NON-EQUILIBRIUM WORLDS
Jean Pierre PETIT – Former Research Director – CNRS FR.
January 12, 2013
When the average person thinks about the equilibrium of a system, they usually picture a ball resting at the bottom of a well, or something similar.
The fundamentals of thermodynamic equilibrium theory contain something more subtle: dynamic equilibrium. The simplest example is the air we breathe. Its molecules are jostled in every direction, exhibiting a mean thermal velocity of 400 m/s. At a tremendous rate, these molecules collide and interact. These collisions change their speeds. However, the physicist translates this into a statistical steady state (the term used is “detailed balance”). Imagine a goblin who, at any time and at any point in a room, can measure molecular speed in a certain direction, within a slight angular uncertainty. At every time increment, our goblin counts V and V + ΔV, algebraic values. He then plots these values on a graph and observes a nice Gaussian curve growing, with a peak mean value near 400 m/s. The faster or slower the molecules are, the smaller their population.
He repeats this work, aiming his measuring device towards any spatial direction, and, surprise, surprise, gets the same result. Molecular agitation in the room is isotropic. Moreover, nothing can disturb this dynamic equilibrium if the temperature remains constant, because gas temperature is exactly the mean kinetic energy arising from this thermal agitation. The physicist describes this gas as being in thermodynamic equilibrium. This state is multifaceted: air molecules do not have spherical symmetry. Diatomic molecules, such as oxygen or helium, are peanut-shaped. Those of carbon dioxide or water vapor have other shapes. All these objects, when rotating, can store energy like tiny flywheels. These molecules can also vibrate. The concept of energy equipartition states that energy must be distributed equally among all these various “modes.” During a collision, some kinetic energy can be converted into vibrational or rotational energy of a molecule. The reverse is also valid. Then all this is statistics, and our goblin can count how many molecules are in such and such a state, have such kinetic energy, or are in such a vibrating state. Returning to our breathing air, this census leads to the stationary status. This medium is then said to be in thermodynamic equilibrium, namely relaxed. Imagine a wizard with the power to stop these molecules, halt their rotation or vibration, modify them at will, create a new statistical law, deform that beautiful Gaussian curve, or even create some anisotropic event, where, for example, thermal speed in one direction is doubled relative to transverse directions. After all, he would let the system evolve following further collisions. How many of these would be necessary for the system to return towards thermodynamic equilibrium? Answer: very few. The mean free travel time of a molecule, between two collisions, gives an idea of the relaxation time in a gas, of its return time towards thermodynamic equilibrium.
Do non-equilibrium media exist, where molecular statistical speeds deviate notably from this comfortable isotropy and the beauty of these nice Gaussian curves?
Oh yes! And it is even the majority case in the universe. A galaxy, this “island universe,” comprised of many hundreds of billions of stars, whose masses are more or less similar, can be seen as a gaseous medium, in which the molecules are… stars. In this precise case, one discovers a disconcerting world where the mean free travel time of a star, before any encounter with a neighboring star, is ten thousand times the age of the universe. What do we mean by encounter? Should it be a collision where the two stars heavily smash into each other? Not even! In the domain of theoretical physics known as the kinetic theory of gases, we consider a collision when a star’s trajectory is noticeably modified while crossing a neighboring star. However, calculation proves these events are extremely rare, and our system of hundreds of billions of stars can be seen as usually non-colliding.
For billions of years, the trajectory of our Sun has been regular, quasi-circular. If our Sun were self-conscious, provided it did not change pace due to encounters, it would ignore having neighbors. It only senses the gravitational field as “smooth.” It walks along its path as if in a basin, without sensing any bump created by other stars. Immediately, the corollary surfaces: place our goblin, now an astronomer, in the vicinity of the Sun in our Galaxy and ask him to build a speed statistic of neighboring stars in any direction. The obvious fact emerges now. The medium, dynamically speaking, is strongly anisotropic. There exists a direction where the stars’ agitation speeds (called residual speeds by astronomers, relative to the mean rotational drive of the galaxy, quite circular and at 230 km/s near the Sun) are practically two times greater than in any other transverse direction. In our breathing air, this was called a spheroidal speed distribution – now, this becomes an ellipsoidal speed distribution. So far, so good? How does this affect our vision, our understanding of the world? It changes everything! Because by far we cannot deal with theories of such drastically non-equilibrium systems.
Leaving aside the paradoxical status where galaxies stand due to this damned effect of dark matter (missing mass), discovered in 1930 by the American, Swiss-originated Fritz Zwicky, and in any case, we could produce any model of self-gravitating, punctual masses (orbiting in their own gravitational field). Our physics always stands near a state of thermodynamic equilibrium. Obviously, any deviation of this or that represents a deviation from equilibrium, for example, a temperature gap between two gaseous areas, which will lead to heat transfer, a transfer of kinetic energy from thermal agitation. In this case, if we put our goblin back to work, he would conclude that the medium, dynamically speaking, is “almost isotropic.” This will be the case of our atmosphere, even when crossed by the utmost windstorms.
Well then, is it impossible to encounter, “to put our fingers on,” situations where a gaseous medium, a fluid, is frankly out of equilibrium? One will find such occurrences when crossing shock waves. They are limited areas, as precisely the thickness of a shock wave is of the order of magnitude of a small number of mean free paths.
When crossed by a shock wave, a gas switches between states very abruptly, considering that a state near thermodynamic equilibrium, in the “shocked” gas, is recovered after some mean free path times.
We reported an observation, forty years ago, in the laboratory where I worked, now dismantled, the “Institut de Mécanique des Fluides de Marseille.” We had then some sort of gas guns we called “shock tubes.” The outline is: using an explosive, we ignited a shock wave, propagating at several thousand meters/sec in a rare gas – initially, this gas was at some millimeters of mercury pressure. The movement of the shock wave recompressed the gas, increasing its density.
We could easily and precisely follow the increase in density using interferometry. At the time, we also measured the heat flow at the surface of Plexiglas mock-ups. As the experiments lasted only fractions of milliseconds, our measuring devices had to have a fast response time. Precisely, they were metallic films of some microns thickness, vacuum-coated on the wall; they acted as thermistors. We evaluated heat flow by recording the resistance of these wall sensors while they heated.
One day we placed a sensor straight on the tube wall. Then we observed that the heat flow was reaching the sensor after a certain delay following the shock wave passage, materialized by an abrupt density jump. Yet we made sure the thermal lag of the sensor was small enough so that this delay was not coming from it. In fact, we put our finger on a return phenomenon towards a quasi-thermodynamic equilibrium, downstream of the shock wave.
We can compare this to a hammer slam. Not only is the density brutally increased, but we also observed a temperature jump, meaning an increase in the thermal speed of molecules. But behind this wave, isotropy is only seen after several mean free path times. Immediately before the density front, the increase in thermal agitation is translated by movements starting perpendicular to the wave direction.
When our sensor collects heat, this results from the impact of air molecules on its surface. Yet immediately before the density front, over some distance, thermal agitation was developing parallel to the wall. The gas was well “heated” but momentarily unable to transfer this heat to the wall. Over the collisions, the “ellipsoid of speeds” was transforming itself into a “spheroid of speeds,” and the sensor ended up giving back the heat flow it received. I believe I remember, with the experimental setup we had, that we recorded this heat flow near one centimeter before the density front.
So, shock waves represent tiny thickness areas where the gaseous medium is strongly out of equilibrium. How do we manage this? By making these areas equal to zero-thickness surfaces. And this has worked for almost a century.
I am old enough to have known almost all of computer history, from the beginning. When I was a student at the “Ecole Nationale Supérieure de l’Aéronautique,” there was no computer in the house. These were installed inside sanctuaries called “calculation centers” which we could not access. We were calculating using slide rules, curiosity objects for today’s generation. In higher school classes, we all had our logarithm books, and every examination included a boring numerical calculation test using these items, which are now displayed in museums.
When I left Sup Aero School, mechanical calculators (FACIT), hand-powered, were just coming out. To multiply numbers, you turned the crank one way; to divide, you turned it the opposite way.
The professors, or department managers, had electrical machines, which were breaking the silence of the office with their cog noise at the Institut Mécanique des Fluides in 1964. Computers had the place of honor, like distant gods only seen through a window, in these calculation centers. These computers, having the power of a today’s pocket computer, were served by priests in white cassocks. You could only communicate with them via a thick stack of punched cards noisily read by a mechanical “card reader.” We bought “calculation time” by the second, so costly it was. It is a Neolithic vision for young people today.
The invasion of microcomputers has changed all this. Moreover, the increase in computer power being eruptive, the Net is now full of pictures where you see vast rooms filled with mysterious black cabinets, managing jaw-dropping quantities of data. Megaflops, gigaflops, petaflops, galore! Back in the seventies, you could easily read the content of an Apple II RAM, which was entirely written as a small booklet.
We are in a Promethean world. Can we say these modern tools increase our mastery of physics? An anecdote comes to mind. In France, I was a pioneer in microcomputing, having managed one of the first centers (based on Apple II) dedicated to this technology. At that time, also a sculpture professor at the Ecole des Beaux Arts of Aix-en-Provence, one day I presented a system with a flatbed plotter which drew master perspective drawings at will. An old professor, raising his eyebrows, said then, “Don’t tell me the computer will replace the artist?”
Paraphrasing this, we could imagine any fellow who, after visiting a mega data center, claimed: “Don’t tell me the computer will replace the brain?”
In spite of unstoppable computing power escalation, and massive multiprocessors, we are far from it. However, in certain areas, these systems have sent our logarithm books and slide rules to the scrapheap, among other things. Who is still playing at calculating integrals, pen and paper? Who is still juggling with differential calculus, apart from pure mathematicians?
Nowadays we believe in “the computer doing everything.” We build algorithms, we supply data, we run until we receive results. If it is to draw any building or nice engineering work, this works so well. The theory of fluids is also a success.
We can place a surface element, of any shape, perpendicularly to some gaseous flow, and compute the whirling flow pattern past it, whatever its aspect. Does it fit the experiment? Not always. Qualitatively, we master the event, for example, we can compute a reliable aerodynamic drag figure as a result of this gas swirling. Similarly, we compute the burning efficiency inside a cylinder, the convection current in an enclosure. Predictive meteorology is gaining fast, providing a time frame of a few days, except for “micro events,” very localized, which are not yet manageable. Is that the case in every domain?
There are bodies that refuse to be kept on a leash by this modern times’ lion tamer, the so-called computer. These are “non-equilibrium” plasmas, title holders, all categories. They also drift away from fluid theory, in spite of a family resemblance, because they are subject to action at a distance, due to the electromagnetic field whose action can only be evaluated by taking into account all ionic particles constituting the system.
Never mind, you say. It is enough to consider plasma as an N-body system. Easier said than done! We spoke earlier about galaxies, as examples of collision-free worlds. Tokamaks are another kind (ITER is a giant tokamak). The gas they contain is extremely scarce. Before starting, the filling pressure inside the 840 cubic meters of ITER would be less than fractions of a millimeter of mercury pressure. Why so low a pressure? Because we are to heat this gas to more than 100 million degrees. Yet you know the pressure is expressed as: p = nkT – k being the Boltzmann constant, T the absolute temperature, and n the number of particles per cubic meter. Plasma confinement is due only to magnetic pressure, this last increasing as the square of the magnetic field.
With a field intensity of 5.2 Tesla, magnetic pressure is 200 atmospheres. In view of plasma confinement, its pressure must remain far below this value. Due to the use of superconductor devices, the magnetic field cannot be increased indefinitely, so the plasma density inside the reactor chamber stays limited to very low values. From these facts, we see a totally collision-free body, escaping any reliable macroscopic definition. Can we manage it like an N-body problem? Don’t even dream about it, present or future – it is not possible to calculate locally, as we could do it with neutral fluid mechanics. Every area is coupled with any other via the electromagnetic field. Take for example the problem of energy transfer from the plasma core to the walls. Besides a mechanism looking like a conduction phenomenon, besides what belongs to turbulence, a third modality comes into play, named “abnormal transport,” using… waves.
In brief, as in the only one, a tokamak is an absolute nightmare for a theorist.
Plasma in itself, apart from its uncontrollable behavior, is not the only one involved. There is everything else: among them is the unavoidable ablation of particles from the wall. Those who practice gliding know the basic parameter of these machines is the lift-to-drag ratio: it expresses the number of meters flown per meter of height lost (the glide ratio). The sailplane wing, at a given speed, is producing a certain lift force. At the same speed, we get a drag force, which is twofold: first is induced drag: a loss of energy due to vortices at the wingtips.
You cannot avoid it unless you have infinite wingspan… It is to decrease it that gliders have such large wingspans, frequently more than 20 meters, associated with an aspect ratio – ratio of half wingspan to mean wing width – larger than 20. The second source of drag is viscous drag. It will be reduced by seeking the smoothest wing surface. Due to a nice polish, we delay the onset of turbulence in the immediate vicinity of the wing surface. This phenomenon is a basic fluid instability; the excellence of surface polish can only delay its coming. Inversely, this turbulence can be triggered by a perturbation. If we look at a line of smoke in a calm atmosphere, it is an upstream of hot gas, colored by its particle content. This thread of smoke, calm at first, will become intensely turbulent after a tenth of a centimeter rise, whatever the quietness of the ambient air. By introducing an obstacle, such as a needle, in this rising flow, we could trigger irreversible turbulence. The same is done by a minute roughness on the polished surface of a sailplane wing, which will trigger turbulent phenomena, locally increasing air friction by an easy factor of a hundred, thus increasing total drag. In modern sailplanes, we succeed in keeping a laminar airflow (non-turbulent, parallel layers) over 60% of the chord line. If by chance a mosquito crashes on the leading edge, this minute asperity will start turbulence in a zone approximately 30 degrees further back. For this reason, in contest sailplanes, whose glide ratio is more than 50, there is a leading edge cleaning device, started automatically and timely, which can be compared to a linear windshield wiper; a sort of brush travels along the leading edge, back and forth, and comes back to rest in a hidden place. Considerable work has been spent to increase the overall glide ratio of airliners, in order to reduce their fuel consumption. Back in the sixties, the “Caravelle,” which was able to fly between Orly and Dijon, had a glide factor of 12. Nowadays, even these monstrous Airbus 380s have a glide factor of more than 20.
That is, when missing propulsive force, with their four motors idle, starting from 10,000 meters height, they can glide over 200 kilometers.
Back to plasmas and tokamaks: in these machines, a micro-volume of turbulence can be triggered by minute particles, torn from the walls, and will invade the reaction chamber. Speaking of turbulence, the range is extremely large and spreads from this micro-turbulence to electrodynamic plasma convulsions involving the entire volume.
As a conclusion, engineers are not at all managing the machine unless using approximated empirical “engineer’s laws,” of weak reliability, about the running system. In this domain where non-equilibrium is king, where measurements are extremely difficult, the computer is of no help. Experiment is the only leader. Also, extrapolation leads to discovering new unforeseen phenomena, such as the vertical plasma movement (VDE, Vertical Displacement Event), which appeared when jumping in size from the Fontenay-aux-Roses TFR to the Culham JET.
The recent fiasco of the NIF (National Ignition Facility, based in Livermore, CA) is a good example of a sounding failure in large and costly facilities, with the help of the most powerful computers in the world. It is the conclusion of the NIC (National Ignition Campaign) after 2 years of trials, from 2010 to 2012. The system, comprised of 192 lasers, delivers 500 terawatts (more than a thousand times the USA electrical grid power) in a handful of nanoseconds, onto a spherical target 2 mm in diameter, filled with a Deuterium–Tritium mixture, itself inserted at the center of a cylindrical box 2 cm long and 1 cm in diameter, called a Hohlraum (oven in German).
The plan is as follows: half of the lasers’ disc-shaped beams burst into the opening on one side of the Hohlraum, the other half barge into the hole on the opposite side. These ultrathin U.V. beams strike the inner walls of the oven, built from gold. This re-emits X-radiation. The laser beams, precisely focused, created three rings of spots on the inner wall. The re-emitted X-radiation then hits the spherical target. We now talk about indirect irradiation. This system has been devised basically to mimic the fusion stage of a hydrogen bomb, where the X-radiation (generated this time by a fission device) strikes the walls of a shell called an ablator, containing the fusion explosive (Lithium deuteride). In the NIF, this last was replaced with a mixture of Deuterium–Tritium, in which fusion starts at a lower temperature, on the order of 100 million degrees. The envelope (the ablator, a thin spherical shell) sublimates and explodes in both external and internal directions. We use this back-compression to create a “hot spot” at the target center, hoping to start ignition in an inertial confinement scheme.
All this had been calculated under the direction of John Lindl. In 2007, a paper devoted to this scientist, during the Maxwell prize-giving, finely described what would happen. The theorists were so self-convinced that Lindl did not hesitate to claim ignition would be the starting point of a vast series of experiments. It is the same for the test manager, who even had fixed a deadline for operational success, October 2012, which was supposed to crown thirty years of efforts, both theoretical and technological.
The result has been an immense fiasco, pinned out by a July 19, 2012 report, emanated from the D.O.E. (US Department Of Energy) and written under the supervision of Davis H. Crandall.
What must remain of this observation report, related to this very material paper, is that in spite of the excellence of this work, both in technology and measurement, nothing that emerged from this experiment exhibited any relation with the computed data and predictions obtained with the help of the most powerful computers.
To the point where some observers were asking if these simulations could represent any investment for further experiments.
The NIF crisis is evident – it is impossible to increase the number of lasers (Neodymium-doped glass) for cost reasons. It is again impossible to increase their unit power – in fact, when they are drenched with energy, above a certain level, they are prone to blow up, whatever the homogeneity and glass quality are.
To succeed in starting ignition and inertial confinement fusion, the implosion speed must be at least 370 km/sec. Not only is this speed not reached, but, seriously, when the shell constituting the ablative device is turned into plasma and pushes its D-T content, “the piston is mixing up with the fuel,” due to a well-known instability, the Rayleigh-Taylor instability. To minimize its effects, we must make the ablator thicker. But then it would increase its inertia, and the implosion speed threshold would not be reached again.
Simulations done on the computer have given false results in all domains. As written in the D.O.E. report, modeling of interactions between the laser and the walls (impact of X-rays on gold walls) is not satisfying, in spite of tens of years of studies spent on this subject, and hundreds of theses and papers. The same thing applies to the interaction between X-ray beams, following a law named “inverse Raman Scattering,” with the gold plasma, coming from the sublimation of the parietal gold inside the chamber. Interaction of X-radiation with the ablator is also not correctly simulated. Lastly, the calculation algorithms (LASNEX) totally underestimated the weight of the Rayleigh-Taylor instability, the deformation of the contact surface of the ablator, Deuterium–Tritium, recalling intestinal villi.
These mishaps show the confidence limits we can set into superb computerized simulation results, once these machines try to attack frankly out-of-equilibrium problems, mostly non-linear, where a bunch of mechanisms, poorly modeled, play a role in the game.
Dr. Jean Pierre Petit