Time is a dimension, but not like space

I only got my undergrad in physics, but I think there is something there to be mined between time as a dimension and the second law of thermodynamics. Why this one?

First, I will render a quote which never failed to amuse me: "The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations -- then so much the worse for Maxwell's equations. If it is found to be contradicted by observation -- well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it to collapse in deepest humiliation." (Eddington)

Why such honor? For one, in statistical physics, you can more or less derive the second law of thermodynamics, from scratch. No need for observation. It's just there the same way the quadratic equation is. Somewhere I have a cheap Dover reprint which contains a relatively easy to follow construction of the second law. It's the math. You can measure things badly, you can find one phenomenon creating the appearance of another, but you cannot fool The Math.

And so the statistical physics you can get from just math gives you this arrow of time, flying only one way, just as we see from spacetime.

To me, and again, I only got a few grad courses under my belt in it, this suggests not just a deep connection between entropy and spacetime, but the inevitability of it from the basic math (really, a talented high schooler could be coached through it) means that there is something about large (for n = ?) numbers of particles losing the reversibility which is so often present in particle interactions where n is smaller. What gives there? How do we go from this "trend" emerging to it being a property of spacetime even if no particles are sitting in said spacetime.

Not that I would have dared write the great Wheeler, but I have wondered if his "geon" concept would have fit in with this sort of thing. It seems so fundamental. One can imagine a universe with a different number of un-unified forces, or gravity dropping as the inverse-cube, or varying physical constants, but the math is still the same in these universes and it then suggests that there's no, uh, room for an option wherein the time facet of spacetime is anything but an arrow flying forever on towards entropy in its many masks.

A great task, or perhaps a very alluring windmill, for someone younger and brighter than I.

The older I am (and I am at my 50s) the more I have this intuition that entropy is a fundamental force driving not only physical phenomena but also social interactions, economy etc.

Formalising this intuition is another story though...

We have the sun to provide us with low entropy energy and the atmosphere to dissipate high entropy energy, so stuff tbat happens on earth can be lowering in entropy possibly. Climate change excepted.

From the Wikipedia page on the second law of thermodynamics:

>For example, the first law allows the process of a cup falling off a table and breaking on the floor, as well as allowing the reverse process of the cup fragments coming back together and 'jumping' back onto the table, while the second law allows the former and denies the latter. The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always tend toward a state of thermodynamic equilibrium where the entropy is highest at the given internal energy.[4] An increase in the combined entropy of system and surroundings accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time.[5][6]

Tools from statistical physics have long been used in sociological and economical models.

It's no wonder, because statistical physics was devised as a tool for the study of complex systems.

For the same reason I don't deem entropy to be a fundamental property of physics, but one of complex systems. As far as I remember from university, the 2nd law of thermodynamics simply arises from the fact, that there are exponentially more unordered than ordered states.

Though information itself may be a fundamental physical property. The recent interest in Quantum computers shines new light on the connection between information and Quantum Mechanics. It remains to be seen, how that point of view is compatible with relativity.

I hope, that one day someone finds out, that the "time dimension" arises in the macroscopic limit from a graph of discrete causal events.

I'm in a similar boat, but I've always felt the opposite! I've always felt the second law is kind of a shoe-in and maybe even shouldn't be a law at all.

The first and third laws, "energy is never created or destroyed" and "for every action there's an equal and opposite reaction" are always true! To my knowledge, no process is ever allowed to break either law. (With exceptions for cosmological process like the expansion of the universe that we really don't purport to understand.)

The second, "entropy can only increase" isn't! That's right, I said it. The processes it describes (a cup unshuttering, or coffee unmixing, or particles all finding their way into the same side of a box) are totally legal process, albeit statistically unlikely. If you restrict your system to few enough particles (say, n=3), random processes that decrease entropy are not only possible, but something that happens with regularity!

Now, I make no claims to be right here. I suspect that Eddington fellow probably knows what he's talking about. But, this has been a longstanding thorn in my understanding of physics, so I'd be interested if anybody has any interesting insights!

Entropy is also defined as the number of different arrangements of particles in a system. We say that entropy is increasing in our universe. But we have also found that space is increasing. If space increases faster than particles move, entropy could even decrease

> With exceptions for cosmological process like the expansion of the universe that we really don't purport to understand

That's not accurate, expansion of the universe (that the standard model of cosmology describes) does not violate conservation of energy. It makes it a little different from the classical view.

In classical mechanics, energy conservation is a well-defined concept in a static or non-expanding spacetime. However, in an expanding universe, especially one described by general relativity (like ours), the energy of the universe is not necessarily conserved in the traditional sense, because the global energy of the universe is difficult to define when spacetime itself is dynamic (expanding)

So GR does not require global conversation of energy in the same way classical (here classical means strictly newtonian mechanics) mechanics does. This dynamic nature of the spacetime allows for energy to appear to "change" due to the expansion. It is more complicated when you add things like dark energy to the equation.

One interesting aspect is the phenomenon of cosmological redshift. As the universe expands, light travelling through space is redshifted. This means that ita wavelength increases and its energy decreases. This "loss" of energy from light is not violating conservation of energy. It is rather consequence of the expansion itself.

Now lets back to dark energy which is driving the accelerated expansion of the universe, the energy associated with the vacuum of space remains constant per unit volume, but as space itself expands, the total energy associated with dark energy increases. This again does not violate the laws of physics because energy conservation is more complex in general relativity than in Newtonian mechanics. And of course the local energy conservation works in a well-defined way if you take a localized region of the spacetime.

Not disputing anything you said but... The issue I see is that without notion of time it is difficult to talk about causality (I guess you could only talk about "entanglement"). It is actually difficult to talk about "events" at all - I guess you can only talk about "facts"?

Could you pls. point to the said book or some other resource for me to learn about this? As such, I follow what you have said, but would love to see the math too. Thanks.

Ilya Prigogine wrote a lot about this, see eg Order Out of Chaos. He won the Nobel prize in chemistry for his work on nonequilibrium thermodynamics.

Boltzmann had originally tried to prove that the second law is a mechanical (statistical) fact and several others tried as well. But Poincaré showed that those systems which, regardless of their initial state, inevitably increase entropy later on inevitably reduce it (his recurrence theorem). There are also in fact reversible processes in nature, so it can't be that mechanics alone implies the procession of time. Something more involved is going on.

Carlo Rovelli has also written a lot about the "thermal time" concept in his books.

My takeaway from them is that you can't really get time out of mechanics by itself (statistical or otherwise). In the same way that you can't get baryon asymmetry. It is intrinsically a selection principle on initial conditions.

Thanks a lot for a detailed and simple answer! I have many keywords now to read more on.

I just read about Poincaré's recurrence theorem here: https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theor...

Even with this, would the Poincaré recurrence time not be (probabilistically) be very large for any complex system, and thereby practically imply the second law of thermodynamics?

I also just re-read about Maxwell's demon (https://en.wikipedia.org/wiki/Maxwell%27s_demon), but see that there are already various arguments against it even before arguments about the demon moving the system towards Poincaré recurrence.

Thanks.