: errors. I mean that it seemed to me (probably wrongly) that if you measure a bunch of variables, and try to make a model from them, then realise you only want a few and the others can be screwed together into a dimensionless ‘thing’, then how do you know the, well, ‘bounds of correctness’ of the dimensionless thing? It was built from imperfect measurements that carried errors in them; where do the errors go when you combine variables into something new? (I mean, it is a silly question, but i haz it.)
(‘ribbon-shaped population’ was my clumsy way of describing a long and narrow, but relatively uninterrupted population of plants that stretches along a certain landscape feature, like a beach. I can’t recall the real word right now.)
Yes, I remember that post. It was ‘almost interesting’ to me, because it is beyond my actual knowledge. So, if you could just maybe make it less scary, we landlubbers would love you to bits. If you’d like.
If you don’t mind, could you highlight which parts you thought were too difficult?
Aside from adding more details, examples, and illustrations, I’m not sure what I could change. I will have to think about this more.
re: errors. I mean that it seemed to me (probably wrongly) that if you measure a bunch of variables, and try to make a model from them, then realise you only want a few and the others can be screwed together into a dimensionless ‘thing’, then how do you know the, well, ‘bounds of correctness’ of the dimensionless thing? It was built from imperfect measurements that carried errors in them; where do the errors go when you combine variables into something new? (I mean, it is a silly question, but i haz it.)
This is an important question to ask. After non-dimensionalizing the data and plotting it, if there aren’t large gaps in the coverage of any dimensionless independent variable, then you can just use the ranges of the dimensionless independent variables.
I could add some plots showing this more obviously in a discussion post.
Here are some example correlations from heat transfer. Engineers did heat transfer experiments in pipes and measured the heat flux as a function of different velocities. They then converted heat flux into the Nusselt number and the velocity/pipe diameter/viscosity into the Reynolds number, and had another term called the Prandtl number. There are plots of these experiments in the literature and you can see where the data for the correlation starts and ends. As you do not always have a clear idea of what happens outside the data (unless you have a theory), this usually is where the limits come from.
Is there a single book or resource you would recommend for learning how group theory/symmetry can be used to develop theories and models?
I work in fluid dynamics, and I’ve mainly seen group theory/symmetry mentioned when forming simplifying coordinate transformations. Fluid dynamicists call these “dimensionless parameters” or “similarity variables”. I am certain other fields use different terminology.
See my response below to WhySpace on getting started with group theory through category theory. For any space-oriented field, I also recommend looking at the topological definition of a space. Also, for any calculus-heavy field, I recommend meditating on the Method of Lagrange Multipliers if you don’t already have a visual grasp of it.
I don’t know of any resource that tackles the problem of developing models via group theory. Developing models is a problem of stating and applying analogies, which is a problem in category theory. If you want to understand that better, you can look through the various classifications of functors since the notion of a functor translates pretty accurately to “analogy”.
I have no background in fluid dynamics, so please filter everything I say here through your own understanding, and please correct me if I’m wrong somewhere.
I don’t think there’s any inherent relationship between dimensionless parameters and group theory. The reason being that dimensionless quantities can refer to too many things (i.e., they’re not really dimensionless, and different dimensionlessnesses have different properties… or rather they may be dimensionless, but they’re not typeless). Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable. I suppose if you can rewrite an equation in terms of dimensionless quantities whose relationships are restricted to have certain properties, then you can treat them like other well-known objects, and you can throw way more math at them.
For example, suppose your “dimensionless” quantity is a scaling parameter such that scale * scale → scale (the product of two scaling operations is equivalent to a single scaling operation). By converting your values to scales, you’ve gained a new operation to work with due to not having to re-translate your quantities on each successive multiplication: element-wise exponentiation. I’d personally see that as a gateway to applying generating series (because who doesn’t love generating series?), but I guess a more mechanics-y application of that would be solving differential equations, which often require exponentiating things.
Any time you have a set of X quantities that can be applied to one another to get another of the X quantities, you have a group of some sort (with some exceptions). That’s what’s going on with the scaling example (x * x → x), and that’s what’s not going on with the !∘sqrt∘ln example. The scaling example just happens to be a particularly simple example of a group. You get less trivial examples when you have multiple “dimensionless” quantities that can interact with one another in standard ways. For example, if vector addition, scaling, and dot products are sensible, your vectors can form a Hilbert space, and you can use wonderful things like angles and vector calculus to meaningful effect.
I can probably give a better answer if I know more precisely what you’re referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Thanks for the detailed reply, sen. I don’t follow everything you said, but I’ll take a look at your recommendations and see after that.
Unfortunately, the subject is rather disjoint. Most fluid dynamicists would have no idea that group theory is relevant. My impression is that some mathematicians have interpreted what fluid dynamicists have done for a long time in terms of group theory, and extended their methods. Fluid dynamicists call the approach “dimensional analysis” if you reduce the number of input parameters or “similarity analysis” if you reduce the number independent variables of a differential equation (more on the latter later)
The goal generally is dimension reduction. For example, if you are to perform a simple factorial experiment with 3 variables and you want to sample 8 different values of each variable, you have 8^3 = 512 samples to make, and that’s not even considering doing multiple trials. But, if you can determine a coordinate transformation which reduces those 3 variables to 1, then you only have 8 samples to make.
The Buckingham Pi theorem allows you to determine how many dimensionless variables are needed to fully specify the problem if you start with dimensional quantities. (If everything is dimensionless to begin with, there’s no benefit from this technique, but other techniques might have benefit.)
For a long list of examples of the dimensionless quantities, see Wikipedia. The Reynolds number is the most well known of these. (Also, contrary to common understanding, the Reynolds number doesn’t really say anything about “how turbulent” a flow is, rather, it would be better thought of as a way to characterize instability of a flow. There are multiple ways to measure “how turbulent” a flow is.)
For a “similarity variable”, I’m not sure what the best place to point them out would be. Here’s one example, though: If you take the 1D unbounded heat equation and change coordinates to \eta = x / \sqrt{\alpha t} (\alpha is the thermal diffusivity), you’ll find the PDE is reduced to an ODE, and solution should be much easier now. The derivation of the reduction to an ODE is not on Wikipedia, but it is very straightforward.
Dimensional analysis is really only taught to engineers working on fluid mechanics and heat transfer. I am continually surprised by how few people are aware of it. It should be part of the undergraduate curriculum for any degree in physics. Statisticians, particularly those who work in experimental design, also should know it. Here’s an interesting video of a talk with an application of dimensional analysis to experimental design. As I recall, one of the questions asked after the talk related the approach to Lie groups.
For an engineering viewpoint, I’d recommend Langhaar’s book. This book does not discuss similarity variables, however. For something bridging the more mathematical and engineering viewpoints I have one recommendation. I haven’t looked at this book, but it’s one of the few I could find which discusses both the Buckingham Pi theorem and Lie groups. For something purely on the group theory side, see Olver’s book.
Anyhow, I asked about this because I get the impression from some physicists that there’s more to applications of group theory to building models than what I’ve seen.
This is an important realization. The Buckingham Pi theorem doesn’t tell you which dimensionless variables are “valid” or “useful”, just the number of them needed to fully specify the problem. Whether or not a dimensionless number is “valid” or “useful” depends on what you are interested in.
Edit: Fixed some typos.
Regarding the Buckingham Pi Theorem (BPT), I think I can double my recommendation that you try to understand the Method of Lagrange Multipliers (MLM) visually. I’ll try to explain in the following paragraph knowing that it won’t make much sense on first reading.
For the Method of Lagrange Multipliers, suppose you have some number of equations in n variables. Consider the n-dimensional space containing the set of all solutions to those equations. The set of solutions describes a k-dimensional manifold (meaning the surface of the manifold forms a k-dimensional space), where k depends on the number of independent equations you have. The set of all points perpendicular to this manifold (the null space, or the space of points that, projected onto the manifold, give the zero vector) can be described by an (n-k)-dimensional space. Any (n-k)-dimensional space can be generated (by vector scaling and vector addition) of (n-k) independent vectors. For the Buckingham Pi Theorem, replace each vector with a matrix/group, vector scaling with exponentiation, and vector addition with multiplication. Your Buckingham Pi exponents are Lagrange multipliers, and your Pi groups are Lagrange perpendicular vectors (the gradient/normal vectors of your constraints/dimensions).
I guess in that sense, I can see why people would make the jump to Lie groups. The Pi Groups / basis vectors form the generator of any other vector in that dimensionless space, and they’re obviously invertible. Honestly, I haven’t spent much time with Lie Groups and Lie Algebra, so I can’t tell you why they’re useful. If my earlier explanation of dimensionless quantities holds (which, after seeing the Buckingham Pi Theorem, I’m even more convinced that it does), then it has something to do with symmetry with respect to scale, The reason I say “scale” as opposed to any other x * x → x quantity is that the scale kind of dimensionlessness seems to pop up in a lot of dimensionless quantities specific to fluid dynamics, including Reynold’s Number.
Sorry, I know that didn’t make much sense. I’m pretty sure it will though once you go through the recommendations in my earlier reply.
Regarding Reynold’s Number, I suspect you’re not going to see the difference between the dimensional and the dimensionless quantities until you try solving that differential equation at the bottom of the page. Try it both with and without converting to dimensionless quantities, and make sure to keep track of the semantics of each term as you go through the process. Here’s one that’s worked out for the dimensionless case. If you try solving it for the non-dimensionless case, you should see the problem.
It’s getting really late. I’ll go through your comments on similarity variables in a later reply.
Thanks for the references and your comments. I’ve learned a lot from this discussion.
Glad to help. I’ll go through your recommendations later this month when I have more time.
Could you guys cooperate or something and write an intro Discussion or Main post on this for landlubbers? Pretty please?
I have glanced at a very brief introductory article on dim.an. in regards to Reynold’s number when I wondered whether I could model dissemination of fern’s spores within a ribbon-shaped population, or just simply read about such model, but it all seemed like so much trouble. And even worse, I had a weird feeling like ‘oh this has to be so noisy, how do they even know how the errors are combined in these new parameters? Surely they don’t just sum.’
(Um, a datapoint from a non-mathy person, I think I’m not alone in this.)
Sure, I’d be interested in writing an article on dimensional analysis and scaling in general. I might have time over my winter break. It’s also worth noting that I posted on dimensional analysis before. Dimensional analysis is not as popular as principal components analysis, despite being much easier, and I think this is unfortunate.
I don’t know what a “ribbon-shaped population” is, but I imagine that fern spores are blown off by wind and then dispersed by a combination of wind and turbulence. Turbulent dispersion of particles is essentially an entire field by itself. I have some experience in it from modeling water droplet trajectories for fire suppression, so I might be able to help you more, assuming I understand your problem correctly. Feel free to send me a message on here if you’d like help.
Could you explain this a little more? I’m not exactly following.
Because dimensional homogeneity is a requirement for physical models, any series of independent dimensionless variables you construct should be “correct” in a strict sense, but they are not unique, and consequently you might not naively pick “useful” variables. If this doesn’t make sense, then I could explain in more detail or differently.
Yes, I remember that post. It was ‘almost interesting’ to me, because it is beyond my actual knowledge. So, if you could just maybe make it less scary, we landlubbers would love you to bits. If you’d like.
I agree about the wind and the turbulence, which is somewhat “dampered” by the prolonged period of spore dissemination and the possibility (I don’t know how real) of re-dissemination of the ones that “didn’t stick” the first time. The thing I am (was) most interested in—how fertilization occurs in the new organisms growing from the spores—is further complicated by the motility of sperm and the relatively big window of opportunity (probably several seasons)… so I am not sure if modeling the dissemination has any value, but still. This part is at least above-ground. It’s really an example of looking for your keys under a lamplight.
re: errors. I mean that it seemed to me (probably wrongly) that if you measure a bunch of variables, and try to make a model from them, then realise you only want a few and the others can be screwed together into a dimensionless ‘thing’, then how do you know the, well, ‘bounds of correctness’ of the dimensionless thing? It was built from imperfect measurements that carried errors in them; where do the errors go when you combine variables into something new? (I mean, it is a silly question, but i haz it.)
(‘ribbon-shaped population’ was my clumsy way of describing a long and narrow, but relatively uninterrupted population of plants that stretches along a certain landscape feature, like a beach. I can’t recall the real word right now.)
Romashka, I appreciate the reply.
If you don’t mind, could you highlight which parts you thought were too difficult?
Aside from adding more details, examples, and illustrations, I’m not sure what I could change. I will have to think about this more.
This is an important question to ask. After non-dimensionalizing the data and plotting it, if there aren’t large gaps in the coverage of any dimensionless independent variable, then you can just use the ranges of the dimensionless independent variables.
I could add some plots showing this more obviously in a discussion post.
Here are some example correlations from heat transfer. Engineers did heat transfer experiments in pipes and measured the heat flux as a function of different velocities. They then converted heat flux into the Nusselt number and the velocity/pipe diameter/viscosity into the Reynolds number, and had another term called the Prandtl number. There are plots of these experiments in the literature and you can see where the data for the correlation starts and ends. As you do not always have a clear idea of what happens outside the data (unless you have a theory), this usually is where the limits come from.