The 2nd Law in a World of Finite Precision

Let there be a World of Finite Precision.

Here is a summary of aspects of the 2nd Law of Thermodynamics discussed in recent posts: 

• 2nd Law gives an arrow of time or direction of time. 
• A dissipative system satisfies a 2nd Law.
• A dissipative system contains a diffusion mechanism decreasing sharp gradients by averaging. 
• Averaging is irreversible since an average does not display how it was formed. 
• Averaging/diffusion destroys ordered structure/information irreversibly. 
• Key example: Destruction of large scale ordered kinetic energy into small scale unordered kinetic energy as heat energy in turbulent viscous dissipation.    

To describe the World, it is not sufficient to describe dissipative destruction, since also processes of construction are present. These are processes of emergence where structures like waves and vortices with velocity gradients are formed in fluids, solid ordered structures are formed by crystallisation and living organisms develop. 

The World then appears as combat between anabolism as building of ordered structure and metabolism as destruction of ordered structure into unordered heat energy. 

The 2nd Law states that destruction cannot be avoided. Perpetual motion is impossible. There will always be some friction/viscosity/averaging present which makes real physical processes irreversible with an arrow of time. 

The key question is now why some form of friction/viscosity/averaging cannot be avoided? There is no good answer in classical mathematical physics, because it assumes infinite precision and with infinite precision there is no need to form averages since all details can be kept. In other words, in a World of Infinite Precision there would be no 2nd Law stating unavoidable irreversibility, but its existence would not be guaranteed.  

But the World appears to exist and then satisfy a 2nd Law and so we are led to an idea of an Analog World of Finite Precision, which possible can be mimicked by a Digital World of Finite Precision (while a possibly non-existing World of infinite precision cannot). 

The Navier-Stokes equation for a fluid/gas with positive viscosity as well as Boltzmann's equations for a dilute gas are dissipative systems satisfying a 2nd Law with positive dissipation. But why positive viscosity? Why positive dissipation?

The Euler equations describe a fluid with zero viscosity, which formally in infinite precision is a system without dissipation violating the 2nd Law.  

We are led to consider the Euler equations in Finite Precision, which we approach by digital computation to find that computational solutions are turbulent with positive turbulent dissipation independent of mesh size/precision once sufficiently small. We understand that the presence of viscosity/dissipation is the result of a necessary averaging to avoid the flow to blow-up from increasing large velocity gradients emerging form convection mixing high and low speed flow. 

We thus explain the emergence of positive viscosity in a system with formally zero viscosity as a necessary mechanism to allow the system to continue to exist in time. 

The 2nd Law thus appears as being a mathematical necessity in an existing World of Finite Precision.   

The mathematical details of this scenario in the setting of Euler's equations id described in the books Computational Turbulent Incompressible Flow, Computational Thermodynamics and Euler Right.

PS A related question is if in a World of Very Limited Human Intelligence a WW3 destruction is necessary?