archery

Coincidental Solar Eclipse?

There will be an annular solar eclipse taking place this evening (5/20/2012) viewable from the western parts of the US and eastern Asia.

This is what happens when the Moon blocks some of the light from the Sun by passing in front of it from our vantage point on Earth. It is an annular eclipse and not a total eclipse because the Moon won’t completely cover up the Sun in the sky as it does during a total solar eclipse. Solar eclipses happen in these two varieties because the distance from the Earth to the Moon varies during its orbit around Earth. Therefore, the apparent size of the Moon looks different depending on how close the Moon is to Earth, and will only cover up as much of the Sun as it can. In fact, the Moon happens to be near its furthest point now.

It is interesting to note how close the Moon comes to completely covering up the Sun, and how this depends on the geometry of the situation.

During a total solar eclipse, the Moon will cover up the Sun almost exactly without much overlap. This happens because the apparent size of the Moon as viewed from Earth is nearly the same as the apparent size of the Sun as viewed from Earth.

This can be measured by comparing the ratio of the Sun’s radius and its distance to Earth to the ratio of the Moon’s radius and its distance to Earth. According to my own calculations they only differ by factor of about 3% of the Moon ratio. This is justified since the the Moon happens to be about 400 times smaller than the Sun, but is also about 400 times closer. Bigger differences in these ratios would imply that the Moon looks smaller or larger than the Sun during an eclipse. This closeness is equivalent to the claim that the two right triangles drawn in this diagram are close to being similar.

Is there any physical reason for why these ratios are the way they are? It seems plausible that the Moon could have had a different size, and orbited a little closer or further away from Earth thereby preventing such a ‘perfect’ total eclipse from happening. The configurations we witness almost seem like some coincidence!

So what? Is there any value to this special orientation during an eclipse?

Actually, the perfect total eclipses we are lucky enough to experience are valuable opportunities for astronomical observation.

In 1919, Arthur Eddington observed a total solar eclipse and was able to experimentally verify the phenomenon of gravitational lensing—one of the theoretical predictions of general relativity, which involved the bending of distant star light due to the Sun’s gravity. In addition, these perfect total eclipses also allow for other observations of solar phenomenon.

If it were not for this ecliptic coincidence and things were any different, then how much more difficult would it have been for scientists to learn about these other astrophysical phenomenon?

PRINT THIS POST Notes: 91 5/20/12 — 6:08pm Filed under: #Science #Sun #math #Order Of Magnitude

Anonymous

asks:

Did you choose this coordinate transformation at random or is the position of the neurons in V1 in someway a basis for this transition?

It was not random. The only choice made by me was to follow the models as they are originally described in the papers cited in the post ( [1], [2]. [3]). Let me elaborate more on how the simplified transformation used in the post differs from the one given in the papers, and also on how specific positions of neurons in V1 play a role in the creation and transformation of neural patterns to visual hallucinations. Keep in mind that I am no neuroscientist.

Cowan and Ermentrout in [1] were amongst the first to do such a modeling. The construction of the formula for the coordinate transformation does indeed seem to be motivated by the actual architecture of V1. They state in the paper: “A variety of experimental observations, anatomical, physiological, and psychophysical, have established in primates, and presumably also in humans, that there is a conformal projection of the visual field, onto the visual cortex.”

The formula for the transformation that was used in the post is a simplified version of the actual formula as it appears in the papers in two ways. The complete formula (first appearing on page 2 of [1]) contains a few numerical constants as multiplicative factors, which represent physical parameters of the retina for things like the sizes and numbers of ganglion cells in the eye. So in this regard, the coordinate transformation isn’t necessarily constructed in terms of the “positions of neurons in V1”, but rather in terms of properties of the retina. The first way, the transformation used in the post differs from the one in the paper is through the omission of these physical constants. This is reasonable for the purposes of illustration because the constants mainly serve to define the relevant size scale, and the overall pictures resulting form the transformation remain essentially the same without them.

Also, the original formula for the coordinate transformation has two limiting cases in which the formula can be approximated by simpler ones. One case is for regions that are close to the center of the retina where the transformation turns to the regular transformation from polar coordinates to Cartesian coordinates. The other limiting case occurs for regions that are sufficiently far from the center of the retina. In this case, the transformation becomes the one used in the post (log-polar coordinates).

The authors in [2], which includes Cowan, generalize and extend the original work done in [1]. Its worth noting that [2] was published about 20 years after [1], which was published in 1979. Therefore, since the field of neuroscience has had some time to grow, [2] is able to offer some deeper insights on the matter. The work done in [2] does take into account the relative positions of neurons in V1.

In the papers, and especially in the post, V1 is simply modeled as a flat two dimensional plane. However, in reality V1 possesses a higher dimensional structure consisting of crinkles and folds like the rest of the brain. V1 does more than simply respond to light/dark regions in the visual field with high/low neural activity as described in the post. In addition, neurons on V1 are responsible for detecting contours and edges present in the visual field—that is, the finer boundaries between light and dark regions. Apparently, the depth to V1 consists of collections of neurons referred to as hypercolumns, and it is the neurons in a given hypercolumn which detect the contours in the visual field. The way in which the neurons do this contour detecting is pretty involved in itself and is a primary focus of [2]. Basically, certain neurons in a hypercolum are there to detect only one particular angle that a contour can have, and each hypercolum contains enough different neurons so that contours of all possible angles can be detected. So for each point in the visual field, there exists a hypercolumn in V1 that allows for the proper detection of an edge at any angle in a way that allows for the continuous variation of contours in the visual field.

In the model, hallucinations arise when certain patterns of neural activity are present in V1. The existence of these hypercolumns, and the way neurons are able to interact with neurons in the same and in other hypercolumns, helps explain why certain patterns of neural activity come about (mainly the doubly periodic patterns like the lattice symmetries). It seems that neurons in a given hypercolumn can interact with one another freely, but neurons in different hypercolumns can only interact with each other if the interacting neurons are ones that detect the same contour angle in the visual field. The dynamics of this kind of interaction in V1 is mathematically modeled in [2] and is the main extension that [2] offers to [1]. It is described with more rigor in the paper than I am able to give here. To be honest, I haven’t read it all and am yet to understand all the details myself.

Notes: 2 3/23/12 — 7:17pm Filed under: #science #form constants

Visual Hallucinations and Form Constants

Whether or not you know about or care to use the mathematical coordinate systems described in the last post, it seems that our eyes and brains already make use of them when seeing and hallucinating. How this is the case will be explained here in some detail. Of course, our senses are intricately correlated with the dynamics of our brains in ways so complex that we can barely understand or even comprehend them at all, but there are still ways to reason and model (however crude they may be) the workings of such phenomenon.

Visual hallucinations are a universal human experience which occur in varying degrees in different circumstances—from when we rub or apply pressure to our eyes to more extreme instances like when we are under the influence of psychoactive drugs. What we “see” when we hallucinate is ultimately a unique subjective experience, but there does seem to be some objective similarities amongst reported hallucinations. Some of these hallucinations appear to be fixed in the visual field and do not change as we look around. Moreover, they can even be experienced when the eye are closed, and have also been reported by people who are blind. This gives reason to believe that the source or cause of these hallucinations may be independent of what we really see and receive as sensory input from the external world, and instead originate on a deeper level within the brain itself.

Studies suggest that a certain region of the brain, the visual cortex (also known as V1), plays a primary role in the processing of visual information. There is a correspondence from what we see in the visual field with our eyes to the neural activity in parts of V1. Spatial input of light and dark on the eyes is translated to “light/dark” regions of high/low neural activity in V1. Basically, images form certain patterns on the retinas of our eyes which are then converted to related patterns on the visual cortex in our brains leading to the perception of the image.

To describe more rigorously the nature of this image translation from the eyes to the brain, we can use mathematics and coordinate systems to define a coordinate mapping between the visual field to the V1 region of the brain. Such a transformation was modeled by J. D. Cowan and G. B, Ermentrout in their 1979 paper “A Mathematical theory of Visual Hallucinations”. A simplified version capturing the essence of their model is described in what fallows.

Interpreting the visual field as a two-dimensional plane, let’s use polar coordinates to label points in the visual field. The center of the visual field is taken as the origin of the coordinate system so that a point P in the visual field is described by a pair of numbers P = ( r , a ), where r is the distance of the point P to the center of the visual field and a is the relative angle the point makes with respect to some fixed axis.

Now we need a way a describe how points in the visual field correspond to points in the V1 region of the brain, but first we need a way to label points in the V1 region. To do this, we model the V1 region as another two-dimensional flat plane. This time, let’s use the normal Cartesian coordinate system to label the points in V1 as pairs of numbers ( x , y ).

The coordinate mapping constructed here will tell us how a point in the visual field, labelled as P = ( r , a ) in polar coordinates, gets translated to the corresponding point in the V1 region of the brain labelled as ( x , y ) in Cartesian coordinates. This ( x , y ) point in V1 is given in terms of the r and a coordinates of the visual field through the relationship

( x , y ) = ( ln r , a ),

where ln r is the natural logarithm of r.

Thus, the point that is given by the polar coordinates ( r , a ) in the visual field is mapped to the corresponding point in V1 whose Cartesian coordinate has a horizontal component of ln r and a vertical component of a. This transformation can be interpreted as log-polar coordinates, and may be recognized as the complex logarithm. We can now use this coordinate transformation to describe the shapes of certain hallucinations.

Normally, we actually receive sensory input through the visual field and corresponding neural patterns are triggered in our brains which result in the perception of whatever we are seeing. The model explained here explains the existence of hallucinations by an opposite mechanism. Without even receiving real sensory input through the visual field, V1 can still be a brain region with high neural activity (perhaps more so when on certain drugs). These patterns of neural activity in V1 may be perceived as if one is actually seeing a pattern in the visual field, but they are really just hallucinations. In the 1920s, psychologist Heinrich Klüver researched himself and others while having ingested mescaline (more info here) in the form of peyote buttons and attempted to classify the visual hallucinations they experienced. The observed hallucinations manifesting themselves as geometrical patterns were classified into four types and were referred to as form constants: 1) tunnels, 2) spirals, 3) cobwebs, and 4) lattices.

The coordinate transformation just described between the visual field and the V1 region of the visual cortex is successful in explaining the occurrence of the form constants.

1) Vertical or horizontal stripes stripes of neural activity on the V1 region may look something like this:

Applying the coordinate transformation in the reverse direction, we see that vertical stripes get mapped to circles in the visual field and horizontal stripes get mapped to rays emanating out from the center of the visual field:

2) Stripes of neural activity in V1 in arbitrary directions such as these diagonals
get mapped to spirals in the visual field:

3) Combinations of stripes such as these vertical and horizontal ones

would then get mapped to patterns that resemble cobwebs:

4) Neural activity in V1 is not limited to stripes. There may be activity with certain lattice symmetries like these checkerboard or honeycomb patterns

which would get mapped to hallucinations that appear like these

This model and these examples are idealized cases, but serve to approximate what may be happening in our brains when certain hallucinations occur. Despite having empirical applications in this context, the coordinate transformation described here can be freely applied to any image resulting in transformed images that still manage to express a psychedelic aesthetic (see here for some examples).

If you ever wondered when or why you would use polar coordinates I hope this post serves as a justifiable application. Conversely, I do not necessarily justify the use of psychedelics to help understand what you are learning in your math lessons.

This post was inspired by these papers, which explain what was attempted here and more in much greater detail:

Get the Mathematica code for these animations as a CDF file here.

PRINT THIS POST Notes: 513 3/22/12 — 4:30pm Filed under: #GIF #Science #Math #drugs! #form constants #logpolar coordinates #Mathematica

Here is another way to justify how humans are “central” in this universe.

Consider the range in size of what is physically relevant in our universe. This range spans about 60 orders of magnitude (10⁶⁰) from the size of the visible universe at about 13.7 billion light years (~10²⁸ cm) all the way down to the Planck length, which is the smallest scale in which notions of size and distance essentially breakdown according to standard theories, at ~10^-33 cm.

As humans, at about 10² cm in size, we happen to exist near the middle of this range. This really is no coincidence, nor is it some kind of trickery with units of measurement.

It can be argued that if humans were any smaller our brains wouldn’t have room to develop their complexity allowing for our intelligence, and if we were any bigger our brains would lose their practical efficiency which depends on the brain’s ability to interact with itself. In fact, this must be true for all intelligent life.

So if you ever happen to feel down about your size relative to the rest of the universe, know that you are much more significant than any other structure in this universe exactly the way you are. Even David Deutsch once wrote: “The size of the universe is no more depressing than the size of a cow.”

However, it does seem rather peculiar that we humans exist and are making these observations of size at the point in time when the Universe has expanded to its current size, which just so happens to place us in the middle of this size range. Couldn’t this have been any different?

On the other end of things, one may even be willing to pass the size scale set by the Plank length as being mere coincidence, since the fundamental constants which determine its value could have possibly taken on different values themselves making things very different.

High-res Notes: 385 2/21/12 — 7:29pm Filed under: #Order Of Magnitude #Science #David Deutsch

If for some reason the existential risks mentioned in the last post made you feel insignificant in anyway, let the following serve as consolation and reason to feel significant (and quite literally “central”) to the universe as human beings existing in this period of time.

From a temporal perspective of the evolution of the universe, we live close to a point in time where the universe is switching over from slowing down its expansion to speeding up its expansion. This is important from an observational standpoint, because as the expansion continues distant galaxies and regions of the universe will be disappearing from sight leaving the Milky Way and some of its closest neighbors in isolation. After a sufficient amount of time, light will no longer be able to reach us in ways that permit practical observation of these distant regions. This also includes light that gives evidence of the “big bang” via the cosmic background radiation. Thus, new astronomers in the future may not even be able to observe other galaxies to convince them that they live in a universe with many others. It is also somewhat peculiar that we exist in a period of time where the expansion is neither too weak nor too strong to even detect in the first place. In fact, this is essentially the only time period in the universe’s evolution where a detection of this expansion and the existence of dark energy could be possible! [Read more about this here]

Moving from this larger cosmic scale to our Solar System, there are still more ways we are living in a central moment in time. Within an order of magnitude, we exist halfway through the main life of our Solar System, which began about 4.5 billion years ago and will end in 5-6 billion years when the Sun turns into a red giant and ultimately a white dwarf.

Moreover, we also exist about halfway through the period of time on Earth which permits the survival of life like us. In the past, until about half a billion years ago, the Sun was too faint and the Earth lacked an oxygen-rich atmosphere. In the next half a billion years, the Sun’s luminosity will be increasing causing temperatures on Earth to rise high enough to where the Earth’s oceans will evaporate away and the atmosphere will escape into outer space. This stage will have some drastic consequence’s for Earth in the future, but goes to show that we are currently in the most suitable period for life on Earth.

Relative to the more recent evolution of humans on Earth, there is one more way we are in a pivotal moment in time. We are at the end of an exponential expansion in human population. In the past century, the human population quadrupled, and doubled twice in just the last 100 years. This is the first time this has ever happened in history, and will likely be the last since there are strong doubts that Earth could even handle such a large capacity.

Whether you want to regard these as mere coincidences, or formulate some kind of anthropic arguments as justification, know that this all really happens to be the case. So while we continue to Tumbl, perhaps we can let this increase our awareness and inspire us to live accordingly.

Most of the ideas mentioned here were brought to my own awareness during a course offered at UCSC, called Cosmology and Culture, taught by Joel Primack and Nancy Abrams. This course has resulted in their informative book, The View from the Center of the Universe, which explores these ideas amongst others more thoroughly.

PRINT THIS POST Notes: 28 2/16/12 — 4:00pm Filed under: #Science #evolution #Order Of Magnitude

Some notable dates in the far future

Compiled below is a selection of estimated dates for some events given certain assumptions in the evolution of Earth, the Solar System, and the Universe. Most events are of an astronomical and cosmological nature though some are geological. A more complete list from which the ones included here were taken can be found on Wikipedia.

In 10,000 years - The end of humanity, according to Brandon Carter’s Doomsday argument, which assumes that half of the humans who will ever have lived have already been born.^[3]
In 50,000 years - Niagara Falls erodes away the remaining 20 miles to Lake Erie and ceases to exist.^[6]
In 500,000 years - By this time Earth will have likely been impacted by a meteorite of roughly 1 km in diameter.^[9]
In 1 million years - Highest estimated time until the red supergiant star Betelgeuse explodes in a supernova. The explosion is expected to be easily visible in daylight.^[10]^[11]
In 50 million years - The Californian coast begins to be subducted into the Aleutian Trench^[15]
Africa will have collided with Eurasia, closing the Mediterranean Basin and creating a mountain range similar to the Himalayas.^[16]
In ~240 million years - From its present position, the Solar System will have completed one full orbit of the Galactic center.^[18]
In 250 million years - All the continents on Earth fuse into a possible new supercontinent.^[19]^[20]
In 1 billion years - The Sun’s luminosity increases by 10%, causing Earth’s surface temperatures to reach an average of 47°C and the oceans to boil away.^[22]
In 1.5 billion years - The Sun’s circumstellar habitable zone moves outwards as its increased luminosity causes carbon dioxide to increase in Mars’s atmosphere, raising its surface temperature to levels akin to Earth during the ice age.^[23]
In 5.4 billion years - The Sun becomes a red giant.^[29]Mercury, Venus and possibly Earth are destroyed.^[30]
In 7 billion years - Collision between the Milky Way and Andromeda galaxies.^[32]
In 14.4 billion years - Sun becomes a black dwarf as its luminosity falls below three trillionths its current level, while its temperature falls to 2239 K, making it invisible to human eyes.^[36]
In 20 billion years - The end of the Universe in the Big Rip scenario.^[37] Observations of galaxy cluster speeds by the Chandra X-Ray Observatory suggest that this will not occur.^[38]
In 100 billion years - The Universe’s expansion causes all evidence of the Big Bang to disappear beyond the practical observational limit, rendering cosmology impossible.^[41]
In 10¹² (1 trillion) years - Low estimate for the time until star formation ends in galaxies as galaxies are depleted of the gas clouds they need to form stars.^[43]^,
In 2×10¹² (2 trillion) years - All galaxies outside the Local Supercluster are no longer detectable in any way, assuming that dark energy continues to make the Universe expand at an accelerating rate.^[44]
In 10¹⁵ (1 quadrillion) years - Estimated time until stellar close encounters detach all planets in the Solar System from their orbits.^[43]
By this time, the Sun will have cooled to five degrees above absolute zero.^[47]
In 3×10⁴³ years - Estimated time for all nucleons in the observable Universe to decay, if the proton half-life takes the largest possible value, 10⁴¹ years,^[43] assuming that the Big Bang was inflationary and that the same process that made baryons predominate over anti-baryons in the early Universe makes protons decay.^[53] By this time, if protons do decay, the Black Hole Era, in which black holes are the only remaining celestial objects, begins.^[46]^[43]
In 10⁶⁵ years - Assuming that protons do not decay, estimated time for rigid objects like rocks to rearrange their atoms and molecules via quantum tunneling. On this timescale all matter is liquid.^[49]
In 1.7×10¹⁰⁶ years - Estimated time until a supermassive black hole with a mass of 20 trillion solar masses decays by the Hawking process.^[54] This marks the end of the Black Hole Era. Beyond this time, if protons do decay, the Universe enters the Dark Era, in which all physical objects have decayed to subatomic particles, gradually winding down to their final energy state.^[46]^[43]
In $10^{10^{50}}$ years - Estimated time for a Boltzmann brain to appear in the vacuum via a spontaneous entropy decrease.^[55]
In $10^{10^{56}}$ years - Estimated time for random quantum fluctuations to generate a new Big Bang, according to Caroll and Chen.^[56]
In $10^{10^{10^{76.66}}}$ years - Scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing an isolated black hole of stellar mass.^[57] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is that in a model in which history repeats itself arbitrarily many times due to properties of statistical mechanics, this is the time scale when it will first be somewhat similar (for a reasonable choice of “similar”) to its current state again.

Source:

PRINT THIS POST Notes: 659 2/12/12 — 5:05pm Filed under: #Science #order of magnitude

"Whether or not God plays dice, I do"

A $100,000 offer is being made by quantum computing/complexity theory researcher Scott Aaronson to whomever can disprove the possibility of scalable quantum computation.

Although very primitive, small-scale quantum computers have been developed and realized in the field, the question of whether or not a large-scale quantum computer can exist to handle arbitrarily large computations is the focus of ongoing research and a matter of debate (at least to some researchers).

This is ultimately a matter of the possibility of quantum computers being able to exist in principle according to the laws of physics, and not one of just theoretical and technological advancement in a given period of time.

The way I see it is that any legitimate attempts to actually disprove the possibility of large scale quantum computation will only end up revealing more details and reasons to believe that large scale quantum computers can and will exist. I would even be willing to go as far as saying that the existence of large scale quantum computation ought to be demanded by the idea of computational universality.

Notes: 27 2/10/12 — 11:37am Filed under: #Science #Quantum Computation #Scott Aaronson #$$$

Here is something to think about that was only casually mentioned in passing in the recent video that was posted.

The sunlight you may or may not have experienced today finally managed to reach you after a ~100,000 year long journey since it was originally created at the Sun’s core!

Since the speed of light is finite, about 300,000,000 meters/second (or about 671,000,000 miles/hour), it takes time for it to travel from one point in space to another.

Given that the distance from Earth to the Sun is about 150,000,000,000 meters (about 93,000,000 miles) it takes about 8 minutes for light to reach us!

But this is just the time it takes light to reach us from the surface of the sun.

The light coming from the surface of the Sun is itself created as a by-product of nuclear fusion occurring deep in the Sun’s core.

Once light is created at at the Sun’s core it begins its journey to the surface of the Sun some 700,000,000 meters (430,000 miles) away from the core.

One might assume that this light takes the shortest path and heads straight to the surface, which would only take a couple seconds of travel time.

However, this is not the case because there is all kinds of star stuff that gets in the way.

An actual photon may only travel a mere fraction of a centimeter (anywhere between .01 and .3 centimeters depending on how close it is to the surface) before it makes a collision with other matter thereby diverting its path to some other random direction.

Photons continue moving in these seemingly random trajectories, bumping into other particles along the way, and don’t actually reach the surface until about 100,000 years later (give or take an order of magnitude)!

This kind of behavior characterizing the photons motion is modeled by something called a random walk, and is illustrated in a few different instances in the animations above.

Random walks have widespread applications through out the sciences and mathematics. The idea of random walks are even used in some computer algorithms to allow for more efficient solutions to some problems.

One particular application of personal interest, and a rather abstract generalization of the idea, is the quantum random walk, in which the superposition principle of quantum mechanics is used to put the trajectory into a combination of multiple possible trajectories to assist quantum computers in solving problems. The workings of Grover’s search algorithm can be thought of in this way. This isn’t the only instance that relates quantum mechanics to the workings of the Sun (see here).

Anyway, next time you are out in the relentless light of the Sun you may wonder what was going on some 100,000 years ago when that light first originated in the Sun, or maybe even where you’ll be 100,000 years from now when the light being created in the Sun at this moment finally reaches Earth.

(GIFs created from this Java app)

Notes: 739 1/12/12 — 5:06pm Filed under: #GIF #Science #order of magnitude #physics #quantum schwantum #stars #quantum computation

spookyactioncollective:

The secret lives of invisible magnetic fields are revealed as chaotic ever-changing geometries . All action takes place around NASA’s Space Sciences Laboratories, UC Berkeley, to recordings of space scientists describing their discoveries . Actual VLF audio recordings control the evolution of the fields as they delve into our inaudible surroundings, revealing recurrent ‘whistlers’ produced by fleeting electrons . Are we observing a series of scientific experiments, the universe in flux, or a documentary of a fictional world?.

(via Magnetic Movie)

This video, by Semiconductor, is an artistic rendering exemplifying the dynamics of electromagnetic phenomenon happening around us and through out the universe. The animations are set to audio from VLF radio-wave recordings produced from real ambient sources such as weather systems, electrical storms, the Earth’s core, the upper atmosphere, and solar wind traveling to us all the way from the Sun.

Also, be sure to check out one of my favorite blogs, spooky(abstr)action, over at spookyactioncollective.tumblr.com where you can find quality posts ranging on topics like science, math, and technology amongst many other intriguing and peculiar things.

Source: semiconductorfilms.com

Notes: 77 1/12/12 — 2:26pm Filed under: #video #animation #science #physics #space

albanhouse:

Hey guys, look what I did!

Which is to say that I wrote a Mathematica notebook which, given a Lagrangian, derives the canonical equations of the system. The test run (pictured) is a pendulum under the influence of gravity (V=mgh) where the mass m can move up and down the massless pendulum rod but is connected to the origin by a spring with spring constant k and rest length R.

The amazing thing is how wild non-linear systems can be; the graph shown is for k=0.8, m=0.8, R=1, g=0.2, released from rest at π/2 from the vertical (where θ=0), but playing around with these constants morphs the trajectory in various different ways and you can get a lot of different but interesting plots.

A quick educational bit: the Hamiltonian dynamics happens in phase space, Γ, but since this system has two degrees of freedom dim(Γ)=4 and you and I can’t visualize it. Nevertheless, there’s a trajectory γ[t] parameterized by the time that runs through Γ which the particle will follow given initial conditions. The graph here is the projection of that curve onto the configuration space, Q, so information about the particle’s momentum is lost here.

Some more background information here.

Source: albanhouse

High-res Notes: 78 11/21/11 — 11:52am Filed under: #Science #Physics #Mathematica

ontheunspeakable:

“Until very recently, general relativity was taught only in postgraduate mathematics or physics courses, because the mathematical foundations of the theory were regarded as much too demanding for undergraduates. But the Liebers possessed an astounding, Promethean faith that a much larger audience could learn Einstein’s theories—the genuine article, not watered-down explanations. They believed that Einstein’s work, the deepest understanding of space and time yet conceived, belonged to all of us and should be made accessible to anyone who wanted to learn it. We share that belief. The first editions of this book were homemade by the Liebers (Hugh Lieber colored many of the illustrations by hand). After some years, a publisher took a chance, and kept the book in print for fifteen years. It has been out of print ever since, despite substantial efforts by the book’s fans to get it republished. This new edition has made the dream of decades come true for us.” (via 50 Watts)

A reason to believe that you don’t have to be good at doing index gymnastics to understand the theory of relativity.

More illustrations at 50 Watts.

Source: 50watts.com

High-res Notes: 29 11/18/11 — 2:17pm Filed under: #Math #Physics #Relativity #Science

tomasorban:

Caption: Field ion micrograph of atoms of iridium. The tiny dots are the locations of individual atoms; the ring-like patterns are facets of a single crystal of the metal. The image was made by superimposing successive micrographs, taken with different col- our filters, to show small changes on the surface. The red dots are atoms which have evaporated or corroded away, while the green ones are probably atoms of gas which have been absorbed. Field-ion microscopy involves placing a tiny needle of a substance like iridium in a gas-filled chamber & passing a high voltage through it. The drifting gas ions hit the charged atoms & are repelled at right angles to form this pattern on a screen.

Credit: PROF. ERWIN MUELLER/SCIENCE PHOTO LIBRARY

Source: tomasorban

Notes: 277 10/17/11 — 4:42pm Filed under: #Science #Physics #atoms

A Vindication of Quantum Locality

The notion of nonlocality in quantum mechanics has been a topic of much interest over the past several decades since the conception of quantum mechanics. Nonlocality is the idea that particular systems which are spatially separated can exhibit instantaneous correlations which seemingly violate certain principles of classical physics such as faster-than-light travel. This phenomenon deeply bothered Albert Einstein, and has even led to some metaphysical claims of quantum mysticism.

Experimental evidence of such “nonlocal effects” is by no means doubted in the field. However, the idea of nonlocality may conflict with some of our models and interpretations of the world.

David Deutsch has recently posted a pre-print in which he argues a claim that he made over a decade ago: that reality is not nonlocal as believed by many.

Einstein’s (1949) criterion for locality is that for any two spatially separated physical systems S₁ and S₂ , ‘the real factual situation of the system S₂ is independent of what is done with the system S₁ ’. A previous paper (Deutsch & Hayden (2000)) included a proof that quantum physics satisfies this criterion. The method was first to prove that every quantum computational network satisfies it, and then to infer the same for general quantum systems by appealing to the universality of such networks.

Note that Deutsch’s proof makes use of the theory of quantum computation, which gives reason to believe that this theory may deserve a more fundamental role in our understanding of the world. It is also interesting to note that Deutsch is a strong advocate for the many-worlds interpretation of quantum mechanics, and thinks that this interpretation is the natural setting to understand quantum computation.

This makes me wonder what Einstein would of thought about later ideas like quantum computers and the multiverse. I, in my opinion, think he would have liked them.

PRINT THIS POST Notes: 9 9/30/11 — 4:51pm Filed under: #Albert Einstein #David Deutsch #entanglement #physics #quantum computation #quantum mechanics #science #multiverse

quantumaniac:

Quantum Tunneling

According to Quantum Mechanics, a particle has a definite probability of being anywhere in the entire universe. Although any real distance from the particle’s expected classical path is infinitesimally small, since Quantum Mechanics is a statistical theory those small probabilities must be counted!

Quantum Tunneling is a fascinating effect that arises out of these small probabilities. This effect allows particles to occasionally pass right through obstacles that it would not normally have the proper energy to overcome, or an electron to escape the ‘pull’ of the nucleus without having enough kinetic energy to do so.

Although it’s fairly obvious that atoms are not just randomly going through every barrier, but this is because the probability of that happening is astronomically small because the barriers in our everyday world are rather thick. If made thin enough (approximately 1-3 nm,) the chance that a particle will spontaneously pass through becomes noticeable. A particle can effectively ‘borrow’ energy from the system that it’s acting in, pass through a barrier, and then spontaneously lose it. Mathematically, this can be explained using Heisenberg’s uncertainty principle, which limits the amount of information that can be known about any particle.

Quantum Tunneling is responsible for many interesting happenings throughout our universe, including enzymes in our own bodies. However, we can thank Quantum Tunneling for the sun’s heat. The sun is massive - very massive. However, in order to produce the temperatures necessary to activate nuclear fusion, the sun should be mathematically be much more massive! Due to Quantum Tunneling, there is a small chance that a Hydrogen atom will spontaneously undergo nuclear fusion without the proper temperature. However, since the sun has a tremendous amount of Hydrogen atoms in it, this small chance is converting about four million metric tonnes of Hydrogen each second!

So basically you would not exist here today because stars would not behave the way they do if it wasn’t for quantum mechanics!

Source: quantumaniac