Morphogenesis by Resonance

                                                       Patterns formed by resonance?

The book Morphic Resonance by Rupert Sheldrake addresses the fundamental problem of how organised structures are formed in physics, chemistry, biology from elementary building blocks seemingly without information about the overall structure. How does a flower, bird or human being develop from a genetic code, which contains recipes for protein building blocks but no information about the whole structure? Sheldrake seeks an answer in the form of morphogenetic fields carrying this information as collective resonance phenomena. 

We are familiar with resonance in physics as the wave harmonics of a vibrating string. We understand that wave patterns develop from instabilities with tendency to increase crests and troughs of certain wave lengths. Watch dropping a stone in a pond.

Thus we expect to see form develop from resonance serving as morphogenesis and find this in particular in the case of fluid flow with turbulent vortices developing from convective instabilities as shown in Computational Turbulent Incompressible Flow. 

The non-radiating stable ground state of an atom is represented by the lowest harmonic of a Schrödinger wave equation, while higher harmonics are triggered for a radiating atom. Real Quantum Mechanics gives a new explanation of the lack of radiation from the ground state, as the mystery Bohr struggled with. 

Computational Black Body Radiation presents a new analysis of the transfer of energy from a source of light to a receiver as an atomic resonance phenomenon carried by standing electromagnetic waves without need to introduce photons as particles of light. A similar transfer is seen between two tuning forks carried by standing acoustic waves. 

Sheldrake's concept of morphic resonance thus comes to expression in physics and may serve also in chemistry and biology in more general forms. Maybe memory is carried by resonance... Maybe the fertilised egg carries the blueprint as a resonance bringing the genetic code to life.

Musical harmony is based on tonal resonance, while musical rhythm represent patterns over time. Singing in a choir unites single souls into one. 

Resonance can have the material form of vibrating strings, or immaterial as a common gravitational potential (recall Neo-Newtonian Cosmology) or more generally beliefs forming a society. An immense subject…

Resonance appears as an expression of instability of a system in the sense that a small periodic forcing causes large oscillations in the system. This happens if the periodicity of the forcing agrees with an eigenvalue of the system and the corresponding eigenfunction represents the shape of the system response.  This allows patterns to develop from small forcing in creation of form as morphogenesis "by itself".

2nd Law vs Finite Precision Computation

Recent posts present an approach to the 2nd Law of Thermodynamics based on a notion of finite precision computation in both analog physical and digital form as it appears in the basic case of slightly viscous fluid flow carrying the phenomenon of turbulence as described in detail in Computational Turbulent Incompressible Flow and Computational Thermodynamics. 

In analog physical form finite precision connects to the smallest physical scale present, and in digital form to the mesh size of a computational mesh.  

In fluid flow the smallest physical scale can be viewed to be determined by the viscosity $\nu$ with normalisation of velocity and spatial dimension, with corresponding Reynolds number $Re =\frac{1}{\nu}$.  A basic case concerns the drag of body as the force of resistance to motion through the fluid, which is captured in a drag coefficient $C_D$ depending on the shape of the body, with $C_D\approx 0.4$ for a sphere. The flow around a bluff body like a sphere attaches as laminar and separates in a wake of turbulent flow.  

A critical question concerns the dependence of $C_D$ on $Re$ as the dependence on the smallest physical scale $\frac{1}{Re}$, with the following typical dependence:

We see that $C_D$ overall varies little with $Re$, but has a substantial dip in the wide interval $10^5 <Re <10^6$ referred to as drag crisis, which covers many cases of practical interest in aero/hydrodynamics. 

We see that whether within the interval of drag crisis or outside, $C_D$ varies little with analog computational precision and so is a robust quantity. In particular, the reduced drag in the interval of of the drag crisis depends to a switch from no-slip to slip boundary condition which decreases the width of the turbulent wake but not its intensity.  

Let us now turn to the the precision in digital simulation form as the mesh size $h$. To capture the physical scale would seem to require $h<\nu$ and so $h<10^6$ in typical aero/hydrodynamics, which in 3d is beyond the capacity of any foreseeable computer. This is the status of standard Computational Fluid Dynamics CFD today: Turbulent flow is uncomputable, because resolution to physical scale is impossible. Turbulence modelling is necessary, but seemingly impossible. 

As concerns the 2nd Law, we could stop here: Drag is roughly independent of finest physical scale and in particular does not to go to zero under resolution going to zero. Turbulent dissipation cannot be avoided. The 2nd Law is valid.

But is it really true that turbulent flow is uncomputable? Is it necessary to resolve the flow to smallest physical scale to capture drag? Maybe not, since the plot above gives hope: Except for the drag crisis $C_D$ is roughly independent of physical scale and so the mesh size can maybe be larger then $10^{-6}$, maybe $h=10^{-3}$ could suffice? 

And yes, this turns out to be true as shown in detail in the book Computational Turbulent Incompressible Flow where in particular drag crisis can be captured using a slip boundary condition along with $h=10^{-3}$ showing that turbulent flow is computable today on a laptop. 

We sum up:

• The 2nd Law is true in the sense that turbulent dissipation is substantial independent of smallest physical scale (Kolmogorov's conjecture confirmed), and cannot be avoided because of inherent instability. 
• Turbulent flow is computable because resolution to smallest physical scale is not necessary. 

By artificially adding viscosity a mathematical proof of a 2nd Law stating energy dissipation is direct. The real challenge is to prove why viscosity must be present with a substantial effect, which is done in the book. 

PS Turbulent dissipation at smallest scale is given by $\nu (\frac{du}{dx})^2\sim 1$ with Reynolds number $\frac{du\times dx}{\nu}\sim 1$ with $du$ velocity variation of length scale $dx$, which gives $du\sim \nu^{\frac{1}{4}}$ and $dx\sim \nu^{\frac{3}{4}}$ reflecting Lipschitz continuity of velocity with exponent $\frac{1}{3}$ in accordance with Onsager's conjecture. Turbulent dissipation takes place mainly at smallest scale because energy is transferred in a cascade from large to smaller scales.  

2nd Law vs Perpetual Motion

A physicist would say that perpetual motion is impossible, because by the 2nd Law of Thermodynamics there is always some positive loss of energy from some form of friction, which even a best possible engineer cannot turn off. In a fluid the friction energy takes the form of turbulent dissipation into heat energy. But on the question why there must be some loss, the answer would be vague probably with some reference to statistics, Boltzmann's H-theorem and entropy as disorder.

Let us see if the Physical 2nd Law I have described in recent posts can give an answer with more physical substance. The basic idea is that real physics is a form of analog computation, which can be mimicked in digital computation and that in both cases the computation has finite precision. In real physics that may be set by the smallest physical scale and in computation by the computational mesh. A most remarkable conjectured by Kolmogorov, is that turbulent dissipation rate is independent of the absolute size of the smallest scale, because turbulent energy is transferred in a cascade to dissipate at smallest scale. 

This is confirmed in the book Computational Turbulent Incompressible Flow, which means that turbulent flow is computable without mesh resolution to physical scale and so opens a new window in fluid mechanics.  

It also shows that it is impossible to decrease the loss/turbulent dissipation by refining the precision into a finer model and finer mesh. In other words, it is impossible to bring loss to zero and realise perpetual motion. Because of finite precision.

The notion of finite precision present in both analog and digital physics thus opens to a new understanding of the 2nd Law and why it makes perpetual motion impossible to realise. More substance is given in Computational Thermodynamics.

The 2nd Law does not apply to the microscopics of a hydrogen atom in ground state with an electronic change in time like a harmonic oscillator without change of electron density and then without friction as described on Real Quantum Mechanics. But a radiating atom with electron density changing over time is subject to radiative loss, which must be balanced by an exterior force in sustained motion. The 2nd Law thus is relevant on all scales, not just for macroscopic ensembles of many.   

Corruption of Modern Physics 9: Misuse of Chaotic Systems

Waterfall as partially predictable chaotic system

The idea that global climate is a chaotic system and as such cannot be predicted, because of sensitivity to small perturbations, is often presented as an expression of deep insight into mathematical modeling. But it may hide a common misunderstanding of the nature of a chaotic system. The basic example of a chaotic system is turbulent flow. The nature of turbulent flow is to be unpredictable pointwise in space and time (because of sensitivity) while being predictable in a mean-value sense (because of insensitivity). 

This is developed in detail in Computational Turbulent Incompressible Flow and Computational Thermodynamics recommended for download. The combination of mean-value predictability and pointwise unpredictability is expressed by the fact that the drag of a car as total resistance to motion through air, is computable/predictable while the pressure at specific points on the car body cannot be computed/predicted. The reason mean-values are predictable is the fluctuating nature of turbulence with high pressure followed by low pressure forming stable mean-values. This is the reason nature can function as a more ot less ordered system even if being a chaotic system, which can be seen as a form of order in chaos.  

The Earth climate system can be described as a turbulent thermodynamic ocean-atmosphere system which forms weather local in space and time and global climate as mean-values over space and time. Experience shows that local weather acts as a chaotic system which is unpredictable pointwise in time over more than a week. The question is then to what extent climate as mean-value weather is computable/predictable? 

If all the equations (Navier-Stokes equations and more) modeling the Earth system were known, we would be able to compute/predict for example global mean temperature year 2100 or the onset of the next Ice Age, because of the fluctuating nature of turbulent flow. But we do not know all the parameters entering in the equations nor the initial conditions. Therefore such computation/prediction for now is impossible, but not because the system in principle is chaotic, rather because present climate models contains unknowns. 

This means that with better climate models it could be possible to predict e.g. the onset of the next Ice Age, or on shorter time scales the Winther weather over Europe depending on jet streams and La Nina and more. Even without climate model we can predict the global mean temperature 2023 to be about the same as 2022. 

In short, global ocean-atmosphere weather system is a chaotic system which as global climate is computable/predictable to a certain degree. Work on better climate models is not meaningless.

What is remarkable is the stability of Earth climate without runaway global warming yet with global cooling into repeated Ice Ages,   

Finite Precision Computation/Physics and Heat Energy

Euler CFD as a parameter free Theory of Everything ToE for slightly viscous incompressible fluid flow is a prime example of the idea of finite precision digital computation capable of simulating physics as a form of finite precision analog computation. Euler CFD captures turbulent flow from a principle of best possible digital solution of Euler's equations in a situation where exact (laminar) solutions are all unstable without permanence over time and so are unphysical and cannot be observed. 

The essence of turbulent flow captured by Euler CFD is the production of heat energy in turbulent dissipation from residual stabilisation in a situation where Euler residuals can be made small only in a weak mean value sense, but blow up in a strong pointwise sense. The dissipative mechanism thus expresses an impossibility to computationally resolve the flow because of finite precision, which in physical terms means production of heat energy as small scale unordered motion. 

Radiative heat transfer also involves an aspect of finite precision in the sense that a body viewed as a set of oscillators is capable of radiating only frequencies below a certain cut-off frequency scaling with temperature because synchronisation of the oscillators necessary for radiation is in finite precision impossible above cut-off. 

There is connection between turbulent flow and radiative heat transfer in that the heat energy generated in turbulent dissipation ultimately is released in radiation, and so gives a meaning to heat energy as unordered motion as unsynchronised oscillatory motion.  

Recall from the blog post 2nd Coming of the 2nd Law that finite precision computation/physics explains why certain processes are irreversible as processes where large scale kinetic energy/ordered motion is transformed into small scale kinetic energy/disordered motion, which cannot be reversed because the precision required to restore large scale order from small scale disorder is not there. This is like restoring all your manuscripts after a tornado has swept them into little pieces, or your hard disk has collapsed. 

Finite precision computation open an approach to the 2nd Law which is different from the standard based on statistics. Small scale disorder is the result of turbulent dissipation as a finite precision resolution of increasingly complex flow arising from flow instability, a resolution which cannot be reversed in finite precision.  It is like necessary (because storage is limited) chopping digits/details, which cannot be retrieved.

Heat energy as internal energy as small scale disordered motion is low quality energy in the sense that transformation to other forms of energy such as large scale motion comes with severe losses.  This puts limits to the efficiency of steam and combustion engines transforming heat energy into piston motion. On the other hand electric energy is high quality energy typically generated from large scale motion in generators allowing efficient electrical motors returning large scale motion. Heating by electricity is thus involves a form of quality degradation, which can be expensive, while heating by burning fossil fuels is efficient and cheap.    

What Is Heat Energy?

Heat energy is a central element in both thermodynamics and radiative heat transfer. But what is in fact heat energy?

The 1st Law of Thermodynamics states that the total energy as kinetic energy plus (internal) heat energy remains constant in a system (without chemistry/fission/fusion) with no energy exchange with its surrounding. The 2nd Law of Thermodynamics states that transformation of kinetic energy into heat energy is irreversible. 

The Planck-Stefan-Boltzman Law (PSB Law) expresses that the transfer of heat energy by electromagnetic radiation from a warmer body of temperature $T_w$ to a colder body of temperature $T_c<T_w$ scales with $T_w^4 -T_c^4$. 

Computational Thermodynamics and Computational BlackBody Radiation present a new approach to uncover the mysteries of both the 2nd Law of Thermodynamics and the PSB Law based on a principle of finite precision computation/physics. In  this setting heat energy takes the form of small scale unordered kinetic motion.

In thermodynamics kinetic energy thus takes the form of large scale ordered motion and small scale unordered motion which is the result of turbulent dissipation into heat energy. The 2nd Law expresses that the process of turbulent dissipation is irreversible because in finite precision unordered small scale motion cannot be coordinated into large scale ordered motion. Heat energy here appears as "internal energy" with limits set by the 2nd Law as concerns transformation to "external energy" as large scale kinetic motion.

In radiative heat transfer the temperature of a body determines a cut-off frequency scaling with temperature with heat energy as atomic vibrations with only frequencies below cut-off appearing in synchronized ordered form capable to generating outgoing radiation. Here the finite precision limit thus scales with the inverse of the temperature and the heat transfer from a warm to a cold body consist only of the frequencies above cut-off for the colder and below cut-off for the warmer. 

In both cases heat energy is a result of an impossibility arising from finite precision computation. In thermodynamics heat energy is unresolvable unordered small scale kinetic motion. In thermodynamics a body absorbs heat energy as unordered kinetic motion for frequencies above cut-off. 

In short, heat energy emerges as a rest product of finite precision computation/physics meeting unresolvable scales of motion. 

Even if now heat energy is a form of rest product, it does not mean that it cannot be recycled into useful energy to some extent. In thermodynamics a gas expanding into a larger volume creates turbulence which is turned into heat energy, which can be used to do work when expanding into an even bigger volume. In radiative heat transfer a colder body when heating up by absorbing heat in unordered form from a warmer body, increases its cut-off and so can radiate higher frequencies in synchronised ordered form.   

Computability of Turbulent vs Laminar Flow

Euler Computational Fluid Dynamics CFD shows that mean values such as lift and drag from the turbulent flow around all sorts of vehicles/bodies moving through air or water are computable at low computational cost, while point values of the fluid flow and body forces in space and time are uncomputable. Euler CFD shows that the mean values are stable quantities insensitive to mesh resolution and small changes of geometry, while point values are very sensitive. 

In Euler CFD this is captured by a dual linearised solution, which in the case of an underlying turbulent oscillating base flow through cancellation can be of moderate size as an expression of mean value stability. This comes out as independence of lift and drag for flow with large Reynolds number beyond drag crisis.  

Laminar flow on the other hand may be less stable because of base flow without oscillation and cancellation, and thus may require large computational cost to correctly capture. It comes out as a possible dependence of lift and drag on smaller Reynolds numbers before drag crisis. An example of laminar flow is potential flow which is unstable/unphysical and thus uncomputable as solution of the Euler equations.

So, in certain (mean value) sense, turbulent flow can be more easy to compute/predict than laminar flow, which can be viewed to be paradoxical, but then in fact is not. 

Similarity Between Fluid Turbulence and Radiative Energy Transfer

In the TNT Radio interview in the previous post I suggest a similarity between fluid turbulence and radiative heat transfer connecting to a phenomenon of high frequency "cut-off".  

Fluid motion transfers large scale ordered kinetic motion/energy over a cascade of successively smaller and smaller scales into a smallest scale depending on viscosity of unordered kinetic motion/energy as turbulence perceived as heat energy with the smallest scale representing the cut-off. The total energy transfer from large to smallest scale shows to depend mainly on the largest scales with thus little dependence on the smallest scales set by viscosity, thus with little dependence on viscosity once small enough. This is Kolmogorovs law of finite rate of turbulent dissipation.

Computational BlackBody Radiation describes radiative heat transfer between two bodies/oscillators $B1$ and $B2$ of temperature $T1$ and $T2$ with $T1>T2$ as an ordered resonance phenomenon with a frequency "cut off" increasing with temperature with energy balance for frequencies below the lower cut-off for $B2$, while frequencies above and below cut-off for $B1$ are absorbed by $B2$ in the form of unordered high-frequency oscillations perceived as heat energy by $B2$. The result is transfer of energy from the warmer body to the colder body. 

In both cases there is thus a split between ordered large scale motion and unordered small scale motion beyond cut-off perceived as heat energy. In both cases the small scale unordered motion perceived as heat energy is the consequence of an impossibility of sustained ordered motion: In fluid motion an impossibility of transferring energy to smaller scales in ordered fashion, and in radiative transfer an impossibility to balance frequencies above cut-off for the colder body, as an effect of finite precision. 

There is a connection to the 2nd law of thermodynamics with the transformation of large scale motion into heat energy as small scale unordered motion, is irreversible. In radiation it means that heat energy transfer is one-way from warm to cold.

The Illusion of a Wing with Laminar Boundary Layer

Aerodynamicists have long and still carry a dream of a laminar wing as a wing of special shape like the P-51 Mustang with special flow characteristics, including less drag as compared to that of a traditional wing as pictured below:

Our New Theory of Flight revealing the true Secret of Flight shows that this dream is built on misconceptions going back to the boundary layer theory of Prandtl/Schlichting based on the idea of a no-slip boundary condition with fluid particles sticking to the surface of the wing with zero velocity thus creating a thin boundary layer, which can be laminar or turbulent, connecting to the free stream flow.

The dream of a laminar wing would then be a wing with a laminar boundary layer on the upper surface of the wing with small skin friction, which hopefully would be the case for a wing with special shape, like that of P-51. The New Theory shows that this can only be a dream or illusion, because flow with a laminar boundary layer separates on the crest of a wing and generates little lift.

The New Theory of Flight builds on the observation that for the high Reynolds number relevant for a wing, the boundary layer is always turbulent with an effect of small skin friction on the main flow, which can be modeled by Navier-Stokes equations with a slip (small friction) boundary condition instead of the dictate of Prandtl to use a no-slip condition.

The New Theory shows that it is the slip boundary condition modeling the action of a turbulent boundary layer, which prevents the flow from separating on the crest and delays separation until the trailing edge (before stall). You can see this in the video below showing computational solution of Navier-Stokes with a slip boundary condition which agrees very well with experiments, as documented in the references in Solution of the Clay Navier-Stokes Problem.

Summary:

1. With a slip boundary condition there is no boundary layer and the flow can stay non-turbulent attached until turbulent separation at the trailing edge and thus give large lift. 
2. With a no-slip boundary condition and laminar boundary layer the flow separates on the crest and gives little lift. 
3. With a no-slip boundary condition the boundary layer is in fact always turbulent with an action on the main flow like a slip boundary condition. 
4. The dream of drag reduction of a laminar wing is already cashed in by a slip/small friction boundary condition. 
5. It is a turbulent boundary layer which can be modeled with slip, and not a laminar boundary layer even though its skin friction is small. 

A laminar wing in the sense of having laminar boundary layer, does not work.

A laminar wing in the sense of a flow with a slip boundary condition without layer and turbulence until the trailing edge, works fine. In this sense a standard wing thus already has made the dream come true, but that is not what aerodynamicists call a laminar wing.

Turbulence Riddle Solved by AI as Automated Computational Mathematical Modeling ACMM

The (super)human intellect of Euler (1707-83) formulated Euler's equation for fluid flow.

The information society is based on computational mathematical modeling with Automated Computational Mathematical Modeling ACMM now emerging as a form of Artificial Intelligence AI.

The FEniCS Project is software for ACMM in a setting of (partial) differential equations (mathematical models of physical systems) offering automation of discretisation and computational solution. 

Unicorn/FEniCS is a unified solver for solid and fluid dynamics, which offers a solution to the major unsolved problem of mathematical modeling of turbulent fluid flow in the form of an automated turbulence model as the product of best possible computational solution of Euler's equations for fluid flow. 

This is presented by Johan Jansson in edX courses on High Performance Finite Element Modeling Part I and Part II. Take the courses and see yourself! This is a high-light of the educational program DigiMat carrying from basic school to university bringing ACMM to the people!

The turbulence model is the result of AI in the form of a computational procedure for finding a best possible solution to a set of partial differential equations formed by the Human Intelligence HI of Euler as the sharpest mind of the scientific revolution. Here HI sets the goal and ACMM is the factual process to reach the goal.

The solution of the riddle of turbulence thus comes out from a combination of HI and AI in a setting where HI showed to be too weak to give an answer, as witnessed by 

Werner Heisenberg:

• When I meet God, I’m going to ask him two questions: why relativity? And why turbulence? I really believe he’ll have an answer for the first.

Horace Lamb:

• I am an old man now, and when I die and go to heaven, there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. About the former, I am really rather optimistic.

Turbulent flow (video) around a jumbojet in landing as best possible solution of Euler's equations.

We here view AI as an extension of HI and not (merely) as a replacement as in self-driving cars,
recalling that after all driving a car does not need much intelligence. But predicting turbulent flow does and requires both HI and AI.

Notice that AI is beyond full understanding of HI since AI is a self-learning system and not a system taught by HI. In particular the automatic turbulence model offered by ACMM is beyond full understanding by HI as a the result of a self-learning adaptive computational procedure, generating the turbulent viscosity which showed to be evasive for HI alone.

Notice further that once turbulent fluid motion can be simulated by ACMM, computational fluid environments can be set up for testing of airplane designs, training of pilots or self-learning of flying vehicles as a combined vehicle-environment application of AI.

Climate vs Chaos and Turbulence

Both climate alarmists and skeptics like to suggest deep understanding by expressing that global climate is a non-linear chaotic system and as such is unpredictable (as discussed by Kip Hansen in a recent sequence of posts):

• The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible. (IPCC TAR WG1, Working Group I: The Scientific Basis)
• It is my belief that most climate variability and even climate change could simply be the result of chaos in the climate system. (Roy Spencer)

But to simply say that a chaotic system is unpredictable is not the entire story. It is true that point values in space/time of a chaotic system are unpredictable, due to strong pointwise sensitivity to pointwise perturbations, but mean values of a chaotic system typically are predictable.

It is certainly impossible to predict the daily temperature of a specific city within one degree one year ahead, but meaningful monthly temperatures are routinely reported in tourist guides.

The book Turbulent Incompressible Fluid Flow presents the following analysis of turbulence as prime example of chaos:

1. Point values are unpredictable due to local exponential instability.
2. Mean values are predictable due to cancellation of instability effects.

It may thus well be possible (with a high degree of certainty) to predict that the global mean temperature will be the same 100 years from now, within a degree up or down.

For the Lorenz system, as a key example of a chaotic system, it is impossible to predict in which lobe a trajectory will be long ahead in time, but the total time spent in each lobe is observed to become nearly equal over long time. About the weather in Scandinavia, we know for sure that it will be variable with alternating low and high pressures, with sunshine following rain and vice versa as a result of the dynamics. 

The Quest for the Ultimate Theory of Time: Physical Stability or Empty Probability?

The question of the direction of time, or the arrow of time, is still haunting physicists with the physicist and cosmologist Sean Carrol expressing state of art in e.g. the book From Eternity to Here: The Quest for the Ultimate Theory of Time, which is basically to say following old Boltzmann: There is a quantity named entropy, which cannot decrease with time and when strictly increasing sets a direction of time motivated by Carroll as follows in an introduction:

• The reason why entropy wants to increase is deceptively simple:
• There are more ways to be disorderly than orderly, so an orderly arrangement will naturally tend toward increasing disorder.

But Carroll is not very happy with this his explanation:

• If everything in the universe evolves toward increasing disorder, it must have started out in an exquisitely ordered arrangeement...a state of very low entropy.
• Why were conditions in the early universe set up in a very particular way? That is the question this book sets out to address.
• Unfortunately, no one yet knows the right answer.

And then follows the rest of the book, without answer. The only attempt to give reason to the tendency of entropy to increase, is to argue following Boltzmann, that things naturally evolve from less probable/low entropy states to more probable/higher entropy states. But of course this is circular: To say that more probable is more probable than less probable is a tautology without actual content.

In the book The Clock and the Arrow: A Brief Theory of Time I argue that there is another way of explaining the arrow of time and that is with reference to the physics of stability instead of the non-physics of probability of Boltzmann. The key point is:

• A system cannot remain in an unstable state because the inevitable effect of small fluctuations will have a major effect and thus transform the system to either a more stable state of more or less rest or to another unstable state of non-rest. 
• The transition from unstable to stable rest is irreversible since the reverse process from stable rest to unstable is impossible without major exterior forcing. 
• The transition from unstable is sensitive to small perturbations along with the formally reversed process, and thus cannot be reversed under any form of finite precision physics.    

Here is a summary of my view and that of Boltzmann/Carroll:

1. An arrow of time is given by physical stability properties of certain systems making them irreversible, without asking any specific order of an early universe.
2. An arrow of time is motivated by an empty tautology stating that systems evolve from less probable to more probable states, asking for a highly improbable highly ordered early universe. 

You may decide yourself between 1. and 2. Which is more probable?

How to Win Any Debate: Claim You Understand Entropy!

John von Neumann (1903-1957) was a very clever mathematician who offered the following advice:

• No one really knows what entropy really is, so in a debate you will always have the advantage (by pretending that you know).

This is still true, and causes a lot of confusion. If you want to improve your understanding then you could consult Computational Thermodynamics, which presents the 2nd Law of Thermodynamics resulting from the Euler equations for a compressible gas subject to finite precision computation in the following integrated form, with the dot signifying time differentiation (see the previous post):

• $\dot K+\dot P = W-D$
• $\dot E = -W + D$,  

where $K$ is kinetic energy, $P$ potential energy, $W$ work, $E$ heat energy and $D\ge 0$ is turbulent dissipation with $W > 0$ under expansion and $W < 0$ under compression. The sign of $D$ sets the direction of time with always transfer of energy from $K+P$ to $E$ from turbulent dissipation.

Here turbulent dissipation is the same as entropy production or the other way around:

• Entropy production is the same as turbulent dissipation. 

This removes the mystery from entropy and you can now win any debate, by really knowing what entropy is! 

For application an application see the recent post on Basic Atmospheric Thermodynamics and the 2nd Law.

New Theory of Flight Presented to the World

Simulation movie of airflow around a jumbojet in landing configuration at large angle of attack.

The revised version of New Theory of Flight has now been submitted to Journal of Mathematical Fluid Mechanics for expected swift publication.

This article together with my former students Johan Hoffman and Johan Jansson represents the summit of my scientific career as a combination of mathematical analysis and computation. The article asks for a major revision of text book aerodynamics and opens new roads to aerodynamic design. And it is not a joke…Finally, The Secret of Flight can be revealed to humanity.

Once the article has appeared in JMFM the new theory will be launched in a press release to media. Stay tuned….

Here is the Summary of article:

• The new theory shows that the miracle of flight is made possible by the combined effects of (i) incompressibility, (ii) slip boundary condition and (iii) 3d rotational slip separation, creating a flow around a wing which can be described as (iv) potential flow modified by 3d rotational separation. 
• The basic novelty of the theory is expressed in (iii) as a fundamental 3d flow phenomenon only recently discovered by advanced computation and analyzed mathematically, and thus is not present in the classical theory. 
• Finally, (iv) can be viewed as a realization in our computer age of Euler’s original dream to in his equations capture an unified theory of fluid flow. 
• The crucial conditions of (ii) a slip boundary condition and (iii) 3d rotational slip separation show to be safely satisfied by incompressible flow if the Reynolds number is larger than 106. For lower Reynolds numbers the new theory suggests analysis and design with focus on maintaining (ii) and (iii).

Fluid Turbulence vs Quantum Electrodynamics

Horace Lamb (1849 - 1934) author of the classic text Hydrodynamics: It is asserted that the velocity of a body not acted on by any force will be constant in magnitude and direction, whereas the only means of ascertaining whether a body is, or is not, free from the action of force is by observing whether its velocity is constant.

There is famous quote by the British applied mathematician Horace Lamb summarizing the state of classical fluid mechanics and the new quantum mechanics in 1932 as follows:

• I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.

Concerning the turbulent motion of fluids I am happy to report that this matter is now largely resolved by computation, as made clear in the article New Theory of Flight soon to be delivered for publication in Journal of Mathematical Fluid Mechanics, with lots of supplementary material on The Secret of Flight. This gives good hope that the other problem of quantum electrodynamics can likewise be unlocked by viewing  The World  as Computation:

• In a time of turbulence and change, it is more true than ever that knowledge is power. (JFK)

Direct Aeroacoustical Simulation in Weakly Compressible Flow

Human speech is generated in fluid-structure interaction of a stream of air with the vocal folds generating a pulsating pressure  which is modulated in the vocal tract into a sound wave from the mouth to reach an ear at some distance, studied e.g. in the EUNISON project. Lighthill derived a model from the compressible Navier-Stokes equations of the form

• $\square\rho =\sum_{i,j}\frac{\partial^2}{\partial x_i\partial_j}(\rho u_iu_j)$,

where $\rho$ is air density, $u=(u_1,u_2,u_3)$ flow velocity depending on space coordinates $x=(x_1,x_2,x_3)$ and time $t$, 

• $\square =\frac{\partial^2}{\partial t^2} -c^2\Delta$,

is the wave operator and $c$ the speed of sound. Here the underlying turbulent flow as source enters the wave equation for a density fluctuation, with simulation proceeding in two steps solving first Navier-Stokes equations for the flow velocity and then a wave equation for the density fluctuation.

Since in speech the flow velocity is much smaller than the speed of sound and thus compressibility effects are small, it is natural so consider the Navier-Stokes equations for a weakly compressible fluid (omitting viscosity effects):

1. $\frac{\partial u}{\partial t} +u\cdot\nabla u +\nabla p = 0$,
2. $\alpha^2\frac{\partial p}{\partial t} - \delta\Delta p = - \nabla\cdot u$,  

which with $\alpha =0$ is the model used in Computational Turbulent Incompressible Flow with $\delta\sim h$ with $h$ a computational mesh parameter. We thus consider a pressure equation augmented by a time derivative as a relaxation of Poisson's equation into a heat equation.

Splitting now $u=\bar u +U$ and $P=\bar p + P$ into components with slow and fast time variation suggests that the model 1-2 contains the following equations for the fast components:

• $\frac{\partial U}{\partial t} +\nabla P = 0$,
• $\alpha^2\frac{\partial P}{\partial t}= - \nabla\cdot U$,  

which corresponds to a homogeneous wave equation for the pressure variation $P$:

• $\square P =0$

with $c=\frac{1}{\alpha}$. The model 1-2 thus emerges as a one-step  alternative to the two-step Lighthill model for direct simulation of aeroacoustics of turbulent flow by solving Navier-Stokes equations for weakly compressible flow.  

Essence of Dynamics 1

                           Computed turbulent flow around an airplane represents Case 3. below.

The dynamics of a physical system can typically be described as an initial value problem of finding a vector function U(t) depending on time t such that

• dU/dt + A(U) = F  for t > 0 with U(0) = G,

where, A(U) is a given vector function of U,  F(t) is a given forcing and G is a given intial value at t = 0. In the basic case A(U) = A*U is linear with A = A(t) a matrix depending on time, which is also the linearized form of the system describing growth/decay of perturbations characterizing stable/unstable dynamics.

An essential aspect of the dynamics is the perturbation dynamics described by the linearized system which is determined by the eigenvalues of its linearization matrix A, assuming for simplicity that A is diagonalizable and independent of time:

1. Positive eigenvalues: Stable in forward time; unstable in backward time.
2. Negative eigenvalues : Unstable in forward time; stable in backward time.
3. Both positive and negative eigenvalues: Both unstable and stable in both forward and backward time.
4. Imaginary eigenvalues: Wave solutions marginally stable in both forward and backward time.
5. Complex eigenvalues: Combinations of 1. - 4.         

Here Case 1. represents a dissipative system with exponential decay of perturbations in forward time making long time prediction possible, but backward time reconstruction difficult because of exponential growth of perturbations. This is the dynamics of a diffusion process, e.g. the spreading of a contaminant by diffusion or heat conduction. 

Case 2. is the reverse with forward prediction difficult but backward reconstruction possible. This is the dynamics of a Big Bang explosion. 

Case 3. represents turbulent flow with both exponential growth and decay giving rise to complex dynamics without explosion, with mean-value but not point value predictability in forward time. The picture above shows the turbulent flow around an airplane with mean-value quantities like drag and lift being predictable (in forward time). This case represents the basic unsolved problem of classical mechanics which is now being uncovered by computational methods including revelation of the secret of flight (hidden in the above picture).  

Case 4 represents wave propagation with possibilities of both forward prediction and backward reconstruction, with the harmonic oscillator as basic case. 

There is a further limit case with A non-diagonalizable with an incomplete set of eigenvectors for a multiple zero eigenvalue, with possibly algebraic growth of perturbations, a case arising in transition to turbulence in parallel flow. 

Scientific Principles of Climate Science?

The scientific method is the basis of modern society with the basic requirement that a scientific theory as an expression of the scientific method, must allow making observable predictions.

The theory stating that the Earth rests on the backs of four invisible turtles the presence of which cannot be observed, is not a scientific theory. Controling an airplane by throwing dice is not a scientific approach and will not work, because the outcome a dice throw is not predictable.

The requirement of prediction connects to the mathematical concept of wellposedness formulated by the French mathematician Jean Leray in the 1930s: A mathematical model with certain output from certain input data, is said to be wellposed if small changes in input data result in small changes in output. The model is said to be illposed  if small changes in input can result in large changes in output.

The rationale is that output from an illposed model in general carries no relevant information, since small changes in input, which cannot be controlled,  can give widely different outputs making prediction impossible.  An illposed mathematical model therefore does not represent a scientific theory.

In principle, climate science can be viewed as a mathematical model of the thermodynamics of the Earth-atmosphere system, in the form of the Navier-Stokes equations. It was this model that led Leray to his study of wellposedness, motivated by the fact that the Navier-Stokes equations in general have turbulent solutions with pointwise values being very sensitive to small perturbations, thus being pointwise illposed. But meanvalues show to be insensitive and thus the Navier-Stokes equations show to be wellposed in an appropriate meanvalue sense, as developed in Computational Turbulent Incompressible Flow.

Climate science based on the NS equations may thus be capable of making predictions, although the computational technology required to model the whole Earth-atmosphere, lies far into the future.

Climate science is obsessed by a task of finding evidence of catastrophic anthropogenic global warming CAGW from CO2 emissions, which means finding a large effect of global warming of 3 C from the small change of an increase of the concentration of atmospheric CO2 from the present 390 ppm to 600 ppm. CAGW is thus based on an illposed model and as such cannot be viewed to have scientific support.  To be able to predict the effect of a change of 0.02% CO2, a very high precision climate model must be used, and such a model is not foreseeable.  

Renormalization in QED, Blackbody Radiation and Turbulence

There is a connection between the cut-off in Computational Blackbody Radiation and the cut-off in Renormalization of QED, namely

• avoiding an ultraviolet catastrophe of infinite energy
• by transforming high-frequency waves into low-frequency waves
• in a form of reinvestment.

In the case of blackbody radiation with an absorbing body with a cut-off proportional to temperature subject to forcing from incident waves, the reinvestment comes out as heating (increasing temperature) from incident wave frequencies above the current cut-off.

There is also a connection to turbulent dissipation with high-frequency kinetic energy being transformed into low frequency internal/heat energy. The total energy as the sum of kinetic and internal energy collects the reinvestment and is thus conserved.

Renormalization thus concerns reinvestment of what is cut off, like collecting the branches cut off from a tree and using them together with the trunk of the tree so that nothing is lost.

The basic question concerns partition of energy between the trunk and branches (on branches)

and its dependence on the size of the cut-off length.

In turbulence, the total turbulent dissipation (= drag in bluff body flow) shows to be independent of the cut-off length.

In quantum mechanics the question is if the world would look the same with a different value of Planck's constant which sets the cut-off length on atomic scales.

Radiative vs Turbulent vs Frictional Heating

Mathematical Physics of Blackbody Radiation and Computational Turbulent Incompressible Flow (or Computational Thermodynamics) show a similarity between radiative heating and turbulent heating, which can be viewed as different forms of frictional heating.

In both cases the heating effect shows to be independent of the small coefficient of dissipation/friction connected to the heating (the viscosity in turbulent flow and coefficient of the Abraham-Lorentz recoil force in radiation).This means that macroscopic quantities such as total radiative and turbulent heating show to be insensitive to the absolute scale of the microscopic dissipation creating the heating.

This allows computational simulation of macroscopical features of radiation and turbulent flow without resolution to actual smallest physical scales. This makes in particular turbulent flow computable contrary to common perceptions of state-of-the-art.

One may compare making fire by friction with a bow and drill: What is important is the total power invested, not the specific dimensions of the bow and drill.