Entropy & Cosmology: Arrow of Time Paradox

[yuiop]

The question of the "arrow of time" and increasing entropy is not as clean cut as one might imagine. I think it it R.Penrose that pointed out that while a simple application of thermodymanis predicts the "heat death" of the universe where everything is cold and in thermal equilibrium, there is a paradox because the initial conditions of the universe just after the big bang is one of thermal equilibrium. This can be seen in the very small degree of thermal anisotropy in the CMB. THe situation for the entropy of the universe os a whole is complicatd by gravity. We know from Hawking that the entropy of a gravitationally collapsed object (a black hole) is very high. This suggests that the expanding universe with reduction in mass energy density is an example of reducing gravitational entropy (the opposite of a black hole). So while the expansion of the universe represents increasing thermal entropy it represents reducing gravitational entropy at the same time. Many quantum cosmological models predict a "bouncing" universe that alternately collapses and expands again. That requires the total entropy of the universe to be constant over time as only constant entropy processes are reversible. That suggests that maybe thermal and gravitational entropy work in opposite directions and cancel each other out. It would be kinda cool it the total energy of the universe is always zero, the total charge of the universe is zero, the total linear and angular momentum is zero and the total entropy of the universe is always zero.

 

[JesseM]

I think it it R.Penrose that pointed out that while a simple application of thermodymanis predicts the "heat death" of the universe where everything is cold and in thermal equilibrium, there is a paradox because the initial conditions of the universe just after the big bang is one of thermal equilibrium.

Actually Penrose argues that for gravitational systems, the equilibrium state would not be a smooth and homogeneous one, but would rather be a clumpy one with all the matter in black holes. He and others considers it a major unsolved problem in cosmology to explain the very low entropy state of the universe immediately after the Big Bang.

Many quantum cosmological models predict a "bouncing" universe that alternately collapses and expands again. That requires the total entropy of the universe to be constant over time as only constant entropy processes are reversible.

I don't know if there are actually any mainstream cosmological models in which the total entropy is constant. I have seen an interesting hypothesis by physicist Sean Carroll that suggests that in the "eternal inflation" scenario a small patch of a high-entropy universe can inflate into a low-entropy baby universe, and where there is actually no upper limit to the entropy of the entire multiverse so that this can keep on happening forever. See here and here for Carroll's summaries of this idea.

 

[George Jones]

THe situation for the entropy of the universe os a whole is complicatd by gravity. We know from Hawking that the entropy of a gravitationally collapsed object (a black hole) is very high. This suggests that the expanding universe with reduction in mass energy density is an example of reducing gravitational entropy (the opposite of a black hole).

Now, some points similar to those posted by JesseM.

Penrose says exactly the opposite. The universe clumps as it expands. According to Penrose, this gravitational clumping causes an increase in entropy that dominates changes in entropy due to other processes.

Why is there a second law of thermodynamics? I don't know if there is agreement on this, but some physicists, including Roger Penrose and Sean Carroll, think that the second law has a cosmological origin. In the blog entry

http://cosmicvariance.com/2007/06/11/latest-declamations-about-the-arrow-of-time/

Sean Carroll concludes

But if you want to describe why the Second Law actually works in the real world in which we actually live, cosmology inevitably comes into play.

 

[yuiop]

Hi George and JesseM,

Actually Penrose argues that for gravitational systems, the equilibrium state would not be a smooth and homogeneous one, but would rather be a clumpy one with all the matter in black holes. He and others considers it a major unsolved problem in cosmology to explain the very low entropy state of the universe immediately after the Big Bang.

If you look at this lecture by Roger Penrose given at the Permiter Institute: http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=a5af1a59-b637-4974-8eb8-c55ef34b9d7f&shouldResize=False# at t=20mins you will see that Penrose quite clearly states that the thermal equilibrium just after the Big Bang represents a state of maximal entropy. If you have the time watch the Penrose video from t=20 mins to t= 30 mins to get more of the context. Penrose does not quite explicity make the connection that total entropy is constant, but given time he will get there ;) especially as he seems to be working on a cyclic model of the universe, and any cyclic model that continues forever requires entropy to be constant.

I don't know if there are actually any mainstream cosmological models in which the total entropy is constant. I have seen an interesting hypothesis by physicist Sean Carroll that suggests that in the "eternal inflation" scenario a small patch of a high-entropy universe can inflate into a low-entropy baby universe, and where there is actually no upper limit to the entropy of the entire multiverse so that this can keep on happening forever. See here and here for Carroll's summaries of this idea.

I guess classical thermodymanics allows any low entropy process to occur if it accompanied by a corresponding or greater increase in entropy elsewhere in the universe. Some people say that the existence of life and the continued existence of humans is an example of low entropy systems funded by an increase in entropy elsewhere, such as the Sun burning up.

Now, some points similar to those posted by JesseM.

Penrose says exactly the opposite.

As I said to JesseM, Penrose quite clearly states "the thermal equilibrium at the beginning of the universe is a MAXIMAL entropy state" and this represents a paradox becasue if the entropy of the universe has been increasing since the beginning of the universe, the beginning of the universe should have been a MINIMAL state of entropy.

The universe clumps as it expands. According to Penrose, this gravitational clumping causes an increase in entropy that dominates changes in entropy due to other processes.

Penrose has refined the clumpiness due to black holes. When the universe has expanded significantly the reduced thermal background of the CMB allows black holes to evaporate away, removing the clumpiness and creating a universe that is all radiation. His latest model requires that there is no mass at the end of the universe as it cycles around to start a new big bang. The biggest problem with his theory, as he sees it, is what to do with electrons. There does not seem to be any evidence that electrons decay into radiation.

Anyway, here are some simple observations:

Take an insulated partitioned box with hot gas on one side and cold gas on the other side. Remove the partition and the gas molecules mix going to state of thermal equilibrium.

(E1) A universe with an initial condition of thermal equilibrium = High Entropy.

Take a gas cloud that has gravitationally collapsed to a high density black hole. Hawking black hole entropy equations show that a state of gravitational collapse is a state of maximum entropy.

(E2) A universe with high initial density = High Entropy.

Take a cloud of gas with a high initial temperature in a small volume that is not gravitationally bound and it will expand, increasing the volume and degrees of freedom and cooling at the same time. So high temperature and low volume is a state of low entropy.

(E3) A universe with high initial temperature and low volume = Low Entropy.

So in calculating the initial entropy of the universe we should not take anyone of the forms of entropy described above (and there are probably others) in isolation, but rather consider and combine all forms of entropy to give a total entropy of the form TE = E1+E2-E3 with the possibility that TE is an invariant quantity possibly equal to zero as far as the universe is concerned.

Jambaugh in the Quantum Physics forum of PF gives an interesting Quantum Mechanical argument for the entropy of the universe being zero here: https://www.physicsforums.com/showthread.php?t=170493

 

[JesseM]

If you look at this lecture by Roger Penrose given at the Permiter Institute: http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=a5af1a59-b637-4974-8eb8-c55ef34b9d7f&shouldResize=False# at t=20mins you will see that Penrose quite clearly states that the thermal equilibrium just after the Big Bang represents a state of maximal entropy.

No, he clearly says that the total entropy of the universe depends on both the entropy of the matter and the gravitational entropy of the geometry of the universe, and that although the matter was near equilibrium (as evidenced by the blackbody spectrum of the CMBR) the smooth geometry of space represents a very low entropy for the universe as a whole at the time just after the Big Bang.

By the way, the audio wasn't working for me on that page--if anyone else has this problem, I found an mp3 of the audio here, you can start it a few seconds after the video starts and it'll be approximately synched up.

 

[yuiop]

One form of entropy I have not really discussed is structural or information entropy. A poplular notion of increasing entropy is increased randomness or chaos with structure representing a low state of entropy. I am not sure how this fits into the overall picture. However a definite pattern emerges from the thress form of entropy I have described above and that is that increasing entropy of anyone type represents a loss of potential energy and the ability to do work. A high temperature gas in a small volume has high pressure and an obvious capacity to do work and the arrow of time moves in the direction of reducing that potential energy so that you end up with a low temperature, low pressure gas in a large volume that has less potential to do work. A gravitationally bound cloud of gas has high gravitational potential energy and the arrow of time dictates that it moves to a lower graviational potential. When the gas cloud collapses it heats the gas and increases the pressure possibly forming a star. The loss of gravitational potential energy and increase in gravitational entropy has resulted in an increase of thermal potential energy and a reduction of thermal entropy. This hints at the interchangeabilty of forms of entropy working dynamically together. A box of hot gas and cold gas going to thermal equilibrium represents a loss of potential energy as the thermal gradient that was available to do work has been lost.

Here is a simple thermodynamic thought experiment. A cylinder is filled with gas and the gas is retained by an internal spring attached to a piston. Outside the cylinder is a vacuum and the cylinder is perfectly thermally insulated. The spring pulling the piston inwards puts the gas under pressure. The system could stay static with the force of the spring exactly equal to the force of the gas acting on the piston. Now if the piston is sharpely tapped to give it inward momentum the pressure and temperature of the gas increasesand this is enough to halt the piston and start it moving outwards. As the piston moves outwards the retaining spring's potential enrgy increases and eventually it stops the piston and sends it back into the cylinder increasing the pressure and starting the cycle again. If the piston was perfectly frictionless and if the cylinder was perfectly insulated this oscillation would continue forever as the thermal potential energy is converted into the potential energy of the stretched spring and vice versa. The total entropy of the system remains constant as an increase in thermal entropy is compensated by the reduction of entropy in the stretched spring. Now a real spring and cylinder could not be perfectly frictionless and perfectly thermally insulated, BUT the universe as a whole can.

 

[yuiop]

No, he clearly says that the total entropy of the universe depends on both the entropy of the matter and the gravitational entropy of the geometry of the universe, and that although the matter was near equilibrium (as evidenced by the blackbody spectrum of the CMBR) the smooth geometry of space represents a very low entropy for the universe as a whole at the time just after the Big Bang...

I guess someone will have to crunch the numbers and figure the contributions of the all the forms of entropy. You are right about the entropy due the smoothness of the geometry of space (lack of clumpiness) and I have added it to my list of entropies as (E4):
-----------------------------------------------------------------------------
(E1) A universe with an initial condition of thermal equilibrium = High Entropy.

(E2) A universe with high initial density = High Entropy.

(E3) A universe with high initial temperature and low volume = Low Entropy.

(E4) A universe with initial smooth gravitational geometry (low clumpiness) = Low entropy.

So in calculating the initial entropy of the universe we should not take anyone of the forms of entropy described above (and there are probably others) in isolation, but rather consider and combine all forms of entropy to give a total entropy of a form something like TE = (E1/E3)-(E3/E4) with the possibility that TE is an invariant quantity possibly equal to zero as far as the universe is concerned.
----------------------------------------------------------------------------

It is interesting to note the dynamic changes in entropy as the universe evolves.

In the initial universe:
Thermal equilibrium (High entropy)
Smooth geometry (Low entropy)
High temperature (Low entropy)
High gravitational density (High entropy)

As the early universe progressed structures such as galaxies formed increasing entropy due to increased clumpiness, the overall temperature cooled increasing entropy, stars formed local hot spots providing thermal gradients reducing entropy and the universe expanded increasing entropy because of increased volume and reducing entropy because of the increasing gravitational potential.

Early universe:
Thermal gradient as stars form (Low entropy)
Gravitaional gradient as structures form (Increasing entropy)
Falling temperature (Increasing entropy)
Reducing mass density (Reducing entropy)

Current universe:
Tendancy towards thermal equilibrium as stars burn up (Increasing entropy)
Increased gravitational gradient as more mass ends up in black holes (Increasing entropy)
Falling temperature (Increasing entropy)
Reducing mass density (Reducing entropy)

Late universe:
Thermal equilibrium (High entropy)
All matter in black holes (High entropy)
Low temperature (High entropy)
Reducing mass density (Reducing entropy)

Very late universe:
Thermal gradient as black holes evaporate (Reducing entropy)
Smooth geometry as the black holes evaporate (Reducing entropy)
Low temperature (High entropy)
Reducing mass density (Reducing entropy)

Final universe?
Thermal equilibrium (High entropy)
Smooth geometry - no black holes (Very Low entropy)
Low temperature (High entropy)
Reducing mass density (Reducing entropy)

I have assumed a universe that contimues to expand forever. If it was to collapse then:

Collapsing universe:
Thermal equilibrium (High entropy)
Smooth geometry - no black holes (Very Low entropy)
Increasing temperature (Reducing entropy)
Increasing mass energy density (Increasing entropy)

Those are of course VERY rough back of envelope estimates of entropy and other forms of entropy such as chemical/structural/chaotic/information entropy might have to factored in. As I mentioned before, it is an interesting possibility worth investigating, that the total entropy of the universe in all it forms is invariant, opening up the possibility of dynamic cyclic eternal models rather than a universe on a one way thermodynamic trip to an inevitable "heat death".

As I mentioned before, the cyclic models that keep turning up in quantum gravity models with alternating expansions and cruches require total entropy to be invariant.

 

[JesseM]

Those are of course VERY rough back of envelope estimates of entropy and other forms of entropy such as chemical/structural/chaotic/information entropy might have to factored in. As I mentioned before, it is an interesting possibility worth investigating, that the total entropy of the universe in all it forms is invariant, opening up the possibility of dynamic cyclic eternal models rather than a universe on a one way thermodynamic trip to an inevitable "heat death".

I just don't think the idea makes much sense. Consider the exact midpoint of expansion/contraction in a Big Bang/Big Crunch universe, and then look at what happens in both the forward time direction and the backward time dimension from that moment--if the entropy was constant, what would account for the very different behavior in both directions, assuming the underlying laws of physics are completely time-symmetric?

As I mentioned before, the cyclic models that keep turning up in quantum gravity models with alternating expansions and cruches require total entropy to be invariant.

The cyclic universe model usually discussed in quantum gravity does not involve alternating expansions and crunches, rather it involves something like two branes that repeatedly collide with each other. As in many other eternal universe models, it might be that one way of accounting for the low entropy Big Bang is to imagine there is ultimately no upper limit to the total entropy...this may be what they're suggesting in the paragraph of the wikipedia article where it says "As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death. However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up." (perhaps this would require the branes to be infinite, I dunno) This also seems to be a possibility Penrose suggests with the diagram in his final slide in the presentation, where the largest region of phase space in each earlier universe is drawn as being a small region of the phase space in the next universe.

 

[yuiop]

I just don't think the idea makes much sense. Consider the exact midpoint of expansion/contraction in a Big Bang/Big Crunch universe, and then look at what happens in both the forward time direction and the backward time dimension from that moment--if the entropy was constant, what would account for the very different behavior in both directions, assuming the underlying laws of physics are completely time-symmetric?

I think we have agreed in another thread that the laws of physics on a large scale are not completely time symmetric. I also want to add that the the cyclic model does not imply any form of time reversal at the midpoint. If a human happened to be around at the midpoint of the universe time line, then he would not start getting younger as the universe heads towards the big crunch.

The cyclic universe model usually discussed in quantum gravity does not involve alternating expansions and crunches, rather it involves something like two branes that repeatedly collide with each other. As in many other eternal universe models, it might be that one way of accounting for the low entropy Big Bang is to imagine there is ultimately no upper limit to the total entropy...this may be what they're suggesting in the paragraph of the wikipedia article where it says "As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death. However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up." (perhaps this would require the branes to be infinite, I dunno) This also seems to be a possibility Penrose suggests with the diagram in his final slide in the presentation, where the largest region of phase space in each earlier universe is drawn as being a small region of the phase space in the next universe.

As far as I know, the branes idea is not the fully accepted mainstream standard model. As far as I can tell, the alternative cyclic models all try to get around the thermodynamic constraint of entropy always increasing. It is obvious that in a "pure" cyclic model the entropy at each sucessive big bang would be the same as the previous big bang and if entropy increases during the evolution of the universe then at some stage entropy would have to get lower to to get back to a new big bang. What I am trying to say, is that if the entropy of the universe is invariant over time, then there is no requirement to modify the models to get around the entropy constraint. Penrose readily admits that his model is not complete and his depiction of the entropy getting ever larger from one cycle to the next may not be the final solution. The laws of thermodynamics state that entropy either increases over time OR stays the same. It clearly does not rule out constant entropy and time does not require entropy to increase, all that it requires is that the entropy of a large thermodynamically insulated system does not get smaller. Staying the same is perfectly acceptable. The big mistake is to think of entropy in one form only.

 

[robheus]

Hi George and JesseM,

If you look at this lecture by Roger Penrose given at the Permiter Institute: http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=a5af1a59-b637-4974-8eb8-c55ef34b9d7f&shouldResize=False# at t=20mins you will see that Penrose quite clearly states that the thermal equilibrium just after the Big Bang represents a state of maximal entropy. If you have the time watch the Penrose video from t=20 mins to t= 30 mins to get more of the context. Penrose does not quite explicity make the connection that total entropy is constant, but given time he will get there ;) especially as he seems to be working on a cyclic model of the universe, and any cyclic model that continues forever requires entropy to be constant.

Seen that lecture.

I think it should be interpreted as follows.
The universe starts out in a maximal entropy state, indeed, despite that it was totally unordered, but for reasons of the geometry.

Now entropy runs down in the universe as one normally would expect.

But he then theoretizes what would happen with the universe in the long run, accelerated expansion and hawking radiotion would eventually tear apart all normal matter, and supposedly lead to a universe with only radiation.

He then states that the universe more or less forgets about time (if there is no matter, there is no way of measuring time anyway), and you can rescale the universe and then *magically* return to the original configuration of the universe in a maximal entropy state.

If there is infinity entropy anyway, entropy can run down eternally without problem.

 

[yuiop]

Seen that lecture.

I think it should be interpreted as follows.
The universe starts out in a maximal entropy state, indeed, despite that it was totally unordered, but for reasons of the geometry.

Now entropy runs down in the universe as one normally would expect.

But he then theoretizes what would happen with the universe in the long run, accelerated expansion and hawking radiotion would eventually tear apart all normal matter, and supposedly lead to a universe with only radiation.

He then states that the universe more or less forgets about time (if there is no matter, there is no way of measuring time anyway), and you can rescale the universe and then *magically* return to the original configuration of the universe in a maximal entropy state.

One small technical point. Entropy is formally stated as increasing over time so that requires the universe to start in a state on MINIMAL entropy and for the entropy to run "up" as the universe evolves.

One idea in relation to Penrose's idea is this. If the universe eventualy reaches a state of being radiation dominated after all the black holes have evaporated, then if the universe was initially finely balanced between expansion and collapse when it was matter dominated it would be tilted in the direction of collapse when it is radiation dominated. This is because the gravitational influence of all the energy of the universe in the form of radiation is greater than than when the energy is the form of mass. At least that is the impression I get from the curvature of the radiation dominated universe being proportional to (t/t0)^(1/2) and the curvature of the mass dominated universe being proportional to (t/t0)^(2/3) for a mass dominated universe.

 

[JesseM]

I think we have agreed in another thread that the laws of physics on a large scale are not completely time symmetric.

Leaving aside CPT symmetry vs. T symmetry, the only reason they appear nonsymmetric on a large scale is because many systems are out of equilibrium and so changing in entropy. But you are proposing that there is no change in entropy in the universe as a whole, so in your proposal, how to explain why the Big Bang/Big Crunch universe behaves very differently going backwards from the midpoint of expansion than it does going forwards?

I also want to add that the the cyclic model does not imply any form of time reversal at the midpoint. If a human happened to be around at the midpoint of the universe time line, then he would not start getting younger as the universe heads towards the big crunch.

Not in the basic GR model of an expanding/contracting universe, no. But this is precisely because the standard view is that entropy is increasing, and would continue to increase in the contracting phase.

As far as I know, the branes idea is not the fully accepted mainstream standard model.

Sure, but no cyclic model is "fully accepted" since there is no evidence for it at present, and the theoretical basis is uncertain without a theory of quantum gravity. But you referred to "the cyclic models that keep turning up in quantum gravity models with alternating expansions and cruches"--what quantum gravity models are you talking about here, if not M-theory and the idea of branes in a higher dimension?

It is obvious that in a "pure" cyclic model the entropy at each sucessive big bang would be the same as the previous big bang and if entropy increases during the evolution of the universe then at some stage entropy would have to get lower to to get back to a new big bang.

How is that obvious? Only if you assume from the start that "cyclic" must mean repeating in every respect, but there is no physical motivation for that. If you just go on GR plus an assumption that quantum gravity effects kick in and cause a bounce right before the singularity, there's no reason entropy shouldn't continually increase with each successive bounce, until it reaches some maximum.

What I am trying to say, is that if the entropy of the universe is invariant over time, then there is no requirement to modify the models to get around the entropy constraint.

But "the entropy of the universe is invariant over time" just doesn't seem to make sense if it's supposed to be compatible with the various arrows of time observed in our universe. There's no reason you should see very different behavior going backward from the midpoint of expansion as you do going forward from it if there is no change in entropy.

It clearly does not rule out constant entropy and time does not require entropy to increase

"Time" does not require it, but asymmetric arrows of time (the forward direction looking different from the past direction, so that you can see 'something's not right' if someone shows you a movie of a physical system but secretly decides to play it backwards) in a universe with time-symmetric fundamental laws do require changes in entropy.