Physical 2nd Law Without Statistics and Entropy

Why is there a 2nd Law of Thermodynamics?

In the book Computational Thermodynamics a 2nd Law of Thermodynamics is formulated as follows in the setting of Euler's equations for a compressible fluid/gas with vanishing viscosity (with quantities integrated in space): 

• $\frac{dK}{dt} - W=Q\ge 0$,               (2nd Law)

where $K$ is kinetic energy, $W$ is mechanical work of variable sign (positive in expansion) and $Q>0$ is turbulent dissipation as a positive quantity adding to internal heat energy. In this formulation all quantities involved have physical meaning and no notion of entropy of unclear physical meaning is present (and so is not needed). Neither is any statistics involved. 

The 2nd Law states a limit to transformation between kinetic energy and work with turbulent dissipation appearing as a loss in the form of heat energy, which according to a stability analysis in the book, always is present locally in space and time. The loss is thus inevitable. The loss is also irreversible, since time reversal violates the sign of (2nd Law) and so gives an arrow of time. Kinetic energy once converted to heat energy cannot be retrieved as an expression of the term internal energy. 

The above formulation of the 2nd Law closely connects to classical formulations preceding that of Boltzmann, who introduced statistics and entropy. A new aspect is that it is based on Euler's equations as a precise mathematical model, in particular on computational solution in a certain precise weak-strong sense reflecting that strong solutions do not exist. This gives the 2nd Law a precise computational mathematical meaning in the presence of finite precision, which can be interpreted in physical terms so can be referred to as a Physical 2nd Law. 

The reason that heat energy as micro-scale unordered kinetic energy once created in turbulent dissipation, cannot be retrieved into macro -cale ordered kinetic energy, is that it requires a very high precision which cannot be met with finite precision physics/computation.  

The Physical 2nd Law thus appears as a resolution in the digital age of a basic problem of physics, which could not be resolved within classical analytical mathematics nor statistics. 

This post connects to earlier posts on Wolfram's recently presented resolution also based on computation but in fundamentally different form. 

Modern physicists have since long left the 2nd Law behind as a trivial no-problem not asking for any resolution although it has been an outstanding open problem of physics, and have so proceeded to new orchards of string theory and multi-versa, which however have not delivered any fruits and so a return to basics could possibly be of some interest to todays fundamental physicists, or not?

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 than 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 fluid such as compressible air or incompressible water, showing an 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.  

This 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 swirling structures 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. Recall that Boltzmann used a dilute gas as his mathematical model in the form of Boltzmann's equation with his $H-theorem$ expressing his 2nd Law assuming that two particles about to collide are statistically uncorrelated, which is impossible to verify experimentally nor is very likely. 

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 of 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. 

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. 

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. 

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?

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 Observer, which 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? 

The 2nd Law vs Progress of Physics

The 2nd Law of Thermodynamics is the main enigma of classical physics left unresolved by modern physicists thus leaving the scene to other scientists. After a 50-year struggle Stephen Wolfram presents a  resolution in the form of computational irreducibility explained in the recent podcast Did Stephen Wolfram Finally Prove the 2nd Law of Thermodynamics?

The role of the 2nd Law is to explain irreversibility in macroscopic processes ultimately based on microscopical processes which are reversible in time. From where does the irreversibility come? Why is there an arrow of time pointing forward as an expression of increasing entropy or disorder?

Wolfram seeks an answer viewing physical processes as forms of analog computation subject to speed/cost limitations, which connects to my own explanation explained in the books Computational Thermodynamics and The Clock and the Arrow.  

The common idea is that the evolution of a physical system over a time step from one time instant to a next, can be viewed as a form of analog computation or processing of information subject to certain limitations forcing destruction of information which cannot be retrieved. 

I complement this general idea by offering a reason why necessarily physics evolves into more complex configurations beyond computational resolution and so require destruction of information, with turbulent fluid flow as key example. 

In fluid flow velocity differences/gradients can increase by advection as form of instability, while sharp gradients are smoothed by viscosity as a stabilising effect.  If the viscosity is small, like in air and water, the resulting flow becomes so complex that computational resolution is no longer possible as feature of turbulence, which forces destruction of information into irreversibility. 

The 2nd Law can thus be given a meaning in terms of computation of finite precision + complexity arising from instability, which can be made precise in mathematical terms with turbulent dissipation replacing the role of the mysterious concept of entropy as a measure of disorder or randomness. This is a meaning not asking for any observer, which still lingers in Wolfram's computational irreducibility.  

It is often heard that there has been no/little progress in modern fundamental physics since 1973, when string theory took over, and of course no/little progress in classical physics since 1900 when modern physics took over. 

It took 2000 years for Pythagoras to take over from Euclide with the development of Calculus forming the scientific revolution. 

Da Vinci pondering the nature of turbulent fluid flow as an expression of the 2nd Law,