lördag 2 mars 2024

Modern Physics as Chaos

Connection between Entropy and Death.

In Greek mythology Gaia as Mother Earth and Heavens emerged from Chaos in a process from disorder to order. 

The science of physics seeking to describe this process in mathematical terms is split into classical physics in the names of Newton, Euler and Maxwell before 1900, and modern physics in the names of Einstein (relativity) and Schrödinger and Bohr (quantum mechanics) after 1900. 

The split is expressed by a shift from determinism of classical physics to indeterminism of modern physics as a process from order to disorder. 

How could this happen? Isn't science about order rather than disorder? Isn't life about order rather disorder? Isn't true that disorder/death can be created in one single blow without precision, while building order/life requires both time and precision?

The reason for the shift was the 2nd Law of Thermodynamics first formulated by Clausius in 1867 as follows:
  • It is impossible to construct a device which operates on a cycle and whose sole effect is the transfer of heat from a cooler body to a hotter body. 
The 2nd Law in this form states that heat energy cannot without losses be transformed into mechanical work, which directly connected to the efficiency of steam engines booming in the 19th century. 

The task of physicists was to rationalise Clausius rather cryptic formulation into an explanation/proof in quantitative physical terms, which was taken on by Boltzmann in a long struggle and eventually made him end his life when realising that he had failed. Boltzmann aimed at expressing the 2nd  Law in terms of microscopic mechanics of colliding gas molecules, which however showed to require an assumption about collision statistics, which was a form of surrender. 

But nobody was able to come up with something better and so Boltzmann's proof came to be the accepted explanation by modern physicists, who carried Boltzmann's statistics into quantum mechanics as if it had a fundamental meaning, albeit a different. But resorting to statistics is a form of surrender and failure of rationality.

So the world is still waiting for a proof of the 2nd Law after more than 100 years which has motivated Stephen Wolfram to step in with his New Kind of Science based on computation, as discussed in recent posts. 

I have also presented a proof of the 2nd Law based on computation, but fundamentally different from that of Wolfram, as also discussed in recent posts. The essence of my argument concerns the relation between order and disorder, between Gaia and Chaos. 

As a basic example I consider the expression of the 2nd Law captured in computational solutions of Euler's equations for a slightly viscous incompressible fluid such as air and water, showing the unavoidable appearance of turbulence transforming ordered kinetic motion/energy into disordered kinetic energy in the form of heat. This transformation is irreversible because the precision required to coordinate disordered kinetic motion into ordered kinetic motion cannot be fully met in physical terms: there is always some loss. Necessarily appearing turbulence thus appears as a loss which cannot be fully retrieved, which is the essence of Clausius 2nd Law formulated without reference to statistics.  

The above is analysed in mathematical detail in the books  Computational Turbulent Incompressible Flow and Computational Thermodynamics and in popular form in earlier posts. The analysis in particular describes how ordered structures such as vortices and streaks can develop in a cascade from large to small scales with increasing velocity gradients, which on some smallest scale have to be destroyed into heat to prevent blow-up. 

The result is a Physical 2nd Law expressed in the physical concepts of kinetic energy, work and turbulent dissipation, which does not require the evasive concept of entropy nor any statistics. I believe that both Clausius and Boltzmann could have been pleased with this form.

Returning to greek mythology we find that the 2nd Law in this form expresses and interplay between Chaos/disorder and Gaia/order, where ordered structures emerge form disordered at the expense of energy as a loss, and ordered fine-scale structures are destroyed into disordered heat to prevent blow-up, all described in a mathematical analysis of fluid flow as a basic feature of the World. As a result time is moving forward as explained in catching terms in The Clock and the Arrow.

Statistics was introduced by Boltzmann in a desperate effort to give the 2nd Law a mechanical meaning and so created statistical mechanics, picked up by Planck in 1900 in an equally desperate attempt to explain blackbody radiation in mechanical terms, which opened Pandoras box to quantum mechanics as statistics leading to the disorder/Chaos or the "crisis of modern physics".  

For a derivation of blackbody radiation without statistics, see Computational Blackbody Radiation.  
For a version of quantum mechanics without statistics, see Real Quantum Mechanics. 

fredag 1 mars 2024

Wolfram Explains 2nd Law to Perplexed Lex

The losses of 64% in a fuel engine partly come from irreversible turbulent dissipation.

In the Lex Fridman interview 2nd Law Explained Stephen Wolfram explains (in my abbreviation): 

  • The 2nd Law is what a computationally bounded observer like us perceives of a computationally irreducible world.
  • Space is probably discrete as a form of particle theory, not continuous as a field theory.  
  • I hope to find an analog of Brownian motion revealing the discreteness of space.
  • If you know all positions of the molecules of a gas, the entropy is zero. 
Let me compare with the explanation I have presented based on 
  • finite precision computation 
  • battle increase difference - decrease difference
with turbulence as the key phenomenon expressing a 2nd Law where large scale coordinated motion (created from increasing velocity differences) is destroyed into small scale uncoordinated motion as heat energy (created from decrease of difference). This is explained in detail in the books Computational Turbulent Incompressible Flow and Computational Thermodynamics and earlier posts. 

There is a connection between Wolfram's computationally bounded and my finite precision computation viewing the evolution in time of a physical system as a form of analog computation which possibly can be mimicked by digital computation. 

A main difference is that I start with a model of physics as a computational form of Euler's equations for fluid flow as analog physics in digital form, and show that this model produces turbulent solutions in close agreement with observations, which satisfies a 2nd Law with entropy taking the form of turbulent dissipation as a quantity which can only increase with time expressing irreversibility. 

Wolfram instead starts with a discrete model as a simple ad hoc Rule without real physics but then misses the key phenomenon of turbulence and so ends up in a lengthy lecture connecting to classical concepts like random, number of microstates, entropy as measure of disorder, coarse-graining, prepared initial conditions, Brownian motion....leaving Lex perplexed. 

In fact, inventing ad hoc a Rule displaying the nature of turbulence, appears to be very difficult, while the computational Euler model presents itself as a Rule expressing Newton's laws of motion.  

I think I would be able to say something a bit more understandable, if invited by Lex...

A key aspect of the 2nd Law is that turbulent dissipation is loss which cannot be avoided in e g a heat engines delivering coordinated kinetic motion from heat or chemical energy, because of unavoidable increase of difference which has to be controlled to avoid blow-up. It is thus not enough to understand that turbulent dissipation generates heat as a loss (which is easy), corresponding to adding viscosity of some form. We also have to understand that turbulent dissipation is necessary to avoid blow-up and so allow continuation in time (which can be understood by a stability analysis). 

Note also that it is necessary to consider a computational form of Euler's equations since exact physical solutions do not exist. Euler's equations formally expressing Newton's laws but lacking exact solutions, thus in computational form turn into real physical laws in agreement with observations, like physical laws describing reality emerging by computation from formal laws of human minds. 

torsdag 29 februari 2024

A Logical Analysis of Swedish Foreign Policy

Swedish State Media and Press serve the Swedish people everyday the following message:

  • Russia cannot be allowed to win the war in Ukraine                                 (A)
supported by information that mercenaries from 46 nations is helping the Ukranian army, because
  • Russia is winning the war.                                                                             (B)
We understand that (A) and (B) cannot both be true and further that there is some ambiguity hidden in "cannot be allowed": Is it USA/NATO/Sweden who will set a stop to (B) or some divine power? 

Let's consider the first option: 

  • USA/NATO/Sweden cannot allow Russia to win the war in Ukraine.      (AA)
The only way to make (AA) come true is by a direct intervention since the present two year proxy war has led to (B). This is what Macron said hosting a Ukraine Summit with European Heads of State

In other words, to stop Russia from winning the war, which is what is now happening,  it is necessary to 
  • escalate to a full scale war between USA/NATO/Sweden and Russia  (C)
which is what now is happening. 

We conclude that (C) is required to make (AA) true. But (C) will lead to WWIII, where 
  • USA/NATO/Sweden will lose                                                                               (D)
since in a nuclear war everybody loses.  

We understand that the only way (A) can be made true is that also (D) is made true: 
  • To stop Russia from winning, requires USA/NATO/Sweden to lose.          (E) 
The trouble is that the politicians running the show of USA/NATO/Sweden are not capable of logical thinking and so to them (E) is incomprehensible far beyond their limits of imagination. This is very unfortunate to all the people understanding very well what (E) means. And you? 

PS1 Sanctions against Russia which hurt West much more than Russia is a similar milder form of self-infliction to stop an adversary. 

PS2 I have followed a line of thought that the lack of logic in modern western politics is a result of the corruption of logic in modern western physics initiated by Einstein. Does it make sense? 

Why Do Physicists Not Buy Wolfram' New Kind of Science?

Stephen Wolfram famous creator of Mathematica is now presenting a new approach to the 2nd Law of Thermodynamics based on A New Kind of Science in the form of fundamental computational rules forming a Ruliad. 

Wolfram does not belong to the inner circles of modern physicists where he is met with skepticism because of "too much computation" with the Ruliad leaving out the fundamental mathematical models of physics such as the Euler, Maxwell and Schrödinger equations, because they do not have Ruliad computational form. 

I have suggested to Wolfram to meet the skepticism by including computational forms of these equations into the Ruliad following an idea I have long been pursuing based on viewing real physics as a form of analog computation which can be mimicked and understood by digital computation. 

A main result of this idea is a new explanation and understanding of the 2nd Law as an expression of (i) finite precision computation and (ii) instability or non-wellposedness as detailed in Computational Thermodynamics and in popular form in The Clock and the Arrow. 

The rule is here a computational form of the Euler equations for fluid flow with solutions exhibiting unavoidable irreversible turn to turbulence as an understandable expression of the 2nd Law. 

Recall that the 2nd Law is the main unresolved mystery of classical physics, which modern physicist no longer care about. 

I thus share the idea of Wolfram to view physics as a form of analog computation, but think his Ruliad is too limited to contain real physics. 

It remains to see if Wolfram is interested to expand his Ruliad to include e g the Euler equations in computational form. That would meet the criticism from main-stream physicists. 

onsdag 28 februari 2024

Classical Universal vs Modern Man-Made Physics

Stopping a roulette wheel by hand as man-made physics.

The recent posts on Wolfram's new proof of the 2nd Law of Thermodynamics exhibit a basic difference between 

  • Classical physics as universal deterministic physics independent of human observation.
  • Modern physics as man-made statistical physics dependent on human observation. 
The change from classical to modern physics was initiated by Planck in 1900 in an analysis of blackbody radiation extending Boltzmann's statistical mechanics of material particles to light particles, and then picked up by Einstein as photons in 1905. 

The break with classical mechanics was the introduction of statistics as a man-made concept used by insurance companies collecting data to compute customer tariffs, without any corresponding agency in Nature. 

The next break came with Einstein's relativity theory giving different observers different conceptions without common ground, and the final break came with Born's statistical interpretation of Schrödinger's wave function of quantum mechanics stating that electrons in atoms play roulette with an act of observation influencing the physics by stopping the roulette wheel by human hand, referred to as collapse of the wave function

These were all breaks with the universality of classical deterministic physics independent of human observation into different forms of man-made statistical physics, thus giving up the basic cause-effect of deterministic physics, as a collapse of physics. 

The reason to make this immense sacrifice, with far-reaching loss of rationality, was a perceived limitation of classical deterministic continuum physics to describe (i) black-body radiation, (ii) atom physics and (iii) speed of light. 

But it is really necessary to pay this major scientific cost into bankruptcy of modern physics? 

Is it possible that (i)-(iii) can be handled within classical deterministic continuum physics? 

I present a positive answer here: 
Take a look! Why pay a high price for something of questionable value ruining your economy, if it is not necessary? 


måndag 26 februari 2024

Man-Made or Universal 2nd Law of Thermodynamics?

Stephen Wolfram presents after 50 years of brooding a resolution of the mystery of the 2nd Law of Thermodynamics, which never got a satisfactory answer in classical physics, nor by modern physicists occupied with other mysteries.

Wolfram's basic idea is that human observers are computationally bounded and so have to reduce a very complex detailed partially random world into something simpler in the form of mean values, which makes evolution in time irreversible and so gives a direction of time. 

The 2nd Law to Wolfram thus emerges as a man-made law of physics resulting from computational boundedness of human beings, to be compared with a universal law of physics independent of human limitations. 

Let me compare Wolfram's resolution with the one I present in this book. We both view real physics as a form of analog computation, which can be simulated by digital computation in mathematical models, to Wolfram taking the form of man-made Rules and to me universal differential equations. 

To Wolfram computational boundedness reflects boundedness of human observers, while I seek to give it a universal meaning in the sense that real physics is a form of analog computation with finite precision.

As a key example the book takes Euler's equations for incompressible flow a fluid with vanishing viscosity from 1755 expressing (i) Newton's 2nd Law and (ii) incompressibility. With vanishing viscosity this mathematical model is a parameter free universal model, which Euler prophetically claimed would describe all of slightly viscous incompressible fluid flow, if only the equations could be solved which had to wait 250 years...

Solving Euler's equations computationally after suitable discretisation, produces solutions which are turbulent with well determined mean values under different discretisation, while point values fluctuate in a seemingly random unpredictable way. 

Turbulence appears from instabilities created by convection into increasingly large velocity gradients which ultimately are controlled by dissipation into heat, without which the flow would cease to exist. This is not a process only in the eyes of humans, but a universal process necessary to allow the world to continue to exist and not come to a stop: The show must go on! 

It is a process which is irreversible since heat energy in the form of small scale kinetic motion once produced in turbulence from large scale kinetic motion, cannot be reversed because of finite precision. 

Computational solution of Euler's equations thus offers a universal model satisfying a 2nd Law, which can be understood to emerge from finite precision computation + convective instability thus without mystery.  The macro world so emerging is independent of the level of finite precision or resolution of microscopics as an important aspect of universality in particular making turbulence computable with laptop power.

Sum up: Wolfram presents a man-made 2nd Law while I present a universal 2nd Law free of human perception. Your choice!

For an explanation of the 2nd Law in popular form, see The Clock and the Arrow.

PS1 The code for computing solutions to Euler's equations can be seen as a Rule in Wolfram's sense, which however is not ad hoc man-made but expresses universal Newtonian physics. It is in fact not easy to ad hoc invent a Rule which expresses the wide range of scales of turbulence captured by Kolmogorov and Euler solutions as a universal phenomenon.  

PS2 The 2nd Law of thermodynamics is classically expressed as an inevitable increase of entropy, however without any convincing specification of this concept in physical terems. The book Computational Thermodynamics presents a 2nd Law in terms of kinetic energy, internal energy and turbulent dissipation all with clear physical meaning, thus without having to invoke the troublesome concept of entropy.

PS3 I have contacted Wolfram asking for a discussion, and received positive response from his entourage but not reached all the way. Wolfram is viewed to be super smart and so I would certainly learn a lot from an exchange of ideas...which I will report once it happens...

PS4 Wolfram is not a main-stream modern physicist (nor am I) and is quite lone in his quest for the truth of the 2nd Law, abandoned with the advent of modern physics in 1900.  

PS5 You may compare with Sabine Hossenfelders Do We Create Reality?

söndag 25 februari 2024

Newton vs Big Bang, Dark Energy and Dark Matter

In her latest post Sabine Hossenfelder asks if we can get energy for free e g in the form of Dark Energy as a main mystery of modern cosmology. Let us see what Newton can bring to this question starting with his law of gravitation: 

  • $\Delta\phi =\rho$ or $\rho =\Delta\phi$          
connecting mass density $\rho (x,t)$ to gravitational potential $\phi (x,t)$ though the Laplacian differential operator $\Delta$ with $x$ a Euclidean space coordinate and $t$ time. 

The standard view is that mass density is non-negative $\rho (x,t)\ge 0$ for all $(x,t)$, but if we expand the scope why not allow $\rho (x,t)$ to also locally be negative, then corresponding to some form of negative mass. If we dare to take this step, we find the following remarkable facts:
  1. With $\rho (x,t)$ an initial vanishingly small perturbation of an initial zero state varying very quickly in space between positive and negative values, the corresponding potential $\phi (x,t)$ will inflate to substantial size, as if gravitational potential is created out of nothing. This may correspond to a Big Bang from which a Universe filled with both positive and negative mass can evolve. 
  2. Regions with negative mass density repel regions with positive mass density and so create an expansion seemingly out of nothing, which may correspond to Dark Energy, while larger regions of small positive and negative mass density can form and then locally contract by gravitational attraction into galaxies with large local density.
  3. Large regions where $\phi (x,t)$ is slowly varying with $\rho (x,t)=\Delta\phi (x,t)\ge 0$ small may correspond to Dark Matter, which is not visible but still has major gravitational effect.     
In one shot, we thus open to new views on both Big Bang, Dark Energy and Dark Matter. Any comment?

More substance to such a scenario is given in blog posts on New Newtonian Cosmology. 

lördag 24 februari 2024

Wolfram: What Is an Observer?

Stephen Wolfram has put forward a new explanation of the 2nd Law of physics based on physics as a form of computation with computational irreducibility as key concept.  Wolfram now complements with a new view on the role of an Observerwhich is highlighted in the modern physics of both relativity and quantum mechanics in contrast to classical physics seeking universality.   

Wolfram starts seeking an answer to the question: 

  • What is an observer like us? 

Wolfram thus focusses on observers as humans with our senses and instruments, and suggests that we as human observers through our observations in some sense are generating laws of the world which fit our minds and so help us to explain and understand the World. Wolfram thus seems to say that laws of physics are not universal but man-made.

In particular, Wolfram suggests that the 2nd Law of thermodynamics is not a truly universal law of physics, but rather a law perceived by us as human beings from observation of things tending to get more random over time. Wolfram recalls that the attempts in the late 19th century to give the 2nd Law a universal meaning/explanation free of human perceptions of randomness by in particular Boltzmann, all failed and so gave a deadly shot to classical physics and so prepared modern physics to accept a new key role of an Observer.

But is it really sure that the 2nd Law cannot be given a universal meaning free of human observation? 

My contribution together with Johan Hoffman to this question is a proof of the 2nd Law in the setting of Euler's Model:

  • (i) the Euler equations for nearly incompressible slightly viscous flow in the form of mathematical equations expressing Newton's law's of motion and incompressibility without presence of any parameter,
  • (ii) combined with a computational algorithm for computing best possible solutions to the equations in the sense of a best combination of strong pointwise solution and weak mean-value solution. 
Euler's Model describes all of nearly incompressible slightly viscous fluid flow such as that of water and of air at medium-high velocities, in the same way Maxwell's equations describe all of electromagnetics, in addition in parameter free form not requiring human input.

A 2nd Law for Euler's Model can be formulated and proved as the necessary appearance of turbulence for which mean-values are computable but point-values are not, which shows irreversibility

Any form of sufficient intelligence using (i) and (ii) would see the same world of fluid flow and the same 2nd Law, and so universality would be present. 

What does Wolfram say? 

fredag 23 februari 2024

Motion vs Appearance or Emergence

This is a continuation of earlier posts on Zeno's paradox as an unresolved mystery of the physics of motion:

  • How can an arrow move, when at each time instant it is still, that is, not moving?
  • Is the arrow jumping from one position to the next in a discrete series of events in increasing time? 
No convincing resolution is offered by either classical or modern physics, and so the question is dismissed as a no-question so obvious that it does not need any explanation: Just look and see how things are moving  or shifting positions in space $x(t)$ with time $t$, with velocity $v(t)=\dot x(t)$ and the dot signifies differentiation with respect to time. 

Given a velocity $v(t)$, the corresponding motion/trajectory $x(t)$ is created by exactly solving the differential equation $\dot x(t)=v(t)$ (as if the arrow is smoothly changing position in time without jumps), or by time-stepping from one discrete time instant to a next (as if the arrow de facto is jumping).   

But a child eager to understand the World may not be satisfied with such an empty explanation, but maybe by the following argument:

Let us compare the concept of motion with that of appearance or emergence. If a certain person appears at a party, invited or not, the question may come up how the person got there, more precisely what trajectory of motion the person had followed? Today the path would be stored in the cloud, but then as a discrete sequence of still-positions just like the arrow, and the basic question would remain: How is motion possible at all? But fact is that the person did appear and so let us shift focus from motion to appearance.

We then take Newtonian mechanics to our help which describes the World by the following conservation laws in Eulerian form:
  • $\dot\rho +\nabla\cdot m=0$                                (conservation of mass)                          (1)
  • $\rho =\Delta\phi$                                          (conservation of gravitational force)     (2)
  • $\dot m +\nabla\cdot (vm)-\rho\nabla\phi=0$        (conservation of momentum)                (3)                                                            
where $\rho (x,t)$ is mass density, $\phi (x,t)$ gravitational potential, $m$ is momentum, $v= \frac{m}{\rho}$ is velocity and $x$ a Euclidean space coordinate.

The standard way of interpreting (1)-(3) is to say that presence of mass at $(x,t)$ creates the gravitational potential $\phi (y,t)$ for all points $y$ different from $x$ by instant action at distance at time $t$, which however lacks physics explanation. Further, trajectories of motion $x(t)$ appear as solutions to $\dot x=v(x,t)$. 

I have suggested a different possibility, which is to view instead the potential $\phi (x,t)$ as primary from which mass $\rho (x,t)=\Delta\phi (x,t)$ is created by differentiation as an instant local action expressed by the Laplacian $\Delta$, which possibly is not inexplicable. The potential $\phi (x,t)$ then changes or evolves in time according to (1) with connection (2), without any need of particle trajectories of motion, 

In this view mass emerges or appears at different locations in space following the evolution of the gravitational potential, and we do not have to speak about particle/mass motion and explain exactly how the motion is realised. It connects to time-stepping corresponding to jumping from one discrete time event to the next. 

So it may be fruitful to think of appearance evolving in time rather than motion. In this perspective motion is illusionary, like a water wave appearing to move in space without corresponding motion of water.  


tisdag 20 februari 2024

Speed of Gravity in a Static Gravitational Field?

To save General Relativity GR assuming that the speed of gravity is finite equal to the speed of light from collapse when confronted with observations apparently requiring a very much larger speed of gravity, it is commonly stated that in a static gravitational field there is no effect of time delay from finite speed of of propagation. And so common wisdom claims that there is no contradiction between GR and observations conforming to a speed of gravity much larger than the speed of light. 

It is this convincing? No problem in GR from finite speed of gravity? Let us see: A gravitational model with finite speed of propagation $c$ in a Newtonian approximation of GR takes the form

  • $\frac{1}{c^2}\ddot\phi -\Delta\phi =\rho $,       (*)
where $\phi (x,t)$ is the gravitational potential corresponding to a mass distribution $\rho (x,t)$, where $x$ is a Euclidean space coordinate, $t$ a time coordinate and the dot signifies differentiation in time. 
Now a static gravitational potential is characterised by $\ddot\phi =\dot\phi =0$ and so the value of $c$ can be anything, in particular as large as desired even larger than the speed of light without changing anything. In other words it is meaningless to speak about speed of gravity in a static gravitational field. 

To state that in a static gravitational field in GR there is no effect of finite speed of gravity does not make sense. There is no speed at all. 

Further, gravitational fields are not static, not even between the Sun and Jupiter, and so this case lacks interest. 

Yet in GR the speed of gravity is viewed to be finite = speed of light c, which requires a theory of quantum gravity to explain finite speed. But no theory of quantum gravity has been found despite intense search for 100 years. Further, gravitational waves in GR are viewed to require merge of black holes to appear... 

The idea of a finite speed of gravity = speed of light is the main road block to a Theory of Everything ToE combining Newton, Maxwell and Schrödinger. What would happen if we simply remove the block by replacing Einstein by Newton? What would be missed? Nothing? And then?