December 2013 - mathfun@mailman.xmission.com

[math-fun] periodic CFs with Round vs Floor
by Bill Gosper 13 Dec '13

13 Dec '13

Superficial observations. How much does rounding (thereby disabling terms =1, and enabling negative terms) shorten the period? In[281]:= rtcfperiod /@ Range[69] - rndcfperiod /@ Range[69] Out[281]= {{0}, 0, 0, {1}, 0, 0, 2, 0, {2}, 0, 0, 0, 2, 2, 0, {3}, 0, \ 0, 2, 0, 2, 2, 2, 0, {4}, 0, 0, 0, -3, 0, 2, 2, 2, 2, 0, {5}, 0, 0, \ 0, 0, 0, 0, 4, 4, 2, 4, 2, 0, {6}, 0, 0, 2, -3, 2, 0, 0, 2, -1, 2, 2, \ 4, 2, 0, {7}, 0, 0, 2, 0, 2} It actually lengthened √29 (and √53) by three! and √241 by seven: In[282]:= rtcfperiod /@ Range[239, 288] - rndcfperiod /@ Range[239, 288] Out[282]= {2, 0, -7, 4, 2, 12, 4, 4, -2, 2, 6, 2, 4, 2, 8, 2, 0, \ {15}, 0, 0, 2, 0, 4, 4, 4, 0, 2, 0, 2, 6, 0, 0, 8, 0, 2, 2, 2, 4, 6, \ 2, -2, 0, 4, 4, 8, 6, 2, 2, 2, 0} The biggest helps seem to outweigh the biggest hurts. The curlybraces mark the terminating (square) cases. In[275]:= cfrnt[29, 9] Out[275]= {{{0, 29}, {1, 0}}, {{5, 1}, {4, -5}}, {{7, 4}, {-5, -7}}, {{3, -5}, {-4, -3}}, {{5, -4}, {-1, -5}}, {{5, -1}, {-4, -5}}, {{7, -4}, {5, -7}}, {{3, 5}, {4, -3}}, {{5, 4}, {1, -5}}, {{5, 1}, {4, -5}}} The actual √29 roundcf, In[276]:= rndtrm /@ % Out[276]= {5, 3, -2, -2, -10, -3, 2, 2, 10, 3} vs the vanilla one, In[278]:= ContinuedFraction[Sqrt[29], Length[%%]] Out[278]= {5, 2, 1, 1, 2, 10, 2, 1, 1, 2} In[279]:= FromContinuedFraction /@ {%276, %278} Out[279]= {315156/58523, 9801/1820} of course converges faster, term for term: In[280]:= % - Sqrt[29.] Out[280]= {-1.35546*10^-10, 2.80303*10^-8} Even faster, period for period. --rwg

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[math-fun] barrier estimation
by Veit Elser 13 Dec '13

13 Dec '13

Here's a question about orthogonal matrices. Consider orthogonal matrices U of size NxN with the following "energy": E(U) = -sqrt(N) sum_{i j} |U_i j| (the negative sum of the absolute values of all the elements, appropriately scaled). The lowest energy matrices are Hadamard matrices, as these have equal-magnitude elements. We'll consider only N where Hadamard matrices exist. Let U_0 and U_1 be any two distinct Hadamard matrices, so two ground states of our energy function. Let U(t) be a continuous curve between these two matrices, i.e. U(0)=U_0 and U(1)=U_1. Here's the question: find a good lower bound on the "energy barriers" between ground states: the maximum over t, of E(U(t))-E(U(0)). You can approach this by first thinking about which kinds of Hadamard pairs have low barriers, and then bounding the barrier for that type of pair. The factor sqrt(N) is just so that the scaled orthogonal matrix elements are +/-1 in the Hadamard ground states (the usual convention). A simple case is where U_0 and U_1 differ just by the swap of two rows (or columns). A curve connecting them could be a Givens rotation; that has barrier energy of order N. Can you come up with a pair/curve that has a lower barrier? -Veit

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Re: [math-fun] coldest place on earth
by Keith F. Lynch 13 Dec '13

13 Dec '13

Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > ... You quoted every word of my post, turning all my sentences into questions, and adding no text of your own. Why? Warren D Smith <warren.wds(a)gmail.com> wrote: > "Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: >> Note that the partial pressure of CO2 on Mars is much higher than >> on Earth, since its atmosphere is 95% CO2. > --the atmospheric pressure on Mars allegedly is about 8.7% of > Earth's so again I agree with Lynch. Unfortunately, your decimal point is off. At that pressure, no spacesuit would be needed, just warm clothing and an oxygen tank. Also, liquid water could exist. But the pressure is actually at most 1/10th of that, and often far less. But the partial pressure of CO2 on Mars is still much higher than on Earth. >> I mean, is there really a gas cloud from some supernova that is >> microkelvin? Or is this just a speculation? No such clouds have ever been detected, for obvious reasons, but I don't think anyone doubts they exist. (They emit far too little to be detectable. And since they're transparent to microwaves, they don't cast a shadow in the cosmic microwave background.) After all, clouds from supernovas definitely rapidly expand without limit. And they lose a *lot* of heat when they're still hot, so the first part of the cooling is much faster than adiabatic. Randall Munroe recently pointed out that when a star goes supernova, it's brighter 1 AU away than a detonating H-bomb held against your eyeball -- brighter by nine orders of magnitude. Most of that light comes from the expanding blown-off cloud. > For example, is that Finnish lab that got 100 picoKelvin, really the > coldest place in the universe? As David Makin points out, we have no idea what aliens may have done. If it is the coldest anything has ever been, at least on our planet, is there any risk of dangerous unforeseen consequences? Bose-Einstein condensate requires very low temperatures, but if I understand boson statistics correctly, the larger the condensate, the less critical the temperature is. Could a nanokelvin condensate "infect" its microkelvin surroundings, which would then spread to gradually larger and warmer surroundings until the whole planet underwent such a phase change? I have no idea what that would be like, but I very much doubt it would be survivable. Perhaps that's the solution to the Fermi paradox: Advanced civilization invariably doom themselves, not with high-energy physics as Fermi may have suspected, but with low-energy physics. Remember, you heard it here first. When I experienced the earthquake here in Virginia two years ago, I wondered if that was what was happening, since we had never had earthquakes here before. > To make a completely crazy scenario, let's say we have a neutron > star that has planets orbiting it. (It is known from pulsar timings > that such exist.) After a long time, the planet gets cold, > approaching 2.7K. That would take a very long time, since neutron stars are very hot, have a very high heat capacity, and gain a lot of heat energy whenever anything collides with them. I suspect that by the time any neutron star has reached room temperature, the cosmic background temperature will have cooled to way below 2.7 K. I don't think there are even any black dwarf stars yet. The universe just isn't old enough. > Now further suppose that the planet is totally magnetized by the > enormous magnetic field of the neutron star, which far exceeds any > magnetic field we can make on Earth. My understanding is that the total flux is no greater than the parent star's. The flux *density* is much higher, but only because the neutron star is much smaller. So a planet at an earthlike distance probably wouldn't experience any more magnetic effects than Earth gets from the sun. And it seems unlikely that the planet could be even that close to the neutron star without having been vaporized when the neutron star formed in a supernova explosion. I suppose a planet could have been captured later, but it's difficult to see how. A planet approaching a neutron star would normally either be flung away at the same speed or would be reduced to gravel by the tidal effects. > Now a rogue planet swoops in from outer space and gravitationally > interacts with our planet, causing it to escape from neutron star. > As a result, it experiences demagnetization cooling. > Maybe in this way temperatures way below 2.7K could be reached > naturally. I'm dubious, but it might be possible. I won't say it's completely impossible, but I think supernova explosions are far more common. An astronomer acquaintance of mine once said there was no mystery about why the sun's photosphere isn't as hot as its chromosphere. He said that the photosphere is where most of the cooling in the solar system is taking place! It's all in your perspective. Seen from outside the sun, the photosphere is a source of heat. But seen from outside a refrigerator, the refrigerator is a source of heat too. (How to use your freezer as an impromptu heat pump: Make lots of ice. Throw the ice outdoors. You'll be heating your house with greater than 100% efficiency!) Seen from the inside, a supernova is just a really powerful refrigerator. So it's not surprising that some of its remains could reach very low temperatures. The same could be said for the whole universe. It didn't start at 2.7 K, you know.

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Re: [math-fun] coldest place on earth
by Henry Baker 13 Dec '13

13 Dec '13

This is correct, but the _amount_ of energy required to run to completion isn't correlated with the number of steps in the computation. Bennett shows that the computation runs more-or-less forward (using a very small push) until a few steps from the end, at which point you need to add a few extra kT's to get it to _stay_ at completion. At 06:41 PM 12/10/2013, Eugene Salamin wrote: >A reversible computer can run without an energy source, but which way does it run? You need some entropy increase to drive it in the desired direction.

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Re: [math-fun] coldest place on earth
by Adam P. Goucher 13 Dec '13

13 Dec '13

> From: Keith F. Lynch > > Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > > ... > > You quoted every word of my post, turning all my sentences into > questions, and adding no text of your own. Why? He certainly quoted your entire post, but didn't turn any of your sentences into questions (perhaps you're referring to the extraneous question mark that was also present in your original e-mail -- but nevertheless that seems to be an interesting definition of `all'). Also, he did append new material in the form of: >Answer: You get the same amount of heat, since each digit on a punch card (in IBM format) removes 1 chad. > -- Gene However, the new material was prefixed by the same number of > symbols as everything else in the e-mail, so it could quite easily have been mistaken to be part of your original message. > Warren D Smith <warren.wds(a)gmail.com> wrote: > > "Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: > >> Note that the partial pressure of CO2 on Mars is much higher than > >> on Earth, since its atmosphere is 95% CO2. > > > --the atmospheric pressure on Mars allegedly is about 8.7% of > > Earth's so again I agree with Lynch. > > Unfortunately, your decimal point is off. At that pressure, no > spacesuit would be needed, just warm clothing and an oxygen tank. > Also, liquid water could exist. But the pressure is actually at most > 1/10th of that, and often far less. But the partial pressure of CO2 > on Mars is still much higher than on Earth. Yeah, there's very little CO2 present in the Earth's atmosphere (<0.04% if I remember correctly). > Randall Munroe recently pointed out that when a star goes supernova, > it's brighter 1 AU away than a detonating H-bomb held against your > eyeball -- brighter by nine orders of magnitude. Astonishing! Wait a moment: the absolute magnitude of a Type IA supernova is -19.3, so its apparent magnitude when positioned at the centre of the Solar system would be -50.9. Now if a H-bomb held against your eyeball is merely 10^-9 of that, then it is 22.5 magnitudes dimmer, i.e. -28.7. But -28.7 is only about six times brighter than our Sun at the moment; that seems to be a severe underestimate of the brightness of an atomic bomb held against your eyeball. Unless you mean `nine magnitudes' rather than `nine orders of magnitude' (the conversion factor is exactly 0.4)...? > > For example, is that Finnish lab that got 100 picoKelvin, really the > > coldest place in the universe? > > As David Makin points out, we have no idea what aliens may have done. Agreed. Can we be sure that no natural phenomena can lead to temperatures this low? > > To make a completely crazy scenario, let's say we have a neutron > > star that has planets orbiting it. (It is known from pulsar timings > > that such exist.) Exciting! Do we know whether there are any (preferably habitable) planets orbiting one of the stars (e.g. S2) that orbits Sagittarius A*? That would be an awesome place to live. > An astronomer acquaintance of mine once said there was no mystery > about why the sun's photosphere isn't as hot as its chromosphere. He > said that the photosphere is where most of the cooling in the solar > system is taking place! It's all in your perspective. Seen from > outside the sun, the photosphere is a source of heat. But seen from > outside a refrigerator, the refrigerator is a source of heat too. That's an excellent point. > (How to use your freezer as an impromptu heat pump: Make lots of ice. > Throw the ice outdoors. You'll be heating your house with greater than > 100% efficiency!) Indeed. Commercial heat pumps can achieve about 600% efficiency, if I remember correctly. Sincerely, Adam P. Goucher

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[math-fun] coldest
by Warren D Smith 13 Dec '13

13 Dec '13

1. sorry, Mars atmospheric pressure is 0.6% of Earth's (says other page http://en.wikipedia.org/wiki/Atmosphere_of_Mars 2. re the claim supernova gas cloud can cool to microkelvins, a simpler test that might be possible to perform, is set off a chunk of chemical explosive in space. Will resulting gas cloud cool to below 1 kelvin? Maybe.

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[math-fun] record coldest phase transition
by Warren D Smith 11 Dec '13

11 Dec '13

>From: "Cordwell, William R" <wrcordw(a)sandia.gov> >Don't know if you want to count this: "During the 1970s, David Lee, Douglas Osheroff and Robert Coleman Richardson discovered two phase transitions along the melting curve, which were soon realized to be the two superfluid phases of helium-3.[21][22] The transition to a superfluid occurs at 2.491 millikelvins (i.e., 0.002491 K) on the melting curve. They were awarded the 1996 Nobel Prize in Physics for their discovery." http://en.wikipedia.org/wiki/Helium-3 >From Veit Elser: In 1991 a Finnish group (Hakonen et al.) discovered spontaneous ordering among the nuclear spins in silver at 560 picokelvins. That may still be the record, for observed phase transitions in ordinary bulk matter. --yes, excellent. Helium-3 at 2491 microkelvins did not beat my earlier record at 325. The 560 picoK transition does, but I actually am not thrilled with it because I forgot to say I'm interested in atomic physics for this purpose, whereas this depended on nuclear physics. The point I wanted to use this record to make, was this. Naively speaking, the atomic physics energy scale is the Rydberg 13.6 eV energy, corresponding to T=105000 kelvin if E=(3/2)*kB*T. So a naive person might figure that any matter phase transitions are going to happen at about 100,000 kelvin. In fact they happen in at least some cases at temperatures 485,000,000 times colder. So the naive estimate can be quite wrong. My question is: just how wrong it can be. Is there any limit to the wrongness? Personally I doubt any limit is in sight. My rhodium superconductivity transition at 325 microkelvin presumably could be brought lower using some appropriate alloy of rhodium and something else, adjusted to optimize (pessimize?) it. On 12/11/13, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: coldest place on earth (Eugene Salamin) > 2. Re: coldest (David Makin) > 3. Re: [EXTERNAL] coldest (Cordwell, William R) > 4. Re: [EXTERNAL] coldest (Veit Elser) > 5. coldest place, CO2 retraction (Warren D Smith) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Tue, 10 Dec 2013 18:53:13 -0800 (PST) > From: Eugene Salamin <gene_salamin(a)yahoo.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] coldest place on earth > Message-ID: > <1386730393.29707.YahooMailNeo(a)web162105.mail.bf1.yahoo.com> > Content-Type: text/plain; charset=iso-8859-1 > > > > > >>________________________________ >> From: Keith F. Lynch <kfl(a)KeithLynch.net> >>To: math-fun(a)mailman.xmission.com >>Sent: Tuesday, December 10, 2013 5:46 PM >>Subject: Re: [math-fun] coldest place on earth >> >> >>Warren D Smith <warren.wds(a)gmail.com> wrote: >>> The interesting thing about this is that dry ice (froen carbon dioxide) >>> forms at -78.5C (-109.3F) at Earth atmospheric pressures. >>> Mars has a carbon dioxide seasonal "ice" cap. >>> It would now appear that the Earth does too. >> >>No.? That temperature is when CO2 frost will form with a partial >>pressure of CO2 of one atmosphere.? But of course only about 1/2500th >>of the air is CO2.? So if someone dropped a block of dry ice in that >>Antarctic valley, it would have sublimated, just like it would in your >>back yard.? Well, okay, maybe more slowly, but it certainly wouldn't >>have grown, much less appeared spontaneously. >> >>Note that the partial pressure of CO2 on Mars is much higher than on >>Earth, since its atmosphere is 95% CO2.? In fact, if you neglected to >>pump the Mars air out of a room on Mars before filling it with Earth >>air, it would exceed the OSHA limit for human CO2 exposure.? If you >>teleported Mars's CO2 ice cap to Earth, keeping its temperature >>unchanged, it would quickly disappear. >> >>Eugene Salamin <gene_salamin(a)yahoo.com> wrote: >>> All of space is filled with the 2.7 K cosmic microwave background.? >> >>True.? That means an inactive object in space far from any other >>source of heat will (eventually) be at that temperature.? An active >>object can of course be warmer. >> >>> It requires shielding and active refrigeration to cool something to >>> a lower temperature. >> >>No.? An active object can also be *colder*.? Adiabatic expansion >>can turn a megakelvin gas cloud from a supernova explosion into a >>microkelvin gas cloud.? Atoms in the same vicinity within such a cloud >>started with the same speed and direction, so their relative speeds >>are much lower than in a 2.7 kelvin cloud.? Of course the cloud will >>eventually absorb the microwaves and warm up, but that could take >>a very long time, as a thin neutral gas is very transparent to >>microwaves and to almost everything else. >> >>Spin temperatures, color temperatures, and information temperatures >>are metaphorical when taken out of context.? As Eugene Salamin pointed >>out, flipping a magnetic field can result in spin temperatures which >>are either negative or hotter than infinity, depending on your >>perspective.? But I don't think they can be used to heat or cool your >>home.? Nor can a microwave oven be used to freeze food, even though >>its radiation has a color temperature of less than one kelvin.? Nor >>can you start a fire with an x-ray machine even though its radiation >>has a color temperature of millions of kelvins. >> >>As for information temperature, would you get more heat by burning a >>deck of punched cards containing the first million digits of pi or by >>burning a deck of punched cards containing a million random digits? >>(This is actually a surprisingly subtle question if you analyze it in >>enough detail.? Can Maxwell's Demon calculate pi?? And does it have a >>Szilard engine to play with?) >> >>Answer:? You get the same amount of heat, since each digit on a punch card >> (in IBM format) removes 1 chad. >>? --? Gene >> > > ------------------------------ > > Message: 2 > Date: Wed, 11 Dec 2013 03:02:35 +0000 > From: David Makin <makinmagic(a)tiscali.co.uk> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] coldest > Message-ID: <D3E87CAC-C3D4-4E43-945B-1151DA990598(a)tiscali.co.uk> > Content-Type: text/plain; charset=us-ascii > > > On 10 Dec 2013, at 23:22, Warren D Smith wrote: >> >> (Was this really the coldest temperature in the universe? Perhaps there >> are natural refrigerators out there somewhere?) > > > Or more sophisticated alien ones.... > > ------------------------------ > > Message: 3 > Date: Wed, 11 Dec 2013 14:17:24 +0000 > From: "Cordwell, William R" <wrcordw(a)sandia.gov> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] [EXTERNAL] coldest > Message-ID: > <D7A6AA7EB875854FB67699AAF1F5F0344011C4B5(a)EXMB03.srn.sandia.gov> > Content-Type: text/plain; charset="us-ascii" > > Don't know if you want to count this: > > "During the 1970s, David Lee, Douglas Osheroff and Robert Coleman Richardson > discovered two phase transitions along the melting curve, which were soon > realized to be the two superfluid phases of helium-3.[21][22] The transition > to a superfluid occurs at 2.491 millikelvins (i.e., 0.002491 K) on the > melting curve. They were awarded the 1996 Nobel Prize in Physics for their > discovery." > > http://en.wikipedia.org/wiki/Helium-3 > > -----Original Message----- > From: math-fun-bounces(a)mailman.xmission.com > [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Warren D Smith > Sent: Tuesday, December 10, 2013 4:23 PM > To: math-fun(a)mailman.xmission.com > Subject: [EXTERNAL] [math-fun] coldest > >>This is the coldest place on Earth under natural conditions.? The coldest >> place on >Earth was in a physics experiment at a temperature of 50 nK.? >> This was touted as the >coldest place in the universe. > > --In year 2000 a Finnish lab cooled a rhodium metal chunk to 100 > picoKelvin.. > http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures > (Was this really the coldest temperature in the universe? Perhaps there are > natural refrigerators out there somewhere?) > > I am interested in the lowest temperature at which a physical system is > known to undergo a phase transition. For example helium liquefying at > 4.2K. > Rhodium supposedly becomes superconductor at 325 microkelvins for example; > that is much lower. I'd be interested in any lower-T candidates you > can name (even speculative ones not yet demonstrated in lab). > > A Bose-Einstein condensate gas phase consisting of approximately 2000 > rubidium-87 atoms was made via laser cooling and trapping below 170 nanoK > effective temperature, but I don't know if we should admit that, since it > really was a highly unnatural form of "matter." > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > ------------------------------ > > Message: 4 > Date: Wed, 11 Dec 2013 14:46:37 +0000 > From: Veit Elser <ve10(a)cornell.edu> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] [EXTERNAL] coldest > Message-ID: <7429BC1A-9115-4212-B365-F2BA113D8A30(a)cornell.edu> > Content-Type: text/plain; charset="us-ascii" > > In 1991 a Finnish group (Hakonen et al.) discovered spontaneous ordering > among the nuclear spins in silver at 560 picokelvins. That may still be the > record, for observed phase transitions in ordinary bulk matter. > > -Veit > > On Dec 11, 2013, at 9:17 AM, "Cordwell, William R" <wrcordw(a)sandia.gov> > wrote: > >> Don't know if you want to count this: >> >> "During the 1970s, David Lee, Douglas Osheroff and Robert Coleman >> Richardson discovered two phase transitions along the melting curve, which >> were soon realized to be the two superfluid phases of helium-3.[21][22] >> The transition to a superfluid occurs at 2.491 millikelvins (i.e., >> 0.002491 K) on the melting curve. They were awarded the 1996 Nobel Prize >> in Physics for their discovery." > > > > > ------------------------------ > > Message: 5 > Date: Wed, 11 Dec 2013 10:16:22 -0500 > From: Warren D Smith <warren.wds(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] coldest place, CO2 retraction > Message-ID: > <CAAJP7Y0xkX6Fy2CHJW4pyVdiNjFW7hvACxa_bGQhpcCYK_LVkg(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > >> Warren D Smith <warren.wds(a)gmail.com> wrote: >>> The interesting thing about this is that dry ice (froen carbon dioxide) >>> forms at -78.5C (-109.3F) at Earth atmospheric pressures. >>> Mars has a carbon dioxide seasonal "ice" cap. >>> It would now appear that the Earth does too. >> >>From: "Keith F. Lynch" <kfl(a)KeithLynch.net> >> No. That temperature is when CO2 frost will form with a partial >> pressure of CO2 of one atmosphere. But of course only about 1/2500th >> of the air is CO2. So if someone dropped a block of dry ice in that >> Antarctic valley, it would have sublimated, just like it would in your >> back yard. Well, okay, maybe more slowly, but it certainly wouldn't >> have grown, much less appeared spontaneously. > > --aha yes, good point... BUT, what we need to know, then, is "what is > the sublimation temperature of dry ice at 1/2500 atm CO2 gas > pressure?" > The phase diagram of CO2 is here > http://www.teamonslaught.fsnet.co.uk/co2%20phase%20diagram.GIF > and a better diagram for us since it uses a log scale for pressure, is > http://www.cgenpower.com/images/other/pop-up/fasendiagram.jpg > and it would appear from extrapolating the latter diagram, that to get > stable solid > CO2 at 1/2500 bar of vapor pressure, you'd need to be at temperature of > about > minus 150 celsius. The antarctic record of minus 94 C was not cold enough. > > So, sorry for my wrong claim; I now agree with Lynch. > >> Note that the partial pressure of CO2 on Mars is much higher than on >> Earth, since its atmosphere is 95% CO2. > > --the atmospheric pressure on Mars allegedly is about 8.7% of Earth's > so again I agree with Lynch. > >>> It requires shielding and active refrigeration to cool something to >>> a lower temperature than 2.7K. >> >> No. An active object can also be *colder*. Adiabatic expansion >> can turn a megakelvin gas cloud from a supernova explosion into a >> microkelvin gas cloud. Atoms in the same vicinity within such a cloud >> started with the same speed and direction, so their relative speeds >> are much lower than in a 2.7 kelvin cloud. Of course the cloud will >> eventually absorb the microwaves and warm up, but that could take >> a very long time, as a thin neutral gas is very transparent to >> microwaves and to almost everything else. > > --really? I mean, is there really a gas cloud from some supernova > that is microkelvin? > Or is this just a speculation? I agree with Lynch on the general > principle that an active object could get below 2.7K, but does it > actually happen someplace? > For example, is that Finnish lab that got 100 picoKelvin, really the > coldest place in the universe? > > To make a completely crazy scenario, let's say we have a neutron star > that has planets orbiting it. (It is known from pulsar timings that > such exist.) After a long time, the > planet gets cold, approaching 2.7K. Now further suppose that the > planet is totally magnetized by the enormous magnetic field of the > neutron star, which far exceeds any magnetic field we can make on > Earth. Now a rogue planet swoops > in from outer space and gravitationally interacts with our planet, > causing it to escape from neutron star. As a result, it experiences > demagnetization cooling. > Maybe in this way temperatures way below 2.7K could be reached naturally. > I'm dubious, but it might be possible. > > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 130, Issue 10 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] coldest place on earth
by Henry Baker 11 Dec '13

11 Dec '13

In theory, it should indeed be possible to measure the actual "temperature" inside the "computer" degrees of freedom. What would we need? 1. We'd need an exquisitely sensitive thermometer with an extremely small heat capacity. This is because almost every real thermometer would be Avogadro-sized in comparison to the relatively small number of bits in our computer. Thus, the heat from the thermometer itself would overwhelm the relatively small heat capacity of our "computer" degrees of freedom. 2. We'd need a way to put the thermometer "in contact" with the internal "computer" degrees of freedom. This will be difficult, since computers are well-designed to keep these internal degrees of freedom well-insulated from the "outside". 3. Our thermometer itself would have to be very well insulated from the rest of the world, since we are trying to measure a temperature lower than the rest of the world; we don't want the thermometer itself to be a conduit for heat flow into the "computer" degrees of freedom. Due to these difficulties, it is simpler to calculate/estimate the actual "temperature". At 01:22 PM 12/10/2013, Eugene Salamin wrote: >You seem to be using the term "temperature" in a different sense from >"that which is measured with a thermometer". All computers that I'm >aware of generate heat, and in fact heat removal is the major obstacle >to faster CPU frequency. In your sense, the coldest place on Earth is a > petroglyph, a stone carving.

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[math-fun] coldest place, CO2 retraction
by Warren D Smith 11 Dec '13

11 Dec '13

> Warren D Smith <warren.wds(a)gmail.com> wrote: >> The interesting thing about this is that dry ice (froen carbon dioxide) >> forms at -78.5C (-109.3F) at Earth atmospheric pressures. >> Mars has a carbon dioxide seasonal "ice" cap. >> It would now appear that the Earth does too. > >From: "Keith F. Lynch" <kfl(a)KeithLynch.net> > No. That temperature is when CO2 frost will form with a partial > pressure of CO2 of one atmosphere. But of course only about 1/2500th > of the air is CO2. So if someone dropped a block of dry ice in that > Antarctic valley, it would have sublimated, just like it would in your > back yard. Well, okay, maybe more slowly, but it certainly wouldn't > have grown, much less appeared spontaneously. --aha yes, good point... BUT, what we need to know, then, is "what is the sublimation temperature of dry ice at 1/2500 atm CO2 gas pressure?" The phase diagram of CO2 is here http://www.teamonslaught.fsnet.co.uk/co2%20phase%20diagram.GIF and a better diagram for us since it uses a log scale for pressure, is http://www.cgenpower.com/images/other/pop-up/fasendiagram.jpg and it would appear from extrapolating the latter diagram, that to get stable solid CO2 at 1/2500 bar of vapor pressure, you'd need to be at temperature of about minus 150 celsius. The antarctic record of minus 94 C was not cold enough. So, sorry for my wrong claim; I now agree with Lynch. > Note that the partial pressure of CO2 on Mars is much higher than on > Earth, since its atmosphere is 95% CO2. --the atmospheric pressure on Mars allegedly is about 8.7% of Earth's so again I agree with Lynch. >> It requires shielding and active refrigeration to cool something to >> a lower temperature than 2.7K. > > No. An active object can also be *colder*. Adiabatic expansion > can turn a megakelvin gas cloud from a supernova explosion into a > microkelvin gas cloud. Atoms in the same vicinity within such a cloud > started with the same speed and direction, so their relative speeds > are much lower than in a 2.7 kelvin cloud. Of course the cloud will > eventually absorb the microwaves and warm up, but that could take > a very long time, as a thin neutral gas is very transparent to > microwaves and to almost everything else. --really? I mean, is there really a gas cloud from some supernova that is microkelvin? Or is this just a speculation? I agree with Lynch on the general principle that an active object could get below 2.7K, but does it actually happen someplace? For example, is that Finnish lab that got 100 picoKelvin, really the coldest place in the universe? To make a completely crazy scenario, let's say we have a neutron star that has planets orbiting it. (It is known from pulsar timings that such exist.) After a long time, the planet gets cold, approaching 2.7K. Now further suppose that the planet is totally magnetized by the enormous magnetic field of the neutron star, which far exceeds any magnetic field we can make on Earth. Now a rogue planet swoops in from outer space and gravitationally interacts with our planet, causing it to escape from neutron star. As a result, it experiences demagnetization cooling. Maybe in this way temperatures way below 2.7K could be reached naturally. I'm dubious, but it might be possible.

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[math-fun] coldest
by Warren D Smith 11 Dec '13

11 Dec '13

>This is the coldest place on Earth under natural conditions.? The coldest place on >Earth was in a physics experiment at a temperature of 50 nK.? This was touted as the >coldest place in the universe. --In year 2000 a Finnish lab cooled a rhodium metal chunk to 100 picoKelvin.. http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures (Was this really the coldest temperature in the universe? Perhaps there are natural refrigerators out there somewhere?) I am interested in the lowest temperature at which a physical system is known to undergo a phase transition. For example helium liquefying at 4.2K. Rhodium supposedly becomes superconductor at 325 microkelvins for example; that is much lower. I'd be interested in any lower-T candidates you can name (even speculative ones not yet demonstrated in lab). A Bose-Einstein condensate gas phase consisting of approximately 2000 rubidium-87 atoms was made via laser cooling and trapping below 170 nanoK effective temperature, but I don't know if we should admit that, since it really was a highly unnatural form of "matter."

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