NON-EQUILIBRIUM WORLDS
Jean Pierre PETIT – Former research director – CNRS FR.
12 January 2013
When the average person thinks about equilibrium of a system, they usually picture a ball sitting at the bottom of a well, or something similar.
The fundamentals of thermodynamic equilibrium theory involve something more subtle: dynamic equilibrium. The simplest example is the air we breathe. Its molecules bounce around in every direction, exhibiting a mean thermal velocity of 400 m/s. At an enormous rate, these molecules collide and interact. These collisions alter their speed. Nevertheless, physicists interpret this as statistical stationarity (the term used here is “detailed balancing”). Imagine a little elf who, at any moment and at any point in the room, can measure the speed of molecules in a certain direction, allowing for a slight angular uncertainty. At each time interval, this elf counts V and V + ΔV, obtaining an algebraic value. He then plots these values on a graph and observes a nice Gaussian curve emerging, with a peak mean value close to 400 m/s. Thus, the faster or slower the molecules are, the fewer there are of them.
He repeats this process, aiming his measuring device toward any spatial direction, and—surprise!—gets the same result. Molecular agitation in the room is isotropic. Moreover, nothing can disrupt this dynamic equilibrium, provided the temperature remains constant; after all, gas temperature precisely equals the mean kinetic energy stemming from this thermal agitation. Physicists describe such gas as being in thermodynamic equilibrium. This state is multifaceted: air molecules lack spherical symmetry. Diatomic molecules such as oxygen or helium have a peanut-like shape. Carbon dioxide and water vapor molecules assume other forms. All these objects, when rotating, can store energy akin to tiny flywheels. These molecules can also vibrate. The principle of uniform energy distribution stipulates that energy must be evenly allocated among all these various “modes.” During a collision, some kinetic energy may convert into vibrational or rotational energy of a molecule; the reverse also holds true. Consequently, statistics come into play: our elf can tally how many molecules are in such-and-such state, possess such kinetic energy, or are in such a vibrational condition. Returning to the air we breathe, this census reveals a stationary condition. This medium is then said to be in thermodynamic equilibrium, i.e., relaxed. Picture a wizard capable of halting those molecules, freezing their rotation or vibration, altering them at will, thereby generating a novel statistical law, distorting that lovely Gaussian curve, even engendering some anisotropic phenomenon wherein, for instance, thermal speed in one direction doubles relative to transverse directions. Ultimately, he would allow the system to evolve via subsequent collisions. How many such collisions would be required for the system to revert to thermodynamic equilibrium? Answer: very few. The mean free path of a molecule between two collisions offers an indication of the relaxation time in a gas, i.e., how quickly it returns to thermodynamic equilibrium.
Do non-equilibrium media exist, wherein molecular statistical velocities noticeably deviate from this cozy isotropy and the elegance of those neat Gaussian curves?
Absolutely! In fact, this constitutes the norm throughout the universe. A galaxy—this “universe-island”—comprising hundreds of billions of stars of roughly comparable mass, can be viewed as a gaseous medium where molecules correspond to... stars. In this specific case, we encounter a baffling realm wherein the mean free path of a star prior to encountering any neighboring star equals ten thousand times the age of the universe. What do we mean by “encounter”? Must it be a collision wherein two stars violently smash together? Not at all! Within the realm of theoretical physics termed kinetic theory of gases, we deem a collision to occur when a star’s trajectory noticeably changes upon passing near a neighboring star. Calculations demonstrate, however, that such events are exceedingly rare; consequently, our system of hundreds of billions of stars generally experiences no collisions.
Over billions of years, the Sun’s trajectory has remained regular, nearly circular. Had the Sun possessed consciousness and remained unaffected by encounters, it would have perceived no neighbors at all. It would merely sense a “smooth” gravitational field. It would progress along its path as though traversing a basin, never noticing any bumps caused by other stars. An immediate corollary follows: position our elf—now an astronomer—near the Sun within our Galaxy and task him with compiling a speed statistic for neighboring stars across all directions. An obvious truth emerges: dynamically speaking, this medium is profoundly anisotropic. There exists a particular direction wherein the stars’ agitation speeds (astronomers refer to these as residual speeds, relative to the galaxy’s mean rotational motion, which is fairly circular and clocks in at 230 km/s near the Sun) are practically double those observed in any transverse direction. In ordinary air, this was termed spheroidal speed distribution; now, it becomes ellipsoidal speed distribution. So far, so good? How does this influence our perception, our comprehension of the world? It transforms everything! Because we simply cannot grapple with theories concerning such radically non-equilibrium systems.
Setting aside the paradoxical condition of galaxies owing to that infernal dark matter effect (missing mass), discovered in 1930 by American-Swiss Fritz Zwicky, we could theoretically devise any model of self-gravitating, punctual masses (orbiting within their own gravitational fields). Our physics invariably operates near a state of thermodynamic equilibrium. Evidently, any deviation from this state constitutes a departure from equilibrium—for instance, a temperature disparity between two gaseous zones, which triggers heat transfer, i.e., transfer of kinetic energy via thermal agitation. In such cases, reactivating our elf would lead him to conclude that, dynamically speaking, the medium is “almost isotropic.” This applies to our atmosphere, even amidst ferocious windstorms.
Thus, is it impossible to encounter, to “pinpoint,” scenarios wherein a gaseous medium or fluid blatantly defies equilibrium? Such instances arise when traversing shock waves. These constitute confined regions, since the thickness of a shock wave roughly equals a handful of mean free paths.
Upon encountering a shock wave, a gas abruptly transitions between states; a condition near thermodynamic equilibrium is restored in the “shocked” gas after several mean free paths transpire.
Forty years ago, at the laboratory where I worked—now defunct, the “Institut de Mécanique des Fluides de Marseille”—we documented an observation. We employed what we termed “shock tubes,” essentially gas guns. The procedure: utilizing an explosive, we generated a shock wave propagating at several thousand meters per second through a rarefied gas—initially at a pressure of a few millimeters of mercury. This shock wave recompressed the gas, boosting its density.
Interferometry allowed us to monitor density increases with ease and precision. Concurrently, we gauged heat flow at the surface of Plexiglas mock-ups. Since experiments lasted mere fractions of milliseconds, our measuring apparatus needed rapid response times. Specifically, these were metallic films a few microns thick, vacuum-coated onto the walls, functioning as thermistors. We assessed heat flow by tracking resistance variations in these wall-mounted sensors as they warmed up.
One day, we affixed a sensor directly onto the tube wall. We then noticed that heat flow reached the sensor after a certain delay following shock wave passage, marked by a sudden density surge. We verified that the sensor’s thermal lag was negligible enough to preclude such delay originating therein. Essentially, we pinpointed a return phenomenon toward quasi thermodynamic equilibrium downstream of the shock wave.
We may liken this to a hammer strike. Not only does density skyrocket, but we also observe a temperature spike, implying heightened thermal speed of molecules. Yet beyond this wave, isotropy emerges solely after several mean free paths transpire. Immediately ahead of the density front, heightened thermal agitation manifests as motions perpendicular to the wave direction.
When our sensor registers heat, this stems from air molecules impacting its surface. Yet just prior to the density front, over a certain distance, thermal agitation proceeds parallel to the wall. The gas was thoroughly “heated” yet momentarily incapable of transferring that heat to the wall. Through collisions, the “ellipsoid of speeds” morphed into a “spheroid of speeds”; ultimately, the sensor registered returning heat flow. To the best of my recollection, with our experimental setup, we recorded this heat flow roughly one centimeter prior to the density front.
Consequently, shock waves represent minuscule regions wherein the gaseous medium starkly defies equilibrium. How do we tackle this? By equating these regions to surfaces devoid of thickness. This method has functioned effectively for nearly a century. I’m old enough to recall virtually all of computer history, from its inception. When I attended the “Ecole Nationale Supérieure de l’Aéronautique,” no computers existed on campus; they resided in sanctums dubbed “calculation centers” to which we lacked access. We performed calculations using slide rules—curiosities nowadays. In advanced classes, we all carried logarithmic tables; every exam entailed tedious numerical computations employing these instruments, now displayed in museums.
Upon graduating from Sup Aero, mechanical calculators (FACIT) powered by hand made their debut. To multiply, one turned the crank one way; to divide, the opposite.
Professors and department heads utilized electric machines, whose clattering gear noises disrupted the silence of the Institut Mécanique des Fluides in 1964. Computers occupied a privileged position, akin to distant deities glimpsed solely through windows in those calculation centers. These computers, possessing the power of today’s pocket calculators, were tended by priests clad in white robes. Communication occurred solely via stacks of punched cards noisily processed by mechanical “card readers.” We purchased “calculation time” by the second; it was exorbitantly expensive. This is a Neolithic mindset for today’s youth.
The influx of microcomputers transformed all this. Moreover, the explosive growth of computing power has rendered the Internet teeming with images depicting vast chambers replete with enigmatic black cabinets handling staggering volumes of data: megaflops, gigaflops, petaflops—they abound! Back in the seventies, one could effortlessly read the contents of an Apple II RAM, which were meticulously transcribed into a slim booklet.
We inhabit a Promethean era. Can we assert these modern tools enhance our command of physics? An anecdote springs to mind. In France, I spearheaded microcomputing, overseeing one of the earliest centers (Apple II-based) dedicated to this tech. At the time, concurrently serving as a sculpture professor at the Ecole des Beaux Arts of Aix en Provence, I unveiled a system featuring a flatbed plotter capable of drawing masterful perspective sketches at will. An elderly professor, eyebrows raised, remarked, “Don’t tell me computers will supplant artists?”
Paraphrasing this, we might envision any individual, after touring a colossal data center, declaring: “Don’t tell me computers will replace brains?”
Despite relentless surges in computing power and proliferation of multiprocessors, we’re nowhere near that reality. Nevertheless, in certain spheres, these systems have rendered obsolete our logarithmic tables and slide rules, among other implements. Who still calculates integrals using pen and paper? Who still dabbles in differential calculus, save for pure mathematicians?
Today, we believe “computers do everything.” We craft algorithms, feed in data, run computations until results emerge. Whether designing buildings or exquisite engineering marvels, this approach functions admirably. Fluid dynamics likewise thrives.
We can position a surface element of any shape perpendicular to a gaseous flow and compute the swirling pattern around it, irrespective of its form. Does this align with experiments? Not always. Qualitatively, we grasp the phenomenon; for instance, we can reliably compute aerodynamic drag arising from such gas swirling. Likewise, we determine combustion efficiency within cylinders or convective currents inside enclosures. Predictive meteorology advances rapidly, delivering forecasts spanning several days, except for “micro-events”—highly localized occurrences still beyond our grasp. Does this hold true across all disciplines?
Certain entities resist subjugation by this modern-day lion tamer called computer; these are “non-equilibrium” plasmas, spanning all categories. They diverge from fluid theory despite superficial similarities, since they respond to long-range forces exerted by electromagnetic fields, whose impact can only be quantified by accounting for all constituent ionic particles.
“Never mind,” you say. “Simply treat plasma as an N-body system.” Easier said than done! Earlier we discussed galaxies as examples of collision-free realms. Tokamaks represent another category (ITER is a colossal tokamak). The gas they contain is exceedingly sparse. Prior to operation, the pressure inside ITER’s 840-cubic-meter chamber would be less than a fraction of a millimeter of mercury. Why such low pressure? Because we intend to heat this gas to over 100 million degrees. Recall that pressure is defined as p = nkT—where k denotes Boltzmann constant, T absolute temperature, and n particle count per cubic meter. Plasma confinement relies solely on magnetic pressure, which escalates quadratically with magnetic field strength.
At a field intensity of 5.2 Tesla, magnetic pressure reaches 200 atmospheres. For effective plasma confinement, pressure must remain far beneath this threshold. Owing to employment of superconducting apparatus, magnetic field strength cannot be indefinitely amplified; hence plasma density inside the reactor chamber stays exceptionally low. These factors render it a wholly collision-free entity, eluding any dependable macroscopic characterization. Can we address it as an N-body problem? Dream on—neither now nor ever; local computation isn’t feasible, unlike neutral fluid mechanics. Every region interconnects with all others via electromagnetic fields. Take energy transfer from plasma core to walls: besides mechanisms resembling conduction and turbulence, a third modality exists, termed “abnormal transport,” reliant on... waves.
In short, a tokamak constitutes an absolute nightmare for theorists.
Plasma itself, aside from its uncontrollable behavior, isn’t the sole culprit. Numerous other factors intervene: foremost is inevitable particle ablation from walls. Glider enthusiasts know the key parameter of their craft is lift-to-drag ratio—expressing meters flown per meter of altitude lost (the glide ratio). At a given speed, a glider wing generates specific lift force. Simultaneously, drag emerges, comprising two components: first, induced drag—energy loss stemming from vortices at wingtips.
Avoidance is impossible unless wingspan were infinite... Hence gliders feature enormous wingspans, often exceeding 20 meters, with aspect ratios—ratio of half wingspan to mean wing width—exceeding 20. Second drag source is viscous drag; minimizing it requires ultra-smooth wing surfaces. Superior polishing delays turbulence onset adjacent to wing surfaces; this phenomenon is a fundamental fluid instability, and flawless surface finishing merely postpones its advent. Conversely, turbulence can be induced by perturbations. Observe a smoke trail in still air: initially calm, it becomes intensely turbulent after ascending a mere tenth of a centimeter, regardless of ambient tranquility. Introducing an obstacle, e.g., a needle, into this ascending flow can trigger irreversible turbulence. Similarly, minor roughness on a polished glider wing provokes turbulent phenomena, locally amplifying air friction by a hundredth factor, thus boosting total drag. Modern gliders maintain laminar airflow (non-turbulent, parallel layers) over 60% of the chord line. If a mosquito inadvertently collides with the leading edge, this infinitesimal irregularity initiates turbulence within a zone roughly 30 degrees distant. Hence competitive gliders, boasting glide ratios exceeding 50, incorporate automatic leading-edge cleaning devices—comparable to linear wind shields; a brush traverses the leading edge back and forth, then retreats to concealment. Immense effort has been invested to elevate overall glide ratios of airliners, thereby curtailing fuel consumption. Back in the sixties, the “Caravelle,” capable of flying between Orly and Dijon, achieved a glide factor of 12. Nowadays, even gargantuan Airbus 380 models boast glide factors surpassing 20.
In essence, bereft of propulsion, with all four engines idling, these aircraft can glide over 200 kilometers from an altitude of 10,000 meters.
Reverting to plasmas and tokamaks: within these machines, microscopic turbulence can be initiated by minuscule particles dislodged from walls, subsequently engulfing the reaction chamber. In terms of turbulence, the spectrum is vast, ranging from such micro-level disturbances to electrodynamic plasma convulsions affecting the entire volume.
In conclusion, engineers lack control over these machines absent reliance on approximate empirical “engineer’s laws” of dubious reliability governing system operation. In this realm where non-equilibrium reigns supreme and measurements prove excruciatingly arduous, computers provide no assistance. Experimentation reigns supreme. Moreover, extrapolation uncovers unprecedented phenomena, such as vertical plasma displacement (VDE, Vertical Displacement Event), observed upon transitioning from Fontenay aux Roses’ TFR to Culham’s JET.
The recent fiasco of NIF (National Ignition Facility, Livermore, CA) epitomizes failure in massive, costly installations aided by the world’s most powerful computers. It concludes NIC (National Ignition Campaign) after two years of trials, 2010–2012. This system, comprising 192 lasers, unleashes 500 terawatts (over a thousand times the U.S. electrical grid capacity) within mere nanoseconds onto a 2-mm-diameter spherical target packed with deuterium-tritium mixture, itself situated at the core of a 2-cm-long, 1-cm-diameter cylindrical vessel termed Holraum (German for “oven”).
Plan: Half of the disc-shaped laser beams pierce one aperture of Holraum; the other half traverse the opposing opening. These ultrathin UV beams strike inner walls fabricated from gold, which emits X radiation. Precisely focused laser beams generate three rows of spots on these inner walls. Re-emitted X radiation then impacts the spherical target. This constitutes indirect irradiation. Fundamentally, this system was designed to emulate the fusion phase of a hydrogen bomb, wherein X radiation (produced by a fission device) strikes the shell termed ablator, housing fusion explosives (lithium deuteride). At NIF, this ablator was substituted with deuterium-tritium blend igniting at temperatures around 100 million degrees. The envelope (ablator, a thin spherical shell) sublimates and detonates outward and inward. This back-compression aims to generate a “hot spot” at target center, hopefully triggering ignition via inertial confinement.
All this was calculated under John Lindl’s guidance. In 2007, a paper honoring this scientist during Maxwell Prize presentation detailed expected outcomes. Theorists were so convinced that Lindl boldly proclaimed ignition would kick off a sprawling series of experiments. The test supervisor even set October 2012 as deadline for operational triumph, marking thirty years of theoretical and technological endeavor.
Result: a monumental fiasco, documented in a July 19, 2012 report issued by D.O.E. (U.S. Department of Energy) under Davis H. Crandall’s supervision.
What remains noteworthy from this report, pertinent to this discourse, is that despite remarkable technical and measurement achievements, absolutely nothing stemming from this experiment correlated with computed data or predictions derived via the world’s most potent computers.
To the extent that some observers wondered whether these simulations warranted further experimentation.
The NIF predicament is unmistakable—due to cost constraints, augmenting laser count (neodymium-doped glass) is impossible. Likewise, upping individual laser power is unfeasible; beyond a certain energy threshold, they tend to explode irrespective of homogeneity or glass quality.
To achieve ignition and inertial confinement fusion, implosion velocity must exceed 370 km/sec. Not only is this speed unattained, but critically, when the ablator shell transforms into plasma and propels its D-T contents, “the piston mixes with fuel” owing to a well-known instability: Rayleigh-Taylor instability. Mitigating its effects necessitates thickening the ablator; however, this raises inertia, preventing attainment of requisite implosion velocity.
Computer simulations yielded erroneous results across all domains. Per D.O.E. report, modeling interactions between lasers and walls (X-ray impact on gold surfaces) remains inadequate despite decades of research and hundreds of dissertations and papers. Same holds for interactions between X-ray beams—governed by “inverse Raman scattering”—and gold plasma generated by wall sublimation. X-ray interaction with ablator also lacks accurate simulation. Lastly, calculation algorithms (LASNEX) grossly underestimated Rayleigh-Taylor instability’s impact, as well as deformation of the ablator-Deuterium-Tritium contact surface, reminiscent of intestinal villi.
These setbacks underscore the reliability limits we may assign to sophisticated computational outputs when these machines attempt to tackle unequivocally out-of-equilibrium problems, predominantly nonlinear scenarios wherein numerous poorly modeled mechanisms exert influence.
Dr. Jean Pierre Petit