Order parameters and symmetry breaking

[Wilmot McCutchen]

Symmetry breaking means order emerging from a state where everything is the same in all directions. That seems to be the opposite of what most would consider symmetry. Something symmetrical is not a blur, in the common understanding. Turbulence, for example, is considered perfectly symmetrical because there is no order anywhere, and the appearance of a vortex (which is axisymmetric) is called symmetry breaking.

 

This opposition of pedantic rigor to common understanding may be what is standing in the way of progress in extracting power from turbulence. Since the steam age, only a fraction (set by the Carnot efficiency) of the energy is considered usable for power generation. Maybe the Carnot limit is like the sound barrier.. Maybe the wasted energy ("entropy") in turbine exhaust steam can be a power source, with some intelligent design based on a clear understanding of order parameter fields. See, e.g. this device for waste heat power harvesting from turbine exhaust steam: http://www.freepatentsonline.com/7987677.pdf

 

For a tutorial on symmetry breaking and topology by James P. Sethna. see http://www.lassp.cornell.edu/sethna/OrderParameters/Intro.html

[MigL]

Your very first statement and others are incorrect.

There are many kinds of symmetries, and your understanding of entropy is also incomplete.

I haven't read your link, but you should probably re-read it.

Edited December 29, 2012 by MigL

[MigL]

If that doesn't help a very good introductory text, ie no math, is Zee's 'Fearful Symmetry'.

Edited December 30, 2012 by MigL

[Wilmot McCutchen]

I just finished A.Zee's Fearful Symmetry per your recommendation, for which, many thanks. Wonderful read. He doesn't say much about symmetry breaking, The heuristic value of symmetry considerations is memorably illustrated in the evolution of particle physics.

 

Sethna says: "To them [condensed matter physicists], the fundamental question is not discovering the underlying quantum mechanical laws, but in understanding and explaining the new laws that emerge when many particles interact." "Usually, the material has lowest energy when the order parameter field is uniform, when the symmetry is broken in the same way throughout space. In practise, though, the material often doesn't break symmetry uniformly. Most pieces of iron don't appear magnetic, simply because the local magnetization points in different directions at different places. The magnetization is already there at the atomic level: to make a magnet, you pound the different domains until they line up."

 

"When the symmetry is broken in the same way throughout space" -- that sounds like symmetry breaking is the appearance of a path of least resistance to energy flux out of the material. Order, not disorder.

 

Zee also discusses magnetic elements aligning:."pretty soon, the zillions of arrows all end up pointing in the same direction. The rotational symmetry inherent in the underlying physics has been broken spontaneously." He mentions the case where symmetry breaking is forced "by hand."

 

So both Zee and Sethna agree that symmetry breaking is order from disorder. So symmetry is the disorder ...?

[altergnostic]

Hi. Order and disorder are not physical properties, but personal taste. You can say that 123123123 is ordered just like 111222333. Which one has the symmetry broken? Is polar magnetization more ordered than null magnetization? These are very abstract ideas, as you point out. You can call the same configuration ordered or disordered, depending on your definition of order.

Using order and disorder as the theoretical basis for new forms of energy production sounds odd. Would you rather start working on a hypothetical solution to use the energy produced in some physical process by studying order and disorder or by studying the mechanics involved? Every physical process transforms energy. It is just a matter of studying ways to tap into such potential energy sources. I see no need for overly abstract ideas.

[imatfaal]

Hi. Order and disorder are not physical properties, but personal taste. You can say that 123123123 is ordered just like 111222333. Which one has the symmetry broken? Is polar magnetization more ordered than null magnetization? These are very abstract ideas, as you point out. You can call the same configuration ordered or disordered, depending on your definition of order.

Using order and disorder as the theoretical basis for new forms of energy production sounds odd. Would you rather start working on a hypothetical solution to use the energy produced in some physical process by studying order and disorder or by studying the mechanics involved? Every physical process transforms energy. It is just a matter of studying ways to tap into such potential energy sources. I see no need for overly abstract ideas.

 

But surely you could say that there are only two ways in which a group of binary switches show no variation (0000... and 1111...) but many ways in which some are 1 and some are 0. These are not abstract or subjective some descriptions have many arrangements that are similar, other have few. There are 2 ways in which all the switches are the same, there are 2n ways that one is different.... this is not subjective

[altergnostic]

 

But surely you could say that there are only two ways in which a group of binary switches show no variation (0000... and 1111...) but many ways in which some are 1 and some are 0. These are not abstract or subjective some descriptions have many arrangements that are similar, other have few. There are 2 ways in which all the switches are the same, there are 2n ways that one is different.... this is not subjective

Deal. But still, I don't think that's the kind of symmetry that is being related to energy production here. A vortex out of an exhaust does not show that kind of objective order-disorder relation, right?

Actually, no deal. What is the physical difference between <<<<< and <<<<>? The fact that five arrows in vaccum point to the left and that four point to the left and one to the right doesn't physically change anything. This is still a subjective impression, and reality couldn't care less. The switches example are different because their positions are implicitly assigned to different physical processes (like on and off current flows) and 4 electric currents is physically very different than 5 electric currents. But the positions of the switches themselves do not matter. For instance, if one of the five switches is upside down, 5 ups = 4 ons and 1 off, while 4 ups and 1 down = 5 ons. Which one is more symmetric or more ordered? You see what I mean? It is the physical processes that matter, not our judgement of order and disorder. 5 switches = 5 switches, their relative* positions or orientations by themselves have no effect on physical quantities.

 

*this should ring a bell.

[Wilmot McCutchen]

Consider the tornado. A strong axial jet -- linear momentum capable of doing work -- when the tornado touches down and streamlines converge on the axis of symmetry as swirl collapses.

 

Angular momentum converting to linear momentum is the opposite of the dissipative conception of turbulence. So what would you call that? Convergence or collapse seems a better word, as it connotes the optimal energy flux as streamlines converge. "Symmetry breaking".doesn't say enough, in my opinion, and I would suppose most non-experts would agree with me that symmetry is what to create. Why break what you should make? Why is "turbulence only considered from the dissipative perspective. Why not the opposite, for waste heat power generation? Overcoming viscous diffusion by an input of organizing energy to collapse turbulence.

 

Shtern and Hussain, http://www2.egr.uh.edu/~ifdt/archivalpapers/arfm/ShternHussain1999.pdf

"Collapse, Symmetry Breaking, and Hysteresis of Swirling Flows" (1999):

"...there are examples in everyday life of swirl development without any obvious forcing, e.g. the whirlpool in a bath sink. Whether the bathtub vortex occurs due to symmetry breaking or external forcing has been widely discussed but not resolved." .

[imatfaal]

Deal. But still, I don't think that's the kind of symmetry that is being related to energy production here. A vortex out of an exhaust does not show that kind of objective order-disorder relation, right?

Actually, no deal. What is the physical difference between <<<<< and <<<<>? The fact that five arrows in vaccum point to the left and that four point to the left and one to the right doesn't physically change anything. This is still a subjective impression, and reality couldn't care less. The switches example are different because their positions are implicitly assigned to different physical processes (like on and off current flows) and 4 electric currents is physically very different than 5 electric currents. But the positions of the switches themselves do not matter. For instance, if one of the five switches is upside down, 5 ups = 4 ons and 1 off, while 4 ups and 1 down = 5 ons. Which one is more symmetric or more ordered? You see what I mean? It is the physical processes that matter, not our judgement of order and disorder. 5 switches = 5 switches, their relative* positions or orientations by themselves have no effect on physical quantities.

 

*this should ring a bell.

 

 

It matters because the log of the number of possible microstates is the entropy - and temperature is inversely proportional to delta entropy over delta energy. These are measureable quantities and not at all subjective

[MigL]

As an example of symmetry breaking consider a round table setting, with plate, silverware and a glass between each of the plates. The table and its setting is symmetric. Now when guests arrive, sit, and one of them chooses a glass, either to the right or the left of their plate ( manners be damned ), they are effectively breaking symmetry and everyone at the table is constrained to pick the same.

 

As for the example you give from Zee's book. In a low energy state the magnetic domains align in a particular direction. Energy ( heat ) has to be added to the system to sufficiently randomize the domain directions and destroy the magnetic properties. In effect, the higher energy system is more symmetric. This is useable energy, and I would argue that a better measure of entropy is that it increases as useable energy decreases, and is maximised when no useable energy is left.

Order and disorder are just labels we use and are not specific enough or quantifiable.

 

Another example would be water, which is very randomized as a liquid, but as it loses energy, it freezes into a lowe symmetry state. Whereas as a liquid it was the same in all directions, as a solid it is only symmetric at 60 deg. intervals ( snowflake ). In effect, as it froze, symmetry was spontaneously broken, and it dropped to a lower energy state.

 

The conclusion ?

Symmetry breaking always leads to a lower energy state.

Edited January 11, 2013 by MigL

[altergnostic]

My god...

 

The table example is classic, every time I read it I think it must be a joke. Of all the analogies to physical processes, this one is by far the most ridiculous. The fact that people used left or right glasses doesn't change any physical parameters, the number of glasses is the same, heat is the same, weight is the same, etc. Left and right, or direction, are purely relative.

And to state that water is more symmetrical than ice is nonsense. Water molecules move faster, but take a snapshot of either one and symmetry is purely visual, relative, and a matter of taste. Again, it is not symmetry that matters, it is the simple fact that water molecules move around more than in ice, hence, more energy. Adding energy, in this case, is adding heat and motion, not adding symmetry! You can't add symmetry to produce energy just like you can't add perspective to produce energy. The words don't even make sense. This is utterly incoherent. If my on and off switches were not enough, and this table exercise makes sense to you, I quit.

Edited January 14, 2013 by altergnostic

[MigL]

But the set table example shows that symmetry is not related to order or disorder, as it is commonly defined. That is what I was attempting to show. Does the table become more disordered or less disordered when the right or left glass is chosen ?

 

And yes, if you consider the true definition of symmetry ( and entropy ) liquid water is more symmetric than ice ( and has more useable, 'free' energy ).

Only consider quantifiable properties, not subjective concepts like order/disorder and your definition of symmetry ( above ).

There is a standard mathematical definition of symmetry where properties are classified into groups. You should look it up.

[altergnostic]

I see, so indeed the table example shows that symmetry is not necessarily related to order/dirsorder, I agree (although I don't see how this relates to energy production).

 

Now, water is more symmetric than ice, period? Shouldn't the configurarion matter? Ice may be frozen into a very symmetric finite group. And speaking of symmetry and groups, the mathematical (or geometrical, really) definition of symmetry, and the distinct groups in which systems can be divided into, are just as abstract as human taste. It is a grouping based on *preferred choice*. We decide what parameters are important. It is a theory of categorizarion, which aids greatly to simplify some problems, but unless the parameters are assigned to physical properties, we can't talk about energy production. Is a cube more energetic then a spiral? Is left less energetic than right? But when applied to a physical system, we must state exatcly what about a system is symmetric. It could be the shape, the energy potentials, charge distribution, surface patterns, weight distribution, motion patterns, etc. What about water is more symmetric than ice? Are you saying that water is absolutely more symmetric than ice, no matter what parameters we analyse, in whatever condition? Is the difference in symmetry between "hot" ice under great preassure and "cold" water in empty space related in any way to the potential energy production of each, or is it preassure and temperature that are directly related to energy?

 

My point is. Symmetry, in itself, is abstract math, not physics. It has mathematical value, but only by applying it to the right parameters will you be able to use the math in your energy production project, and if that's the case to begin with, you'd have to study the physical properties themselves before working with symmetry. In any case, symmetry is no energy source, nor is fundamentally related to it. Only the assigned physical properties can produce energy, and these can be arbitrarily chosen, allowing us to define it such that less symmetric systems become either less or more energetic than more symmetric ones.

Furthermore, two systems can have, for example, different spatial geometries and temperature distribution, either one of them being more or less symmetric than the other one depending on our choice of parameters, divorcing symmetry from entropy completely.