## lördag 19 augusti 2023

### On Wolfram's Concept of Computational Irreducibility

This is a continuation of the previous post.

Computational Irreducibility CI is used by Stephen Wolfram to describe that prediction of the state of a system requires full resolution of the evolution of the system by computation; to predict the outcome of tossing a coin by computation requires precise resolution of the motion of the coin from toss to landing. Wolfram seeks an explanation of the 2nd Law in terms of CI as an expression of Computational Impossibility.

Wolfram's agenda (initiated when he was 12) connects to mine (since 90s), both serving to give an explanation of the 2nd Law or irreversibility or direction of time, based on computation, although fundamentally different in details.

As a key example I consider turbulent fluid flow which is (mean-value) computable forward in time but not backward in time, thus computationally reducible forward in time but computationally irreducible backward in time, which gives time a direction. More precisely, consider an airplane wing moving through air from left to right meeting still air and leaving a wake of turbulent air. Drag and lift of the wing as force mean values are computable without resolving the flow to physical scale, thus expressing computational reducibility as shown in the book Computational Turbulent Incompressible Flow.

However, reversing the flow with thus incoming turbulent flow from the right, will not give back the still air to the left, but instead a turbulent wake. To return to still air at the left after reversing the flow, would require solving an exponentially unstable problem without any cancellation (in associated linearised dual problem), thus suffering from CI.

The flow of air around a wing is thus an example of irreversible flow, with the irreversibility resulting from lack of cancellation in laminar flow, as carefully explained in the book. The forward problem is computationally reducible, while the backward problem is computationally irreducible = Irreversibility = 2nd Law.

Turbulent dissipation of large scale kinetic energy into small scale heat energy can serve as a prime example of an irreversible process, which is open to study by computational solution of the Euler/Navier-Stokes equations, see post on Euler CFD and Euler's Dream. Construction of such a model from scratch in the spirit of Wolfram's New Kind of Science has not been made. More generally, processes involving some form of friction or other losses are irreversible for the same reasons as presented.

A key fact is that these losses cannot be avoided, because forward evolution without losses requires infinite precision/infinite computational yet is necessary, while the show must go on and the show is not just heat diffusion.

Recall that heat diffusion is easy forward but difficult backward and so is an example of irreversibility, but a trivial example. Turbulent flow is a non-trivial example.

Wolfram's computational physics has been met with skepticism by the physics community, and so his version of the 2nd Law may not be applauded. My version stays closer to real physics in the form of real turbulent flow...

It is remarkable the the simple code of Euler CFD (as an extended version of the 5-point scheme for the 2d Laplacian) generates the whole spectrum of turbulent flow. Remarkable.

Recall the essence of the 2nd Law: Physical systems with nearby particles moving with different velocities, typically arising in slightly viscous flow turning turbulent or moving parts mechanics under friction, develop increasing velocity gradients controlled by turbulent/friction dissipation into heat energy as small scale unordered kinetic energy. This is an irreversible process which cannot be avoided.

In other words: Perpetuum mobile in the form of a system with moving parts is impossible: There will always be some turbulence/friction transforming motion into heat energy in an irreversible process.

On the other hand an atom in ground state is a form of perpetuum mobile, but then in the form of a system without moving parts, as explained in Real Quantum Mechanics: Electrons in ground state do not move around the kernel, but rather turn around like someone unable to get to sleep, while the electron density is stationary. On the other hand, in a radiating atom, electron densities do shift between energy levels under energy exchange.