Saturday, December 26, 2009
spontaneous symmetry breaking=the Mexican hat potential?..Or.
Not a straight Hand-Job Rip-Off Cut-n-Paste but "oh so very close" type close
This is what happens when I simply try to read a Fucking A.A.Attanasio Blog
I mean really now over 500 pages of very advanced modular mathematics & some fair to middling hi-tech reference material only to finally understand that this only covers the smallest & most unimportant Sub, components, Hence the Neopartisan Non-Chaotic and somehow therefor Classicist version, of Applied physics as they relate(or in some particulars herein noted specifically, its failure to relate to) Theoretical physics, Both of which serve only to provide a frame work from which i am given to understand that any and all frameworks fail to bring one fully into the "fold" of understanding, An absolute(good pun huh?) requirement just so you can be certain that none of it applies to a certain specious "spiritual" framework. I used to feel that i was a "Spiritual Being" but upon further reflection Blah Blah Blah
So, it goes without saying that it goes something like this, In physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some symmetry group(and not just any or all) goes into a vacuum state that is not symmetric. When that happens, the system no longer appears to behave in a symmetric manner. It is a phenomenon that naturally occurs in many situations.
Here is part of the A.A. Attanasio Blog that started it all
..........................
.......................................................
---------------------------------------------------------------------------------
The Hawaiian story of Koko Crater actually begins 150 miles away on the Big Island. There, the lusty Pig God, Kamapua'a, assaulted the goddess of fire, Pele. The frenzied Pig God would have ravished his victim if not for the intervention of Pele's sister, the sorceress Kapo, who distracted Kamapua'a from his lewd advances by detaching her vagina and tossing it from the Big Island to O'ahu. The Pig God chased after it. The flying vagina slammed into the earth. And Kamapua'a, unable to stop quickly enough, crashed into the earth, gouging out Kalama Valley, now a suburban community –-- but, until 1975, a pig farm! Kapo retrieved her portable genitals, leaving behind this impression in the land, which is the tuff ring of Koko Crater.
I’ve retold this myth as “Sex, Love and the Mighty Fine Structure Constant,” which you can read here.
Who is the Goddess? Who dropped her vagina here? We are not mythic people, we who invented the printing press and the hydrogen bomb. In the 21st century, the Goddess is where She always was. Her presence behind the mundane experience of life is the creatio continua: the continuous creation of the Higgs Field, the physical process by which the particles of spacetime, including all the atoms of the human body, take on mass (and inertia) --- from emptiness. [You can read a concise and accessible definition of the Higgs Field here: http://hepwww.ph.qmw.ac.uk/epp/higgs1.html] except that this is complete bullshift and no longer an active url
------------------------------------------------------------------------------
..............................................................................
The symmetry group can be discrete, such as the space group of a crystal, or continuous (e.g., a Lie group), such as the rotational symmetry of space. However if the system contains only a single spatial dimension then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.
It behooves the reader at this point(not just the cognitively challenged) to come up with a common example.And so to help explain this phenomenon let us pretend that there is a ball, and this ball is sitting on top of a hill. This ball is in a completely symmetric state. However, its state is unstable: the slightest force of perturbation could cause the ball to roll down the hill in some particular direction. At that point, symmetry has been broken because the direction in which the ball rolled has a feature that distinguishes it from all other directions. It does not "at this point" help to remember that the direction of thrust opposes direction of travel.
Mathematical example: the Mexican hat potential
Graph of a Mexican hat potential function.
In the simplest example, the spontaneously broken field is described by a scalar field theory. In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system will behave, can be split up into kinetic and potential terms
(1) \qquad \mathcal{L} = \partial^\mu \phi \partial_\mu \phi - V(\phi).
Even my pointed little head can circumnavigate some baseline formula however, Geometrically induced discrete spectrum in curved tubes? (i may have ditched class that day) Differential Geometry and its Applications is not so bad
But my impoverished education stops at the fucking 6th grade(Last class i actually graduated from was Harry M. Wegaforth Elementary School)But i Digress
It is in this potential term (V(φ)) that the action of symmetry breaking occurs. An example of a potential is illustrated in the graph at the right.
(Sorry no graph possible here, Just imaging a big floppy wire frame sombrero)
(2) \qquad V(\phi) = -10|\phi|^2 + |\phi|^4 \,
This potential has many possible minima (vacuum states) given by
(3) \qquad \phi = \sqrt{5} e^{i\theta}
Bone Head Note # 1: In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square (the result of multiplying the number by itself) is x.)
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (corresponding to a choice of θ) this symmetry will be lost or spontaneously broken.
Higgs mechanism
Main article: Higgs(HIGGS=HELP-IVE-GIVEN-GEOMETRY-SOUL) mechanism
Bone Head Note # 2: regarding ((Phi)) I ripped this directly from Wikipedia
But Please read this if you you are having issues with understanding, ϕ or Φ
(((Phi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced [ˈfi] in modern Greek and /ˈfaɪ/ or sometimes /ˈfiː/ in English,[1] is the 21st letter of the Greek alphabet. In modern Greek, it represents [f], a voiceless labiodental fricative. In Ancient Greek it represented [pʰ], an aspirated voiceless bilabial plosive (from which English ultimately inherits the spelling "ph" in words derived from Greek). In the system of Greek numerals it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) arose from Φ.)))
In the Standard Model, spontaneous symmetry breaking is accomplished by using the Higgs boson and is responsible for the masses of the W and Z bosons. A slightly more technical presentation of this mechanism is given in the article on the Yukawa interaction, where it is shown how spontaneous symmetry breaking can be used to give mass to fermions.
Broader concept: The required particle if it ever truly existed disappeared from our universe a few fractions of a second(measured in the tens of billionths of a nano second or even smaller increments Ago, So where were we? Oh yea if the required particle can be recreated in the latest & greatest collider/accelerator we have no equipment to ever be able to actually observe & record or in any direct way even be aware of events that occur in so brief a period of time on such a minute scale however, i am led to believe that we can become aware of these events in a sort of Ex post ipsos facto technique or by the Decay products that result directly from these so called events but again i pre-digress
More generally, we can have spontaneous symmetry breaking in nonvacuum situations and for systems not described by actions. The crucial concept here is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter which forms a "frame of reference" to be measured against, so to speak.
If a vacuum state obeys the initial symmetry then the system is said to be in the Wigner mode, otherwise it is in the Goldstone mode You fucking choose, Based of course upon observation that is resplendant in divergence (Byrne-Fripp & Eno)
Remain in hell, without despair
O Urizel
Remain in hell, without despair
O Urizel
Stop!
I am resplendent in divergence
I am resplendent in divergence
I am resplendent in divergence
* For ferromagnetic materials, the laws describing it are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well) nonzero value which points in a certain direction. The residual rotational symmetries which leaves the orientation of this vector invariant remain unbroken but the other rotations get spontaneously broken.
* The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
* The laws of physics are spatially invariant, but here on the surface of the Earth, we have a background gravitational field (which plays the role of the order parameter here) which points downwards, breaking the full rotational symmetry. This explains why up, down and the vertical directions are all "different" but all the horizontal directions are still isotropic.
* General relativity has a Lorenz gauge symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
* Here on Earth, Galilean invariance (in the nonrelativistic approximation) is broken by the velocity field of the Earth/atmosphere, which acts as the order parameter here. This explains why people thought moving bodies tend towards rest before Galileo. We tend not to be aware of broken symmetries.
* For the electroweak model, as explained earlier, the Higgs field acts as the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries explains why it took so long for us to discover electroweak unification.
* For superconductors, there is a collective condensed matter field ψ which acts as the order parameter breaking the electromagnetic gauge symmetry.
* In general relativity, diffeomorphism covariance is broken by the nonzero order parameter, the metric tensor field.
* Take a flat plastic ruler which is identical on both sides and push both ends together. Before buckling, the system is symmetric under the reflection about the plane of the ruler. But after buckling, it either buckles upwards or downwards.
* Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
* Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead must choose between the two new equilibrium points, thus breaking symmetry. Note: This can easily be tried at home with an electric drill, a marble, and a pot cover, (or any other combination you can think of) and is the two-dimensional, mechanical analogue of the symmetry breaking that occurs in the Higgs Boson field.
[edit] Nobel Prize
On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to two Japanese citizens and a Japanese-born American for their work in subatomic physics. American Yoichiro Nambu, 87, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry. Japanese physicists Makoto Kobayashi and Toshihide Maskawa shared the other half of the prize for discovering the origin of the broken symmetry.[1] The trio shared the 10 million kronor (1.25 million USD) purse, as well as a diploma and an invitation to the prize ceremonies in Stockholm on December 10, 2008
Catastrophe theory
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the study of dynamical systems. For other meanings, see catastrophe.
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.
Catastrophe theory, which originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s, considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function).
Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.
See also
* Autocatalytic reactions and order creation
* Catastrophe theory
* CP-violation
* Dynamical symmetry breaking
* Explicit symmetry breaking
* Gauge gravitation theory
* Goldstone boson
* Grand unified theory
* Mermin-Wagner theorem
* Second-order phase transition
* Symmetry breaking
* Vacuum fluctuation
* J. J. Sakurai Prize for Theoretical Particle Physics
* Higgs mechanism
* Higgs boson
* Overview and Differences of 1964 PRL Symmetry Breaking papers
--
Dave Brackeen
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment