- mathfun@mailman.xmission.com - mailman.xmission.com

[math-fun] Heat leaps through empty space -- quantum weirdness
by Henry Baker 17 Dec '19

17 Dec '19

FYI -- Remind me again how we're planning to insulate qubits from the environment ? This report doesn't even mention the Aharonov-Bohm effect, which also enables information transfer across "empty" space w/o utilizing electromagnetic radiation. No computation (classical or quantum) and proceed w/o heat insulation and "refrigerators" to protect the critical degrees of freedom of the computation itself. Quantum error correction could be a kind of "refrigeration" which rejects "heat"/"error" back to the environment, but it's going to require an enormous degree (!) of subtlety to pull this off. And even if we could build more reliable qubits, we won't be able to put them very close together. https://phys.org/news/2019-12-energy-space-quantum-weirdness.html Heat energy leaps through empty space, thanks to quantum weirdness by Kara Manke, University of California - Berkeley December 11, 2019 In a new study, University of California, Berkeley, researchers show that heat energy can travel through a complete vacuum thanks to invisible quantum fluctuations. In the experiment, the team placed two gold-coated silicon nitride membranes a few hundred nanometers apart inside a vacuum chamber. When they heated up one of the membranes, the other warmed up, too, even though there was nothing connecting the two membranes and negligible light energy passing between them. If you use a vacuum-insulated thermos to help keep your coffee hot, you may know it's a good insulator because heat energy has a hard time moving through empty space. Vibrations of atoms or molecules, which carry thermal energy, simply can't travel if there are no atoms or molecules around. But a new study by researchers at the University of California, Berkeley, shows how the weirdness of quantum mechanics can turn even this basic tenet of classical physics on its head. The study, appearing this week in the journal Nature, shows that heat energy can leap across a few hundred nanometers of a complete vacuum, thanks to a quantum mechanical phenomenon called the Casimir interaction. Though this interaction is only significant on very short length scales, it could have profound implications for the design of computer chips and other nanoscale electronic components where heat dissipation is key. It also upends what many of us learned about heat transfer in high school physics. "Heat is usually conducted in a solid through the vibrations of atoms or molecules, or so-called phonons--but in a vacuum, there is no physical medium. So, for many years, textbooks told us that phonons cannot travel through a vacuum," said Xiang Zhang, the professor of mechanical engineering at UC Berkeley who guided the study. "What we discovered, surprisingly, is that phonons can indeed be transferred across a vacuum by invisible quantum fluctuations." In the experiment, Zhang's team placed two gold-coated silicon nitride membranes a few hundred nanometers apart inside a vacuum chamber. When they heated up one of the membranes, the other warmed up, too--even though there was nothing connecting the two membranes and negligible light energy passing between them. "This discovery of a new mechanism of heat transfer opens up unprecedented opportunities for thermal management at the nanoscale, which is important for high-speed computation and data storage," said Hao-Kun Li, a former Ph.D. student in Zhang's group and co-first author of the study. "Now, we can engineer the quantum vacuum to extract heat in integrated circuits." No such thing as empty space The seemingly impossible feat of moving molecular vibrations across a vacuum can be accomplished because, according to quantum mechanics, there is no such thing as truly empty space, said King Yan Fong, a postdoctoral scholar at UC Berkeley and the study's other first author. "Even if you have empty space--no matter, no light--quantum mechanics says it cannot be truly empty. There are still some quantum field fluctuations in a vacuum," Fong said. "These fluctuations give rise to a force that connects two objects, which is called the Casimir interaction. So, when one object heats up and starts shaking and oscillating, that motion can actually be transmitted to the other object across the vacuum because of these quantum fluctuations." Though theorists have long speculated that the Casimir interaction could help molecular vibrations travel through empty space, proving it experimentally has been a major challenge. To do so, the team engineered extremely thin silicon nitride membranes, which they fabricated in a dust-free clean room, and then devised a way to precisely control and monitor their temperature. They found that, by carefully selecting the size and design of the membranes, they could transfer the heat energy over a few hundred nanometers of vacuum. This distance was far enough that other possible modes of heat transfer were negligible--such as energy carried by electromagnetic radiation, which is how energy from the sun heats up Earth. Because molecular vibrations are also the basis of the sounds that we hear, this discovery hints that sounds can also travel through a vacuum, Zhang said. "Twenty-five years ago, during my Ph.D. qualifying exam at Berkeley, one professor asked me 'Why can you hear my voice across this table?' I answered that, 'It is because your sound travels by vibrating molecules in the air.' He further asked, 'What if we suck all air molecules out of this room? Can you still hear me?' I said, 'No, because there is no medium to vibrate,'" Zhang said. "Today, what we discovered is a surprising new mode of heat conduction across a vacuum without a medium, which is achieved by the intriguing quantum vacuum fluctuations. So, I was wrong in my 1994 exam. Now, you can shout through a vacuum." More information: Phonon heat transfer across a vacuum through quantum fluctuations, Nature (2019). DOI: 10.1038/s41586-019-1800-4 , https://nature.com/articles/s41586-019-1800-4

2 1

[math-fun] Identity transformation
by françois mendzina essomba2 16 Dec '19

16 Dec '19

Hi, Pi/2==sum((2*n)!*(1+x^(4*n+2))/(2^(2*n)*(n!)^2*(1+x^4)^(n+1/2)*(2*n+1)) ,n=0..inf); an identity transformation gives this result for any real number. I wonder what other process can lead to this formula Best regards.

4 3

[math-fun] Another Ellipse Certificate
by Brad Klee 16 Dec '19

16 Dec '19

With[{r2x = 1/(1 - x*Cos[z])^2}, ReplaceAll[ Factor@Plus[ Dot[D[r2x, {x, #}] & /@ {0, 1}, {3 x, (x - 1) (x + 1)}], D[(2*Sin[z] - x*Cos[z] Sin[z]) r2x, z]], Sin[z]^2 -> 1 - Cos[z]^2]] This is no more fundamental a definition than the offering of yesterday; however, it may be easier to work with because r^2*dz is an area from, so A(x) = Pi/(1-x^2)^(3/2) , Should also follow from Cartesian definitions, i.e. from an x*dy integral of algebraic function. --Brad

1 0

[math-fun] OEIS A016825 has 28(?) characterizations,
by Bill Gosper 15 Dec '19

15 Dec '19

e.g., In[214]:= ContinuedFraction@Coth@.5 Out[214]= {2, 6, 10, 14, 18, 22, 26} e.g., the oddly even numbers. A 29th: Polar rose (r = cos(k t)) unachievable petal counts. (https://en.wikipedia.org/wiki/Polar_coordinate_system#Polar_rose) —rwg

1 0

[math-fun] Shew that PolarPlot[1/(√2 - Cos@u), {u, 0, 2 π}]
by Bill Gosper 15 Dec '19

15 Dec '19

is an ellipse with focus at the origin. —rwg

2 1

[math-fun] Check your color vision.
by Bill Gosper 14 Dec '19

14 Dec '19

http://www.tweedledum.com/rwg/squares.htm problem 2, can you see the color boundary between "18" and "14"? Maximal zoom. Slide "10" and "4" off the left edge to show only the middle right portions of "14" and "18". Slide up and down. Suspicion: Girls can see it better. Color vision is on the X chromosome. —rwg ("8" and "10" are identical shades of light blue. Proof: TextEdit>Format>Font>Show Colors> <eyedropper>.) I used to grumble at color-enhanced astronomy photos. Until I realized I am colorblind compared to most space aliens.

4 4

[math-fun] Does this define sin(z)?
by Bill Gosper 14 Dec '19

14 Dec '19

1) sin(0) := 0 and 2) |sin(x) - sin(y)| ≤ |x - y| and 3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2). Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg

4 5

Re: [math-fun] Does this define sin(z)?
by Keith F. Lynch 14 Dec '19

14 Dec '19

Bill Gosper <billgosper(a)gmail.com> wrote: > 2) |sin(x) - sin(y)| ? |x - y| and The digest option of this list turns all non-ASCII characters into question marks. I have no idea what was intended to go there, so I can make no sense of the question.

2 1

[math-fun] A problem about chords of Jordan curves
by James Propp 13 Dec '19

13 Dec '19

Given a rectifiable simple closed curve, can one always find points P and Q on the curve that cut the curve into two arcs of equal length, neither of which crosses the chord joining P and Q? Here’s a proof that works if the curve is starlike from an interior point O: Every line through O determines a chord that cuts the curve into two pieces, one longer than the other (unless they’re the same length in which case we’re done). As you rotate the line by 180 degrees, the short piece becomes the long piece and vice versa. Since the lengths vary continuously, there must be some intermediate position of the line that makes the two pieces have equal length. For general Jordan curves, I can find either a proof nor a counterexample. (Seems like the kind of problem that Polish mathematicians in the 1920s would have looked at.) Jim Propp

2 2

[math-fun] "Self-avoiding spacefilling curve" is an oxymoron.
by Bill Gosper 11 Dec '19

11 Dec '19

Besides completely covering the closure of its range, a spacefilling function doubly covers an uncountable subset of it, and triple-covers a dense subset of that. (Can we define a finite "fill factor" to describe *how* dense are these subsets?) The useful idea that we're trying to express is perhaps "noncrossing", meaning that there are arbitrarily fine polygonal samplings that do self-avoid <http://gosper.org/hexpicules.png>. Could we say "self avoidable"? —rwg Another one of those "Hey, wait a minute" math posters. Julian: If you feel like it, please explain how on Earth these two brain hemispheres are born horizontally separated by exactly 10 units. Talk about agenesis corpus callosum.

1 0