# Entropy of a vacuum and heat death of the universe

• Maximise24
In summary, according to the third law of thermodynamics, one could argue that a vacuum has zero entropy, since it has only one ground state and a temperature at absolute zero. However, assuming the accelerated expansion of the universe to result in a 'heat death', i.e. a state of absolute thermal equilibrium with maximum entropy (following from the second law of TD), how is this state different from that of a vacuum? If the universe expands infinitely, energy density will near zero, resulting in absolute zero temperature and the universe basically having one ground state.
Maximise24
According to the third law of thermodynamics, one could argue that a vacuum has zero entropy, since it has only one ground state and a temperature at absolute zero.

However, assuming the accelerated expansion of the universe to result in a 'heat death', i.e. a state of absolute thermal equilibrium with maximum entropy (following from the second law of TD), how is this state different from that of a vacuum? If the universe expands infinitely, energy density will near zero, resulting in absolute zero temperature and the universe basically having one ground state.

Could one not argue that, since the thermal equilibrium of a vacuum is (near-)perfect and since there is little 'usable' information, its entropy is actually at a maximum level?

In short: which thermodynamical definition of a vacuum or vacuum-like state (such as the infinitely expanding universe) is correct?

Why would the vacuum have zero entropy? Use the statistical definition
$$S=k_B~log(W)$$
where $W$ is the number of micro-states compatible with a vacuum. Now, you're saying there's only one of these micro-states. What does this mean?

Since there is no real energy in a vacuum, very little configurations or micro-states are possible. There may be some virtual effects, but can it not be argued that, due to the absence of real energy and temperature, vacuum entropy is, at least, very low? Or would you regard virtual energy as a valid source of micro-states?

Maximise24 said:
Since there is no real energy in a vacuum, very little configurations or micro-states are possible. There may be some virtual effects, but can it not be argued that, due to the absence of real energy and temperature, vacuum entropy is, at least, very low? Or would you regard virtual energy as a valid source of micro-states?

Consider a volume V containg N particles bouncing around in it. If V is large compared to N, that's a pretty decent vacuum. The N particles can be arranged in some number of microstates. Now let's make the vacuum better by increasing V while holding N constant. What happens to the number of available microstates?

Nugatory said:
Consider a volume V containg N particles bouncing around in it. If V is large compared to N, that's a pretty decent vacuum. The N particles can be arranged in some number of microstates. Now let's make the vacuum better by increasing V while holding N constant. What happens to the number of available microstates?

Surely that depends on what effect an increased V has on the particles? If, such as in the case of the expanding universe, particles get pulled away from each other and eventually disintegrate into photons, there will be a lot less interactions and thus less microstates will be possible (increasing entropy). One could say, in the heat death scenario, that, since there are only photons and since all photons are the same, only one microstate will ultimately be possible, namely that of the non-interacting free photon.

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Maximise24 said:
Surely that depends on what effect an increased V has on the particles? If, such as in the case of the expanding universe, particles get pulled away from each other and eventually disintegrate into photons, there will be a lot less interactions and thus less microstates will be possible (increasing entropy). One could say, in the heat death scenario, that, since there are only photons and since all photons are the same, only one microstate will ultimately be possible, namely that of the non-interacting free photon.

The possibility or not of interaction has nothing to do with the number of microstates.

The end state isn't just a single photon, and not sure why you think it might be. The bolded text suggests that you may have misunderstood the statement that photons are indistinguishable. That's true, but it doesn't mean that they're all the same. It just means that if I have two identical photons in a given microstate and I exchange them, I don't get to count that as a new microstate.

Maximise24 said:
Since there is no real energy in a vacuum, very little configurations or micro-states are possible. There may be some virtual effects, but can it not be argued that, due to the absence of real energy and temperature, vacuum entropy is, at least, very low? Or would you regard virtual energy as a valid source of micro-states?

Let's consider the macroscopic state "1 particle in a volume V". This is a pretty good vacuum. What is the number of microstates compatible with this?
Counting: the particle can be anywhere within the volume V, so it W has to be proportional to V, in my opinion.

Nugatory said:
The possibility or not of interaction has nothing to do with the number of microstates.

The end state isn't just a single photon, and not sure why you think it might be. The bolded text suggests that you may have misunderstood the statement that photons are indistinguishable. That's true, but it doesn't mean that they're all the same. It just means that if I have two identical photons in a given microstate and I exchange them, I don't get to count that as a new microstate.

I did indeed not express myself very well on that point, but - still in the heat death scenario - you cannot deny that the entropy of a sea of photons is very high, at least higher than in any other state (since it is the last macrostate of the universe): this is a direct application of the second law of thermodynamics, which says that entropy always goes up.

If not the expansion of the universe and the associated reduced effect of gravity, then what does, according to you, produce this entropy?

And, related to that, how would the heat death macrostate be different from that of the classical vacuum, which is sometimes said to have zero entropy or indeed none at all?

kevinferreira said:
Let's consider the macroscopic state "1 particle in a volume V". This is a pretty good vacuum. What is the number of microstates compatible with this?
Counting: the particle can be anywhere within the volume V, so it W has to be proportional to V, in my opinion.

Yes, and what conclusions might one draw from that with respect to the entropy of the vacuum? Or do you think that a vacuum has no entropy?

Maximise24 said:
Yes, and what conclusions might one draw from that with respect to the entropy of the vacuum? Or do you think that a vacuum has no entropy?

I don't know what to consider as 'microstates' when there is nothing to construct microstates with! A classical vacuum, with no particles nor anything, may not even have an assigned entropy! An alternative (instead of letting the number of particles $N$ go to zero) is to demand instead that the relation $N/V$ goes to zero, i.e. large volume/low number of particles...

kevinferreira said:
I don't know what to consider as 'microstates' when there is nothing to construct microstates with! A classical vacuum, with no particles nor anything, may not even have an assigned entropy! An alternative (instead of letting the number of particles $N$ go to zero) is to demand instead that the relation $N/V$ goes to zero, i.e. large volume/low number of particles...

But what about the quantum information in the vacuum? If you look at the Casimir effect or virtual energy/particles in general, could you not consider them as information (and thus entropy) in the vacuum?

And, in the inflationary model, what about the vacuum quantum fluctuations that are enlarged by inflation, creating real structures? If an initial vacuum has no entropy at all, how could its fluctuations be the basis for systems that do have entropy?

Maximise24 said:
And, related to that, how would the heat death macrostate be different from that of the classical vacuum, which is sometimes said to have zero entropy or indeed none at all?

The classical vacuum contains zero particles and zero energy. However, in post #4 I suggested that you start with a different concept: You have N particles within a volume V; consider what happens to the number of microstates as you increase V while holding N constant. You can get the density as low as you wish by increasing the volume sufficiently; but no matter how much you increase the volume it will always be an N-particle system, never a classical vacuum.

Nugatory said:
The classical vacuum contains zero particles and zero energy. However, in post #4 I suggested that you start with a different concept: You have N particles within a volume V; consider what happens to the number of microstates as you increase V while holding N constant. You can get the density as low as you wish by increasing the volume sufficiently; but no matter how much you increase the volume it will always be an N-particle system, never a classical vacuum.

You would indeed never get a complete vacuum, but by infinitely expanding the volume you would come as close to it as is possible with an N-particle system (lowering the density and temperature to almost zero), right? So then my question would be: could you compare this high-entropy heat death scenario with an actual vacuum? Might we conclude from this that a classical vacuum also possesses entropy or not?

Maximise24 said:
But what about the quantum information in the vacuum? If you look at the Casimir effect or virtual energy/particles in general, could you not consider them as information (and thus entropy) in the vacuum?

And, in the inflationary model, what about the vacuum quantum fluctuations that are enlarged by inflation, creating real structures? If an initial vacuum has no entropy at all, how could its fluctuations be the basis for systems that do have entropy?

There are no quantum fluctuations in the classical vacuum, which is what you've been discussing above; it contains no particles and no energy.

Nugatory said:
There are no quantum fluctuations in the classical vacuum, which is what you've been discussing above; it contains no particles and no energy.

Okay, a quantum state vacuum, then. I'm trying to refer to the vacuum that is hypothesised in inflationary theory. What about that?

Anyone? Or is the question a bit too hard? :-)

Nugatory said:
There are no quantum fluctuations in the classical vacuum, which is what you've been discussing above; it contains no particles and no energy.

No energy? Is there somewhere that this is defined? Wiki describes it as a reference vacuum for electromagnetic effects which means it does have energy, doesn't it?

The vacuum at the universe's heat death would have energy and this macrostate would have microstates associated with it. No good?

ModusPwnd said:
No energy? Is there somewhere that this is defined? Wiki describes it as a reference vacuum for electromagnetic effects which means it does have energy, doesn't it?

Indeed: a "classical vacuum" with completely no energy does not actually exist - there are inevitably quantum effects. I am indeed referring to a vacuum with quantum effects and I was wondering whether it has entropy or not.

Maximise24 said:
You would indeed never get a complete vacuum, but by infinitely expanding the volume you would come as close to it as is possible with an N-particle system (lowering the density and temperature to almost zero), right?
Yes, if I start with a system of N particles and expand the volume that it occupies enough, then the overwhelming majority of its microstates will closely resemble the macrostate known as "classical vacuum". But that doesn't mean doesn't mean that it is a classical vacuum, nor that all of its properties must necessarily approach those of a classical vacuum - and the entropy is one of the properties that is most interestingly different.

Maximise24 said:
Indeed: a "classical vacuum" with completely no energy does not actually exist - there are inevitably quantum effects. I am indeed referring to a vacuum with quantum effects and I was wondering whether it has entropy or not.

I don't even think you need to appeal to quantum effects. A "classical vacuum" is not broken when you shine a light on it, at least not how I conceptualize the phrase. Thats why I was wondering where the phrase's definition comes from.

Maximise24 said:
Indeed: a "classical vacuum" with completely no energy does not actually exist - there are inevitably quantum effects. I am indeed referring to a vacuum with quantum effects and I was wondering whether it has entropy or not.

There's a reason why the moderators moved this thread into Classical. I'm not one of them, but I expect that the reason is that until the classical picture is clearly understood, speculating about the further quantum corrections can only lead to confusion.

Nugatory said:
Yes, if I start with a system of N particles and expand the volume that it occupies enough, then the overwhelming majority of its microstates will closely resemble the macrostate known as "classical vacuum". But that doesn't mean doesn't mean that it is a classical vacuum, nor that all of its properties must necessarily approach those of a classical vacuum - and the entropy is one of the properties that is most interestingly different.

Thank you. And could you say something more about the entropy of a quantum vacuum (such as in inflationary theory)? Don't you agree that it must have some entropy, since the structures of our universe are supposed to stem from enlarged quantum fluctuations?

Nugatory said:
There's a reason why the moderators moved this thread into Classical. I'm not one of them, but I expect that the reason is that until the classical picture is clearly understood, speculating about the further quantum corrections can only lead to confusion.

I actually put it in Classical in the first place (because of the thermodynamics of it), but I agree that it is a fuzzy boundary. So you cannot help me with regards to the entropy of a quantum vacuum, then?

Maximise24 said:
I actually put it in Classical in the first place (because of the thermodynamics of it), but I agree that it is a fuzzy boundary. So you cannot help me with regards to the entropy of a quantum vacuum, then?

It depends - do you think you have a solid understanding of the classical notion of entropy and the heat death of the universe? (I got to say, the first post in this thread did not seem reassuring).

If so, it might be time to start a new thread in QM or Cosmology to ask a specific question about entropy and the quantum vacuum.

First let formulate the question precisely.

Classical vacuum or quantum vacuum?
And 'heat death universe' vacuum or idealised vacuum?
That is, very few particles/energy in a bigger and bigger volume or one fixed volume with zero particles/energy in it? (I don't know how much equivalent are these two views, but if we wish to work with a quantum vacuum and consider vacuum energy then this energy would depend on the volume we take, I think).

If we have a particle in a volume V; let's suppose we know as much as we can about this particle (thus minimising entropy), that is $\Delta p\Delta x\sim\hbar$. Then $\Delta p\sim \hbar V^{-1}$. Count the number of microstates compatible with it: it will be proportionnal to V and to $\hbar V^{-1}$, and thus to $\hbar$...

Nugatory said:
It depends - do you think you have a solid understanding of the classical notion of entropy and the heat death of the universe? (I got to say, the first post in this thread did not seem reassuring).

If so, it might be time to start a new thread in QM or Cosmology to ask a specific question about entropy and the quantum vacuum.

I greatly appreciate your concern, but I do not quite see how my perceived (in)ability to grasp classical entropy would be dependent on your (in)ability to answer my question, unless, of course, you only wish to provide answers when you have a full guarantee that they will be understood completely, in which case you are in the comfortable position of being able to ignore a vast majority of queries that are put to you.

However, I shall heed your wise words and stride forth now to the QM section, where I shall phrase my question utilising the invaluable terminological nuances you have bestowed upon me.

## 1. What is entropy of a vacuum?

The entropy of a vacuum refers to the measure of disorder or randomness within a vacuum or empty space. It is a concept derived from the second law of thermodynamics, which states that the total entropy of a closed system will always increase over time. In the case of a vacuum, since it is considered a closed system, the entropy will continue to increase until it reaches its maximum value.

## 2. How is entropy of a vacuum related to the heat death of the universe?

The concept of heat death of the universe is based on the idea that as the universe continues to expand, the available energy will become more and more spread out, resulting in a state of maximum entropy. This means that all the energy in the universe will eventually be evenly distributed, resulting in a state of complete disorder and no available energy for any kind of work. Therefore, the entropy of a vacuum is closely related to the heat death of the universe, as both concepts describe the ultimate fate of the universe.

## 3. Can the entropy of a vacuum be reversed?

According to the second law of thermodynamics, the entropy of a closed system can only increase or remain the same, but it cannot decrease. Therefore, the entropy of a vacuum cannot be reversed or decreased. The only way to decrease entropy is to introduce external energy or work into the system, which is not possible in the case of a vacuum.

## 4. Is the heat death of the universe a certain event?

The heat death of the universe is a widely accepted scientific theory, but it is not a certain event. It is based on our current understanding of the laws of thermodynamics and the behavior of the universe. However, new discoveries and advancements in science may change our understanding of these concepts and potentially provide alternative theories for the ultimate fate of the universe.

## 5. How long will it take for the universe to reach the state of heat death?

The timeline for the heat death of the universe is currently unknown as it depends on various factors such as the rate of expansion of the universe and the amount of available energy. Some estimates suggest that it could take trillions of years for the universe to reach the state of maximum entropy, while others propose that it may happen much sooner. However, it is important to note that these are all just theoretical predictions and the actual timeline could be different from our current understanding.

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