A 50-Year Quest: My Personal Journey with the Second Law of Thermodynamics
writings.stephenwolfram.comI remember being thrilled by his 2015(?) piece (the really long "cellular automata as universal foundation" one).
Felt like a proper stab at combining physics + CS, thin on details, but fertile ground. Wolfram is an explorer there, barely charting the edges of an unknown continent.
The speed of light, light cones, and speed/time equivalence make a ton of sense through the lens of updates propagating through a grid of cells.
Don't remember the QM part so well, but from what I do remember, he proposes that probability/alternate timelines are subject to the same computational constraints over probability space, that physical objects have in real space.
As an aside, entangled particles were only ever a conceptual issue, right? From an engineering perspective they seem completely practical.
Suppose you and I depart from point A simultaneously. We both travel in spaceships to point B and arrive there at the same time. We both took round-a-bout paths, but you travelled at relativistic speeds and I went much slower. The clock on your ship shows less elapsed time than mine. You didn't slow your clock or something, your time-distance through space/time was actually shorter than mine. That is... very weird. We were both at co-ordinates (Ax, Ay, Az, At) and (Bx, By, Bz, Bt), but Bt - At is larger for me than for you. We can't agree on what the co-ordinate values are or how much they differ. I don't think a simple cell grid is going to capture that.
Entanglement is also deeply weird. I'm not a physicist, but believe it works this way: Particles A and B are entangled. Later, particles A and C become entangled. That doesn't mean that A and B are no longer entangled. Schroedinger's Cat becomes entangled with the particle which does/doesn't decay. When I open the box I become, in a very complex way, entangled with the Cat. Either it is dead and I become depressed, or it is alive and I start looking for milk. But the Cat and the particle are still entangled. If someone, unaware of my existence, later measures the particle they will absolutely know the state of the Cat. Or they could measure me; if I'm hunting for milk then the particle state is "didn't decay." Eventually, everything becomes entangled with everything else. How is that kept track of? How massive (literally) would that amount of information be?
Wolfram's "grid cell" concept is based on cellular automata (similar to Game of Life), where each cell has simple rules that, when taken together, add up to complex overall behavior (emergent phenomena).
Following it further:
The speed of light is due to the refresh rate of these cells. The passage of time (and relationship b/t it and velocity), is because of computational limits on those cells. A cell has limited processing power that can either be spent on updates (the passage of time) or I/O (velocity). Objects also appear to shrink when moving at relativistic speeds, right? Another computational optimization, smaller means less cells are used.
It's a pretty neat mapping. A spaceship is moving at relativistic speeds, which has a large I/O cost on the cells it exists in. That cost is made up in different ways, depending if you're inside or outside the spaceship.
On the inside:
time moves slower, saving time cycles for I/O cycles
On the outside:
the spaceship physically shrinks, using less cells total
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Not too sure about entanglement, but I believe it's tied to eventual consistency. You have the direction backwards though, yeah? The point of Schoedingers cat is that while it's in the box, it may as well not exist. Everything being entangled is just the regular universe.
I'm likely wrong in some subtle (or not subtle) way, but here's how I think of time dilation:
We, and everything on Earth, are traveling through 4D spacetime at the speed of light. Most of that is in the time domain, because the Earth, Solar System, Galaxy, etc., are all moving a very small fraction of the speed of light, so the X,Y,Z components are small. (Kind of like being on a Bloch sphere in quantum computing.)
Any time you move in X,Y,Z, to keep the total magnitude of of spacetime the same, the time component is necessarily smaller. You have to push other particles in the opposite X,Y,Z directions for this to happen, a paradox occurs to me
Does the total momentum through time get conserved? If so, how does that work?
I think it's pretty obvious that the second law comes from reversibility of the microscopic laws.
In classical mechanics, Liouville's theorem makes phase space density incompressible, so you get that entropy always increases. In quantum mechanics, unitarity gives the same incompressibility - the density matrix change with time is multiplying by a unitary matrix left and right, which preserves the eigenvalues, so the entropy, being sum of the log of eigenvalues, stays the same.
The opposite is also obvious: in a system which is irreversible, look at an ensemble of two states which evolve to a single state. The entropy before was 1 bit, after the evolution it's 0 bits. So the system violates the second law.