# B Big Crunch, Big Bang and information loss

1. Jul 18, 2017

### kimbyd

This discussion of information and entropy, by the way, highlights just how difficult it is to nail down "information". Information is a nebulous concept that can refer to a large variety of physical properties.

2. Jul 18, 2017

### jonk75

You misunderstand the information theoretical meaning of "information." The higher the entropy of a system, the more information it contains, because it takes more information to describe it exactly. A room full of gas is not completely described by its pressure, temperature, & volume - e.g. you know nothing about the state of any particular molecule in that example. A full description would require the state of every single molecule to be described individually, which is a lot of information. If the gas was in the ground state (low entropy), it could be described easily by saying all molecules are in their ground state - i.e. low information.

3. Jul 19, 2017

### BernieM

Hold up a second here. Let me get this right. You are saying that a disordered system should be in a higher state of entropy than an ordered system, that the disordered system has more information contained in it, because it takes more information to describe it.

You have a room full of atoms that are identical and at absolute zero.
You have another room full of atoms that are not all the same and are at varying temperatures.

What would be the additional information that would differentiate the disorderly chaotic similar atoms from the orderely atoms?

Would the atoms in the room with high entropy have 2 spins per electron? More charges? No temperature?

But even zero temperature is a temperature. The only difference is that in one case we can make rules about all the atoms and so we don't have to write down say the temperature of every individual atom, as they are all at absolute zero.

So we save some space in our book that we are writing this down in. That's it. Each atom still has a temperature, even if it is absolute zero. Still has a spin, mass, charge, motion, etc. (yes it has motion because everything in the universe is moving, even if particular atom or group of atoms has no thermal vibrations.)

In a real world model you already have motion, of which, the atoms' thermal motion is merely a minute moderation of that movement vector. But it doesn't really change the magnitude of the information needing to be stored, since the atom's macroscopic motions are many magnitudes larger than the thermal vibrations. Again, no extra information is needing to be stored about the atom in a chaotic state as opposed to one that is in an ordered state.

Entropy = information? Or does entropy = the complexity of recording the information.

4. Jul 19, 2017

### Staff: Mentor

This is not quite right. The complete microstate of the system takes the same information to specify no matter what the macrostate is. It's difficult to be more specific in a "B" level thread, but a more technical way of stating what I just said would be that the dimensionality of the system's phase space is the same regardless of its macrostate. Macrostates are just a way of picking out regions in the phase space and saying that they are all "the same" according to some macroscopic criterion, such as temperature, pressure, etc. This is called "coarse graining" the phase space, and it has to be done before we can even define entropy.

Once you have a coarse graining of the phase space, the entropy of the system is, heuristically, $\ln N$, where $N$ is the number of microstates that are in the same coarse-grained category as the system's actual microstate. A system exactly in its ground state--zero temperature--has lower entropy than a system at some finite positive temperature because $N$ is smaller. But that doesn't change the amount of information needed to specify the system's microstate at all--it's a point in a phase space of some number of dimensions, and the number of dimensions, which is what determines the "amount of information" needed to specify the state, never changes.

5. Jul 19, 2017

### kimbyd

That's not at all true. The information you're talking about is the information that has been used as a definition for most of this thread: the full microscopic description of the system. As PeterDonis notes, this is unchanged as entropy changes.

I'd like to go a little bit deeper as to why it's unchanged: it's unchanged because the number of particles in this classical system is unchanged. If you're going to describe the full state, you have to describe the position and momentum of each and every particle in the system. The complexity of that description is completely independent of its configuration.

In quantum mechanics, we have a similar effect going on, even though the number of particles does change. This brings us back to the concept of unitarity, which I'd like to try to explain again in different words.

A unitary operator has a simple definition:

$$U^\dagger U |\psi\rangle = |\psi\rangle$$

That is, if I operate on a state by an operator $U$, and then operate on it again by what is known as the "complex conjugate" $U^\dagger$, then I get the original state back again. Fundamentally, this means that the state $|\psi\rangle$ and the state $U|\psi\rangle$ contain the exact same information.

To try to take this back down to Earth, the operator that lets you see what a state looks like at a different point in time is a unitary operator. So I can look at a state at a future time by operating it with the right unitary operator, and I can then use the complex conjugate to get the original state back.

As long as the "time translation operator" is unitary, then information is conserved.

6. Jul 19, 2017

### Staff: Mentor

To be more precise, it's the complex conjugate transpose--i.e., if you have a representation of $U$ as a matrix with complex entries, then $U^\dagger$ is the matrix you get by transposing $U$ and then taking the complex conjugate of all entries.

7. Jul 19, 2017

### jonk75

That is the crux of it. If it takes more space to describe it in a book, that is more information. A large book contains more information than a small book.

This is probably getting too deep for a discussion here though. You should read up on information theory. A good pop-sci book is James Gleick's "The Information: A History, A Theory, A Flood". A good Wikipedia discussion of the link between Shannon entropy (quantifying information) & thermodynamic entropy is here: https://en.m.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

8. Jul 19, 2017

### jonk75

This is not correct. In a high entropy state, each dimension has a seemingly random value, & every value needs to be specified individually to fully describe the system. In a low entropy state, say the ground state, each dimension has the value 0, & is described that simply.

e.g. If I represent the state as a vector with a million dimensions, to write down the exact state of the system when it has high entropy would take many pages - a lot of information. On the other hand, if the system is in its ground state (low entropy), I can simply describe it by saying, "The value of each dimension is zero." It takes almost no space at all - it has very little information.

If you were correct, then compression of information in software wouldn't be possible.

9. Jul 19, 2017

### Staff: Mentor

Sorry, but your bare assertion is not enough. You're going to need to find some valid references (textbooks or peer-reviewed papers) that support your position. I think you will be unable to do that (see below), but you're welcome to try.

This is nonsense. The "dimensions" don't have values. The number of dimensions in the phase space just tells you how many numbers you need to specify a point in the phase space, i.e., a microstate. This is the same for every microstate.

I think you need to actually look at some textbooks. Your understanding of how the microstate of a system is specified is incorrect.

The ground state of a system has lower entropy because there are fewer microstates that have the same values for some chosen set of macroscopic variables (temperature, pressure, etc.). It has nothing to do with the amount of information needed to specify a given microstate.

This is not correct. I strongly suggest that you take some time to learn the correct physics from a textbook.

Software compression is irrelevant to what we're discussing here.

10. Jul 20, 2017

### BernieM

Well now that that is cleared up.

If I were to go back to the big bang (just a moment after) when the state of the universe at that point is essentially calculable (say at some super hot point that is yet too hot for matter to exist yet) and assign a value to how much information was contained in this universe at that moment, then move forward in time until precipitation of matter occurred, and assign a value then to the quantity of information in the universe at that moment, and compared the two, what would I see? Would I see an increase in the information, a decrease, or would it have remained the same.

When the universe is in a pure energy state, the magnitudes of things are much higher, but I don't think there are a lot of features. So it's more like a 1d array at this point.

Enter a particle and now the array is 2d or 3d perhaps, but the magnitude has been reduced (temperatures went down) and some expansion of the system occurred.

Intuitively I feel that the information in the system is maintained and doesn't increase or decrease, even with the change in state, but I can't prove that. Where do I turn to prove or disprove this?

11. Jul 20, 2017

### kimbyd

Depends a bit upon what you mean by information.

If by information you mean the full configuration of the wavefunction of the universe, then as long as the laws of physics are unitary the two points in time necessarily contain the exact same amount of information. This means that if you had the full state at the early time, you could calculate the late time knowing the laws of physics. If you had the full state at the late time, you could calculate the early time.

It comes down to whether or not the laws of physics are unitary.

12. Jul 20, 2017

### BernieM

By information, I mean all the relevant data and conditions regarding a particle that would provide me a clear enough picture that I could solve that particles prior or subsequent motion, action, interaction, and nature with certainty.

I'm guessing that proving if the laws of physics are unitary, or not, is probably not going to be able to be determined in this thread, nor by anyone in the near future, right?

13. Jul 20, 2017

### kimbyd

Yes, that's more or less the definition I assumed.

Correct. Unitarity is currently unknown, though many physicists suggest the fundamental laws must be unitary to have a sensible notion of causality. I gave an overview of what current physical laws are/aren't unitary in post #13 of this thread.

14. Jul 20, 2017

### jerromyjon

I'd say way back when all the forces were unified would be the most likely bet at having a coherent picture of things, after gravity separated you get into what we have now, spin foams and such to deal with...

15. Jul 20, 2017

### BernieM

I think I have the answer I asked for, thank you everyone.

16. Aug 1, 2017

### PeterKinnon

"What I am trying to get at is if there really is any new information being generated in the universe, or if the entire cycle of the universe, including a potential future big crunch or big whimper wasn't already predetermined at the moment the big bang came into existence or 'occured.'"

So were the works of Shakespeare predetermined at the "big bang"?

The notion that information simply "crystallizes out" as the universe evolves is seductive but our observation that much of the machinery of nature is essentially probabilistic really rules it out.

Furthermore we can safely assert that information is not conserved since simply burning a CD destroys the all extrinsic information impressed upon in as well as most of the intrinsic information inherent in its molecular structure.