Time-reversal invariance and irreversibilities

[saim_]

We know basic classical mechanics is time-reversal invariant while there is a concept of irreversibility in thermodynamics. Is there a simple (by which I mean undergrad level and more preferably lower undergrad level) explanation for this apparent paradox? Someone please either explain this or, better yet, refer me to some good literature on this topic. Thanks.

 

[Vaal]

saim,

The behavior, on the microscopic level, of most thermodynamic systems is effectively random. As a result of this some macroscopic states of the system are drastically more likely than others, this arises more so out of the math than physics. So it is possible for all thermodynamic systems to reverse, it's just usually extremely unlikely.

I would recommend reading more on entropy, it can be a tricky concept at first but I think it will help answer your question.

 

[GeorgeRaetz]

Classical thermodynamics starts with a system of many variables--the positions and momimtua of each molecule in a gas for example; that's called the microstate. From the microstate a contracted description in terms of far fewer variables is calculated that's called the macrostate. The macrostate is then used to describe the system. (The contraction is usually based on averaging the microstate variables. Temperature of a gas, for example is proportional to the average kinetic energy of the molecules).
The contraction of the description by itself introduces the apparent time irreversibility.
Simple example: Start with a bunch of particles each moving with some arbitary and constant velocity. (interactions may be ignored) Draw a sphere which initially surrounds all the particles. Let the macrostate description be the number of particles ,N inside the sphere. N will decrease with time. Reversing the velocities will cause N to momemtarily increase, but in the long run N will always decrease with increasing time.

 

[Andrew Mason]

We know basic classical mechanics is time-reversal invariant while there is a concept of irreversibility in thermodynamics. Is there a simple (by which I mean undergrad level and more preferably lower undergrad level) explanation for this apparent paradox? Someone please either explain this or, better yet, refer me to some good literature on this topic. Thanks.

In classical mechanics, in order to have true time reversal, ie a return of everything in the universe to the same state they were in at a previous time, t0, the universe would have to have the following characteristics:

1. the current state of the universe would have to cause the return to time t0
2. the state of the universe at time t0 would have to cause the return to time present

Now this can certainly happen if the universe consisted of one particle in gravitational orbit around another particle. The two particles would return to their exact same state at time t0 and that state would inevitably lead to the present state.

But in the real (classical approximation of the) universe, energy tends to disperse. It is simply statistically not possible to get things back in the same state once an event occurs. A particle with high kinetic energy impacting a stationary group of particles will impart some of its energy to each of the particles in the group. Those particles will never give all their kinetic energy back to the incident particle to restore its original motion.

In thermodynamics thermal energy can flow in only one direction naturally: hot to cold. This principle is found in the second law of thermodynamics.

AM

 

[saim_]

Thank you for the replies. I have done entropy from a thermodynamics view point. I haven't done any statistical mechanics; just brushed up a little on the internet.

Draw a sphere which initially surrounds all the particles. Let the macrostate description be the number of particles ,N inside the sphere. N will decrease with time. Reversing the velocities will cause N to momemtarily increase, but in the long run N will always decrease with increasing time.

I don't completely understand this but I think you are saying if you run the system in reverse eventually the second law will hold in this direction as well. If I understand you correctly, consider this: two particles, far removed from everything, move towards each other, collide, emit a photon and part. Let's reverse the time. The particles collide, absorb photon, and part. In this example, with reversed time direction, the entropy of the system decreases and never increases again, given the initial conditions. Even take larger systems, a hot and cold slab, far removed from anything. They move closer, collide, a little heat transfers from hot to cold body and move apart. In reverse, they collide, heat moves from cold to hot body and they move apart. Again entropy decreases. How can one explain this? I don't know how it would be different for any larger a system.

As for the explanation that a simple system could be made to do stuff even in violation of the second law, let's take Maxwell's demon as an example. If 2nd law is just a statistical law, why is there a consensus that no matter how simple the system of particles is and no matter how simple the demon is, there is no way it could perform its prescribed function without an overall increase in entropy (I cannot be completely sure of this, but, the popular discussions I read all seem to suggest such a consensus).

So no matter how careful we are, if we give all our power and attention to making entropy constant, it would still increase. Why? How can this happen in time-symmetric dynamics?

Also, there is a reason there is a huge literature out there on this problem. Don't you think there ought to be more to it than this with all those papers with explanations. I just cannot find stuff that's appropriate for me. The papers I found seemed way too tough.

 

[GeorgeRaetz]

I don't completely understand this..

Perhaps I didn't explain it clearly. Motion is unbounded, so particles end up at infinity. Here's another example. Box with a partition has vacuum in one half, air in the other half. Remove the partition and density equalizes. But wait long enough (Poincare recurrence time)and the system will come back to original state or close to it. So one could say that the system is not only time reversible, but periodic as well--but only if you choose to describe the system over a time comparable to the recurrence time and you choose to single only that one episode where the gas just happened to rush over to one half of the box.

two particles, far removed from everything, move towards each other, collide, emit a photon...

A simpler example is spontaneous emission of one atom which seems irreversible, but is explained by statistical averaging. Also reversible spontaneous emission has been observed see the Wikipedia article: http://en.wikipedia.org/wiki/Spontaneous_emission

..Maxwell's demon as an example...

A Maxwell demon (or at least some component of it) would have be (thermodynamically) microscopic to sense an individual molecular velocity. The microscopic part must be coupled to the macroscopic world to have a macroscopic effect. It would now be subject to fluctuations from the macro-world and could no longer communicate the information needed to produce the desired irreversible macroscopic behavior.

... How can this happen in time-symmetric dynamics?

One way to reconcile all this is study the fluctuation dissipation theorems--Whenever a quantity appears to dissipate irreversibly in time,there will be a random and time reversible behavior in that quanitity. Johnson noise from a resistor is an example. Note that the average amplitude of the noise is bandwidth dependent and even vanishes with zero bandwidth; again the time scale you choose determines the degree of irreversibility.

Also, there is a reason there is a huge literature out there on this problem. Don't you think there ought to be more to it than this with all those papers with explanations. .

Some of those papers seem almost mystical in their tone. But time is certainly mysterious and so far there is no rigorous mathematical proof that simply averaging the Liouville equation will lead to observed irreversibility. I can only support the assertion via example.

I just cannot find stuff that's appropriate ...

Try Chapter 27 of Road to Reality by Roger Penrose

 

[saim_]

Thanks for the suggestion from RTR. I had no idea it had an entire chapter on this stuff. I'm currently working on it. I'm stuck on the phase-space argument; the one concerning the gas filled in a small cavity with a valve, in a larger box, and then the valve is opened to let the air out in the larger volume of the entire box. I think I'm having linguistic problems since English is not my first language. Does the argument end with an explanation of how the entropy still really increases in the reverse direction but we have to take into account a much larger phase space or is the problem left unresolved? I'm talking about the last part of section 27.5, that is, pages 668 and most of 669. The writer's wording is a little confusing for me. Does he mean to say that taking into account the much larger phase space, including the entire solar system, the entropy increases even in the reverse time direction? If so, please explain his arguments here.

simpler example is spontaneous emission of one atom which seems irreversible, but is explained by statistical averaging.

How is spontaneous emission reconciled with time reversal symmetry? Even statistically? Please do explain this. The wiki page you mentioned says nothing about reversibility of the phenomenon. In fact, if the process is reversible, as you say, then it really doesn't relate to the problem at hand since reversible processes are already consistent with time symmetry. It's the irreversible processes that have a problem with this symmetry.

A Maxwell demon (or at least some component of it) would have be (thermodynamically) microscopic to sense an individual molecular velocity. The microscopic part must be coupled to the macroscopic world to have a macroscopic effect. It would now be subject to fluctuations from the macro-world and could no longer communicate the information needed to produce the desired irreversible macroscopic behavior.

I miss the point of this completely. Maybe you can dumb it down further to help me understand this :( I gave Maxwell's demon as an example to illustrate that no matter how simple a system is, you cannot really decrease its entropy over a significant period of time; a position that is, as I just found out, supported by Penrose in that same chapter, contrary to popular learned opinion.

As for Poincare recurrence theorem, I have read that the time period is too long to be considered an adequate solution to the problem. Personally, I don't really see how it solves the problem at all; if the system is going to act reversibly over millions of years, there is still a statistical anomaly between the close past and future, isn't it?

I don't understand fluctuation dissipation theorems at all so can't comment on that; can only hope that you'll explain further. Thanks.

 

[GeorgeRaetz]

Experiments with vacuum Rabi oscillation http://en.wikipedia.org/wiki/Vacuum_Rabi_oscillation suggests that spontaneous emission is reversible, but that goes off topic. Need a new thread for that in a different sub-forum.

Penrose argues as follows: reverse exactly the velocity of every single molecule in the gas after it has expanded into the large tank. The resulting motion of the gas shrinking back into the bulb would obviously violate the 2nd law of thermodynamics, so the contradiction you mention seems real. He goes on to say the only way to resolve the contradiction is to start with a low-entropy state. However such an event would occur over a long enough time (Poincare Recurrence time)
He is intrigued by the present low entropy state of the present Universe, and the even lower entropy during the early history which certainly establishes a distinction between past and future. But that's a separate (and still unresolved) issue.

He seems to miss the point that even after thermodynamic equilibrium has established one still observes small fluctuations about the equilibrium values, and about half of these fluctuations, taken individually, will violate the law of increasing entropy. And that's what the fluctuation dissipation (FD) theorems are about. FD theorm is not easy to follow, and is seldom mentioned in beginning statistical mechanics courses. That's too bad, because not only do the FD theorems resolve the apparent contradiction between micro-reversibility and the 2nd law, but they also suggest experiments to confirm the ideas. The amount of flucuation is predicted and also the time scale needed to observe it.

It's all about the time scale: over very short times reversible molecular motion can be discerned. Over the unimaginable long Poincare recurrence time(>2¹⁰²³ sec(?)), the motion is quasi-periodic. In between, law of increasing entropy prevails.

Maxwell Demon could also use a new thread. Proofs depend upon the assumption that at the microscopic level, all process are time reversible; therefore no "micro ratchets" would be available. Now it's hard to establish a negative statement like that. Most compelling evidence, I believe, against the existence of "micro ratchets" is that living systems would have discovered and used them long ago.