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?