Aerobic and Anaerobic Metabolism: A Symmetry

The symmetry between aerobic carbohydrate respiration and photosynthesis is widely appreciated. In the presence of Oxygen, many living things (including photosynthesizers) burn carbohydrates, producing useful energy and CO2. Photosynthesizing creatures, using the energy of sunlight, manufacture carbohydrates from CO2 for their own use; and Oxygen. By outrageous good fortune, photosynthesis has been so wildly successful that has produced a surplus of carbohydrate and Oxygen sufficient to balance the evolution of hungry multicellular respirers; like ourselves.

In physics, this is called a symmetry, a conserved quantity. The reason the quantity is conserved is unclear and taken on faith as a fundamental property of the universe. For fun, we go out on a limb here and propose that physics can take a lesson from biology.

The basic idea of conserved quantities is that they remain constant amid the mayhem around them. When we say energy or baryon number is conserved, we mean only that at the end of a long day of collisions, phase transitions, etc., we get the same quantity we started out with. If we wish to take the lesson from biology, the continuous symmetry of aerobic respiration and photosynthesis means that all the powerful disruptive forces continue cancel out, leaving the relationship the same within the bounds of inevitable fluctuation.

Less well appreciated than the symmetry of aerobic respiration and photosynthesis, is the symmetry of aerobic and anaerobic metabolism. In the guts of hungry multicellular respirers from ourselves to termites, beneath the water surfaces, and as deep in the crust as we have yet drilled; lives a vast anaerobic world.

The vast anaerobic world is ancient and mostly microscopic Archaea, the first living things. The anaerobic world developed some internal symmetries of its own during a billion years or more of evolution before cyanobacteria got around to producing much Oxygen.

The primary modes of anaerobic metabolism are methanogenesis and fermentation. Fermentation produces CO2 and alcohol. Alcohol itself is valuable low entropy carbon (carbohydrate) used by other critters. Methanogenesis can occur  from carbohydrate, producing methane and CO2; or it can combine CO2 and Hydrogen, producing methane and water.

A few arcane critters use methane anaerobically, but methane from CO2 is essentially the final step in anaerobic metabolism. The resulting methane is waste to the Archaea, but an important gift of low entropy Carbon to the aerobic world.

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Cold Smoke

Santa Rosa is comfortably away from the fires this year. Last year’s baptism cleaned out the tinder around here. Yet the Camp Fire has given us the peculiar combination of dense smoke and hard frost.

A lot of the particles in wildfire smoke are about the size of the wavelength of visible light, .4 to .7 microns, so they block sunlight effectively, but are relatively transparent to the long wavelength radiation to space from the surface that fosters frost on otherwise dry, clear nights.

Dust masks are designed to filter out particles larger than .3 microns, but this is the average size of the particles in smoke, so the only filter about half of the particles, and the smaller particles they do not filter are considered more dangerous. Most people I have seen have not even bothered to pinch the masks around their noses, which lets all of the particles in. The air would much rather go around than through the mask.

They closed the schools. More of the foolishness that led to the closure of schools for weeks after the fires here last year. Kids live in houses subject to the same conditions as the schools, and schools are better places to reduce activity levels for many kids with working parents. These kids are just turned loose when schools close.

Like cold smoke, foolishness and superstition seems to drive much human behavior.

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The Flat Universe Society

While it is clear that the earth is sort of round, rather like a baseball that has absorbed too many home runs, to the best of our understanding, the universe is flat.

By flat, we do not mean that you could walk to the edge and fall off, like pre Columbian notions of the earth. Rather, we mean that despite all of Mr. Einstein’s distortions of spacetime, at the largest scales we can manage, the universe follows the geometry of Euclid drawn flat in the sand.

If the old Greek guys had done their geometry on a sphere or a saddle (ignoring the impracticality of sand on these shapes for the sake of argument), the sum of angles in a triangle would not be 180 degrees. The same is true for any surface not perfectly flat.

Astronomers measure the distances to stars and such and can calculate the angles of triangles connecting them. Always 180 degrees.

This seems as peculiar as energy being a function of the speed of light squared.

Nevertheless, the flatness of the universe has become as embedded in our cosmology as pre Columbian notions of the earth. The critical density to achieve flatness is the cornerstone of many equations.

If you could fall off the edge of the universe, would you have found the multiverse?

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Energy and Matter

An interesting upshot of Mr. Einstein’s famous equation is that in units of the speed of light, energy and mass are equal. Mr. Poincare once observed that mathematics is the exercise of making different things the same (equal), and sophistry above with the square of the speed of light might be just the sort of trick he was referring to.

Nevertheless, the human concept of squareness derives from carefully drawn figures in the sand, subdivided and counted. A square divided in two each way yields four smaller squares, so it is natural to say that four is two “squared”. In this frame of reference, when you don’t subdivide your square at all, each side remains one, and you have but one square. That one squared equals one is unassailable.

We have noted before that equating energy to matter times the speed of light squared seems peculiar. Here we explore an alternate notion that the huge asymmetry between energy and mass in units smaller than the speed of light, disappearing to unity at the speed of light, is precisely why the equation works.

Mr. Einstein did not know about fermions. Fermions are subatomic particles that take up space, have mass, and constitute the matter in the universe. Electrons, quarks, and the triplets of quarks we call protons and neutrons are fermions.

Mr. Einstein knew about photons. He was instrumental in their discovery. Photons have no mass, take up no space, represent force rather than matter, and travel at the speed of light. In our standard model photons are bosons, and we see the difference between fermions and bosons as a fundamental division of the universe.

Complicating the picture, the divide between fermions and bosons is not about mass. The divide is about the ability (even propensity) to condense into or occupy the same space. Many bosons have mass derived from Mr. Einstein’s equation because they do not travel at the speed of light. Of the bosons, only photons, gluons (carriers of the force holding quarks into protons and neutrons), and the (as yet) hypothetical gravitons have no mass and travel at the speed of light.

This brings us to an important distinction between mass and matter, often loosely interchanged as the “m” in E=mc^2. All matter has mass, but not all mass is matter. While we can write E=f(ermions)c^2 and the expression is true, we cannot write E=mb(osons)^2.

If we use units of the speed of light, in some sense we pose our subject energy, mass, and matter to be travelling at the speed of light. This is not possible, as mass and matter cannot travel at the speed of light. There is still something the matter with our conception.

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Entropy and Watermelons

What is the entropy of a watermelon? This question is not entirely fair, because living things are singularities swimming upstream against the tide of entropy. We can’t merely count the ways the invisible molecules in a watermelon can be rearranged and still be a watermelon. We don’t even know all of the molecules in a watermelon.

We do know that a watermelon is usually over 90% water. We can weigh the watermelon and calculate the number of water molecules with 90% accuracy. We can then take the ideal gas constant and divide it by Avogadro’s number (this is the Boltzmann constant), multiply this by the log of the enormous number of ways we could exchange water molecules throughout the watermelon without changing its appearance, and derive a number for the entropy of 90% of the watermelon. This would be foolishness, because the reason a watermelon can swim upstream against the tide of entropy to exist at all is a result of information; the information in its DNA. Information can create singularities of negative entropy.

A watermelon is not a gas. Most gasses are invisible to us even at the macroscopic levels we chunk their molecules up, like pressure, temperature, and wind. Our conceptions of thermodynamics were really developed for steam engines, machines designed to extract work from the disequilibrium of pressurized gas. The molecules in a gas want to spread out. The molecules in a watermelon do not, at least not as quickly. The watermelon seems content, as if it has reached some (temporary) quantum of negative entropy equilibrium.

Steam is the gas phase of 90% of a watermelon. Not the steam we think we can see, that is actually liquid water that has condensed from the invisible gas phase. Most of the water in a watermelon is liquid or bound up in solid organic molecules. Solids, by definition, do now want to spread out as much as liquids or gasses.  The very property of being solid constrains the spread. The information coding for particularly the solid rind of the watermelon temporarily defeats the tendency to spread out.

Boltzmann entropy was the intellectual beginning of the notion that a probability field extends through the universe from the point human perception fails. The existence of the watermelon in negative probability phase space makes the value of W (the number of ways the invisible components can be rearranged without perceptible difference) negative. The fundamental reason for this negative entropy is information. Information implies purpose.

Entropy is perplexing because while we easily understand why it increases toward the future, since all the fundamental laws of physics seem reversible, entropy should also increase toward the past. Boltzmann wrote this off to probability, simply maintaining as a brute fact that probability flows to the future. Yet probability fails to explain why an entire universe should evolve with billions of galaxies each containing billions of stars, just to get to a human brain. It is far more likely that a human brain, whose perceptions reputedly define the line between the classical and the probabilistic, would randomly fluctuate into existence.

So while we can derive a number for the entropy of a watermelon based on the statistical mechanical properties of the high proportion of water it contains, we really have no way to evaluate the probability of the information it contains. In some sense all living things are watermelons. In some sense our planet is a watermelon.

 

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God and Dice

We don’t spend much time thinking about what God does for recreation these days, but the great scientists from Newton to Einstein who have framed our Western cosmology were often deeply religious men. It is understandable that when they found three term equations that explained nearly everything, they believed these were insights to the mind of God.

1/r^2 Newton’s inverse square diminution of gravity.

F=ma Newton’s second law. Describes the relations between force, mass, and acceleration.

S=klogW Boltzmann’s formula for entropy.

E=hv Planck-Einstein energy of a photon, Planck’s Constant x  frequency.

E=mc^2 Einstein’s General Relativity. In units of the speed of light, energy=mass.

These are a few famous examples, and except Boltzmann, they are classical mechanics. In classical mechanics there is no need for probability, and there is no margin of error (except in measurement). The rules are absolute. Space is a field, or reference frame with fixed Cartesian coordinates, x,y, and z. Even today, you need nothing but classical mechanics to send a rocket to the moon. Except for a small deviation in the orbit of Mercury, classical mechanics describes the motions of the planets in our solar system.

The flamboyant Monsieur Laplace seems to have been the first to observe that using Newton’s laws, if you knew the position and momentum of every particle in the universe in the present moment, you could extend this information indefinitely into both the past and the future. You would know everything there was to know. (Momentum is a vector quantity that includes direction). Asked by Napoleon why his work contained no mention of God, he replied that he had no need.

In Special Relativity Einstein argued the concept of spacetime, where every xyz coordinate becomes a clock that is inseparable from the point. He showed that in a curved trajectory different observers would record different amounts of time passing. Space and time are relative.

In General Relativity Einstein argued that spacetime itself is warped by gravity, and that not only space and time, but energy and mass are relative. The only constant is the speed of light.

Newton’s and Einstein’s theories are classical, there is no probability, no statistics, and no dice. The title of this post refers to a quote from Albert Einstein made in a letter discussing statistical mechanics with Max Born. He said (in German),  that he believed God does not play dice with the universe. We will argue in favor of Einstein in this regard. We will argue that Boltzmann’s introduction of probability in his study of entropy set the stage for probabilistic notions in quantum mechanics. We will argue that both probabilistic approaches founder on an anthropocentric philosophical error.

It is ironic that the probabilistic approaches to entropy and quantum mechanics both begin with a wildly improbable assumption: that what humans can perceive somehow matters at a fundamental level in the universe. The basic problem is that the nanoscale world of molecules, atoms, and sub-atomic particles is largely invisible to us.

For Boltzmann, the solution was to count the ways the invisible components can be arranged and remain visibly the same. Entropy becomes the log of this number of ways the invisible components can be arranged (w) multiplied by Boltzmann’s Constant (k); which is the ideal gas constant divided by Avogadro’s number. Boltzmann’s insight was that the behavior of gasses well understood at a practical level by engineers of the industrial revolution, was dictated by the interactions of the invisible molecules and atoms.

We can see above that it works very well for the density of the earth’s atmosphere. The molar concentrations of the invisible (to us) gasses in the atmosphere are well-known. If we suddenly developed the ability to see the different gasses, and they became distinguishable to us, nothing would change. Entropy here is the separation between the molecules in the air, density–regardless if we can see the molecules or not. Is it simply more probable that the spaces between molecules seek a higher entropy equilibrium as pressure is relieved, or is it certain they will do so?

Probability is a tool for the blind. If we can see it, it is classical; if not, we would have God play dice. We are blind at the scale of particles, atoms, and most molecules. Quantum mechanics is fundamentally the notion that the energy of electrons is not a smooth linear progression, but rather a stairway with discrete intervals or steps; quanta. Our efforts to locate electrons and other subatomic particles are hindered by the unfortunate circumstance that when we ping them we unavoidably alter them. Werner Heisenberg formulated this as his uncertainty principle. We can know either the positions or  momenta of invisible particles, but not both. This quandary evolved into the probabilistic notion of a wave function, where waves of probability are our only information until we collapse the wave to a particle by pinging it. It is even maintained that there is no such thing as where the particle actually is until it is observed.

Can you imagine anything more anthropocentric? If a tree falls in a forest and nobody hears it, did it really fall? Of course it did. It is not a matter of phase space between probability waves. It is easy to go and see many fallen trees that very likely nobody heard. Trees fall and  particles have positions and momenta regardless the perceptions of naked apes, or even – aliens capable of somehow garnering information on both position and momentum.

Take a macroscopic object, a watermelon. They differ somewhat in color, size, and skin pattern. These differences arise from somewhat different arrangements of components, but people will generally agree that a watermelon is a watermelon. A watermelon is composed of all manner of quarks held together in Protons and Neutrons by the strong nuclear force. These may be surrounded by electrons in quantized shells held to the protons by the electromagnetic force (and to both neutrons and protons weakly by gravity). These atoms may be bound to others by ionic or covalent bonds in complex molecules. Add these layers up and you sometimes get a watermelon.

When we observe a watermelon, we change the universe. Photons of visible light reflect from the surface of the watermelon and our retinas absorb these photons, which otherwise would have travelled to a different fate. Did the watermelon exist before we observed it? Did the photons exist before our retinas absorbed them, or were both the watermelon and the photons mere probability waves before we intervened? When we turn our back to the watermelon, our eyes no longer absorb the photons. Some may be reflected from our clothing, and others may be absorbed by our skin without sending information about the watermelon to our brain. We change the universe even when we don’t observe the watermelon, but the watermelon is still there.

Imagine that our vision deteriorates and we can no longer see individual watermelons, but can barely discern a large pile of watermelons. We will have lost a lot of information. Our degraded retinas absorb a different set of photons.We could then argue that the positions of the watermelons in the stack is a probability function, and there is no such thing as exactly where a watermelon is in the pile until we reach in blindly and grab one. This would be foolishness, but it is exactly what the probabilistic  Copenhagen Interpretation of quantum mechanics would have us do. If the pile of watermelons is subject to gravity and forms a cone, we could even develop formulas that determine the probability of grabbing a watermelon as being much higher when we reach blindly toward the bottom of the pile rather than the top.

By having God play dice, we elevate probability to the status of a force of nature, joining the strong, weak, electromagnetic, and gravitational forces. The field and waves are established in the form of the wave function. A probability particle can’t be far behind.

 

 

 

 

 

 

 

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Entropy and the Grand Canyon II

We left off in the last post having boated 78 miles, about five days on a typical trip, and having encountered the last of the suite of three books that comprise the geological story revealed in the Grand Canyon. We also concluded that the fluctuations in the altitudes of land and sea evident in the story appears to be the breathing of a great beast deep within our planet.

In the 78 miles we have navigated 16 rapids with holes big enough to turn our 18′ rafts upside down. These holes result from the water’s defiance of gravity and momentum to curl back upstream to fill voids in the flow when the water is forced to pass around rocks. It curls back upstream to reduce the gradient in potential energy from the disturbance of the water’s cohesive force, and increase the entropy.

Fortunately, age and experience can influence luck, and this was our first trip where nobody was even out of the boats in a rapid.

The Vishnu metamorphic complex brings us to a really staggering unconformity, called the Great Unconformity. We may recall that the Tapeats sandstone began to be deposited about  545 mya, just slightly before the Cambrian period, the first period of the Paleozoic era 540 mya.

The Tapeats was deposited variously on top of late Proterozoic Super Group members where they occur, and on top of  the early Proterozoic Vishnu complex where the Supergroup is absent in much of the canyon. We may also recall that the Vishnu complex includes both highly metamorphosed sedimentary rocks and magmatic plutons which injected and overprinted any age information in the sediments. Our best overall dates for last crystallization are 1.7 billion years ago.

Where the Tapeats sits on the Vishnu, a little arithmetic reveals that the Great Unconformity spans 1.16 billion years. This is almost exactly 1/4 of the time since our planet is thought to have coalesced from stardust 4.6 billion years ago.

Above is the unconformity in Blacktail Canyon. The bedding within the Tapeats above the unconformity is highlighted by the different lime content and erosion resistance. If you look carefully, you will see some very thin bedding just above the contact, that based on color, seems to be erosion from the Vishnu. These redder bands alternate with lighter material from a different source, possibly the same source as the lighter bedding seen generally in the Tapeats above.

Really though, the Great Unconformity between the Tapeats and Vishnu overstates the case a bit. The Supergroup is absent here, but it exists elsewhere in the Grand Canyon. The Vishnu was folded nearly vertical at some point. The Supergroup was tilted against the Vishnu at some point, and both were eroded to a nearly flat surface during the Great Unconformity. This flat surface eventually received the Tapeats, and the Tapeats therefore overlies the Vishnu and each of the tilted layers of the Supergroup in turn.

Above we show a more fair representation of the Grand Canyon unconformities. These were assembled using date ranges for members from Wikipedia and Macrostrat. The two sources do not always agree, so a measure of our own judgement was required. This cannot be considered the last word, but the general trend for longer unconformities the further we go back in time is clear.

It may be that as you go back in time, the probability of having had a long unconformity that erases a lot of little ones increases. Or it may be that the breathing of the great beast has sped up.

Ludwig Boltzmann was the pioneer of statistical mechanics, a notion that although some important things are unknowable to us, like the position and velocity of every atom in a watermelon; we are not entirely powerless. He showed us that in situations where we can count the possible ways the atoms could be rearranged and still appear to be a watermelon, we can calculate the probability of a watermelon. Unfortunately, neither watermelons nor great breathing beasts are among these situations.

Unconformities are the most fundamental and largest scale changes evident in the Grand Canyon, but they are by no means alone. Every layer shows some level of fluctuation between more marine, calcerous, well sorted periods; and more terrestrial, silicious, and poorly sorted periods.

More Resistant Lighter Layers Form Cavern “Cieling”

We saw the fluctuations in the Redwall above in the last post.

Muav Cycles, ~1,000 year Clastic Intervals

Above is some detail of the Cambrian Muav from National Canyon. If you measure the average distance between the bands and factor the section thickness and duration of the Muav you can calculate the intervals. This is a crude exercise. It can tell you it is probably not  100,000 years, and probably not 10 years.

One Chock too many for wet feet

Above is some context of the finely bedded Muav.

If you measure the spacing of these fluctuations within the Muav, or the Redwall or any other large scale stratum, you discover that despite the appearance of regularity, the spacing is never exactly the same.

One could take a section of aside creek like the one above that cuts through only the Muav, and imagine it as a miniature Grand Canyon. Sequences could be identified , grouped, and dated, and possibly unconformities could be found. This has been done for the more wildly varying members like the Supai.

Ultimately our groupings, and how sharply we focus our scope becomes somewhat arbitrary. It is our nature to rationalize and categorize, but at some point we are always left like Ludwig Boltzmann; trying to count the possible ways this all could have happened to assign a probability, when all we can see is the apparent result a great breathing beast.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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