Arrow of time/ entropy probability

[Tsunami]

In Popper's autobiography, published in his contribution of Library of Philosophers, I read the following, after Popper's presentation of Boltzmann's ideas about entropy:

All this is highly convincing; but in this form it is unfortunately wrong. Boltzmann at first interpreted his H-theorem as proving a one-directional increase of disorder with time. But as Zermelo pointed out, Poincaré had proved previously (and Boltzmann never challenged this proof) that every closed system (gas) returns, after some finite time, to the neighbourhood of any state in which it was before. Thus all states are (approximately) recurring for ever; and if the gas was once in an ordered state, it will after some time return to it. Accordingly there can be no such thing as a preferred direction of time - an "arrow of time" - which is associated with entropy increase.

A bit further this is formulated as such:

The situation looks like this: every closed system (a gas, say) spends almost all its time in disordered states (equilibrium states). There will be fluctuations from the equilibrium, but the frequency of their occurences rapidly decreases with their increasing size. (...) Accordingly, if we want to predicts its future, we can predict (with high probability) an entropy increase, and a precisely analogous retrodiction of its past can also be made. It is strange that it is rarely seen that with Zermelo a revolution occurred in thermodynamics.

Is this all accepted? Any additional comments? (The arrow of time always sounded valid to me.)

 

[Tsunami]

eh your link doesn't work, sorry

edit: via search functions I did find the two pages of that paper.

It seems to me though as if Prigogine seems to discard Popper's point. What Popper tried to show, was that states of more entropy are more likely, regardless of the direction of time. If you imagine a curve of entropy states in function of time, it is easy to imagine a state of low entropy being surrounded by states of higher entropy... thus, you get a sort of gaussian curve with the most optimal result in the center. (It doesn't have to be gaussian, just an high medium which is lower in both directions starting from this point.)

Popper's idea is, I suppose, that if we could track what happens to this situation of entropy states when it moves from low entropy to high entropy, then this situation could be reversed by backtracking everything that happens.

One should note that Popper also refused the idea of having information as a counter-force against entropy. (If people want to use the information argument to tackle Popper's point, I'll look up his ideas regarding this so this can be tackled as well if necessary)

 

[Andrew Mason]

In Popper's autobiography, published in his contribution of Library of Philosophers, I read the following, after Popper's presentation of Boltzmann's ideas about entropy:
...
Is this all accepted? Any additional comments? (The arrow of time always sounded valid to me.)

I have no idea what the scientific basis for this could be. I am not aware of a principle of physics that:

every closed system (gas) returns, after some finite time, to the neighbourhood of any state in which it was before. Thus all states are (approximately) recurring for ever; and if the gas was once in an ordered state, it will after some time return to it.

Not only that, I think it is an incorrect statement. A gas that expands from a smaller to larger space will not compress by itself back to the smaller space. Nor will the forces of nature suddenly combine (eventually) to compress it.

Now, this does not necessarily mean that the conclusion (that the arrow of time is not the result of ever-increasing entropy) is incorrect. Popper was certainly at the top of his field in philosophy, but he was not a scientist. Philosophy should not be confused with science.

It seems to me that the arrow of time is partly about probability (of the universe self-organizing itself to a precise state that existed). Entropy is partly about probability. To say there is no connection seems to be more than a little presumptuous.

I would say that the arrow of time is simply the result of logic. Time can only go in one direction by definition. If the universe is in a certain state, changes and then returns to the precise previous state, the initial state is still in the past.

AM

 

[Physics Monkey]

The theorem being referred to is a famous one by Poincare called the Poincare Recurrence Theorem which roughly states that an isolated and bounded mechanical system will eventually return arbitrarily close to any initial state. This theorem does quite explicitly state that if you had a perfectly isolated classical gas in which the gas started in one side of the container, you would find at a later time that the gas was again all in one side of the container. There is no conflict with the laws of thermodynamics since such a recurrence would take far too long to occur even if one had happened to have on hand a perfectly isolated classical gas.

 

[Physics Monkey]

And to address Tsunami's original question

Is this all accepted? Any additional comments?

Yes, this stuff, by which I mean the reconciliation of the reversible laws of mechanics with the apparent irreversability of thermodynamics, is all pretty much accepted. One particular interesting and transparent test of these ideas is to look at "gases" of cellular automata that have reversible dynamics. One can watch these purely reversible systems start in one side of the container, expand, and eventually fill the whole container. With a sufficiently large number of the little guys running around, you never see them all return to one side of the container. In short, the laws of thermodynamics appear to hold good. However, if you go and perfectly reverse all the velocities, sure enough, the gas goes dutifully right back into one side of the container.

 

[SizarieldoR]

Philosophy should not be confused with science.

Hay, physics isn't the only science (although it is the basic natural science), if memory serves, philosophy can be called a science as well.

Does entropy decrease with the action of forces? I mean, does the entropy in a given amount of lava decrease, when the lava crystallizes?

Also, thermal energy is the vibration of atoms/molecules hitting each other (correct me if I'm wrong). So, if we can calculate how two billiard balls hit each other (and the change in their momentums), we should be able to calculate how two atoms/molecules hit each other as well in the limits of the uncertainty principle. Of course, we will need some freakin-powerful supercomputers and stuff, not to mention taking the London dispersion force into account, but doesn't that mean that entropy isn't an increase in disorder?

 

[vanesch]

I
Is this all accepted? Any additional comments? (The arrow of time always sounded valid to me.)

There is some controversy over this, but the most "down to earth" view is this: the arrow of time (= the increase in entropy) comes about from the special initial condition (low-entropy state of the early universe).

In fact, the initial state of the universe (just after the big bang) is a *highly peculiar state* of very low entropy, and we're still evolving towards equilibrium in this view.

The asymmetry is not in the time symmetric evolution laws (which, to our knowledge, are time reversible), but in the initial condition.
This displaces the question of course to why there was this special initial condition :-)

Now, to come back to Poincare recurrence, this comes in fact down to saying that in a closed classical system, motion is "essentially" periodic ; however, this period is extremely long! If you take any initial state, you will have the motion essentially divided in 3 parts:
a part where there is "increase in entropy" ; a part where there is "decrease in entropy" and a VERY LONG PART where there is equilibrium ; this then cycles on and on, and depending on where you are on this trajectory, you observe increase, decrease, or equilibrium.
In the first case, the "special state" is closer in the past than in the future. In the last case, the "special state" is closer in the future than in the past (just before a "big crunch" ?). And in between there is an extremely long part of equilibrium.
Now, it seems that we are now in a phase of "increase of entropy", because the big bang (special state) is closer to us in the past than in the future.

Now, you could even say that even if it were different, and the special state were closer to us in the "future" than the past, we would probably EXPERIENCE time in the other sense, and call this future the past, and vice versa. But that's less of a standard view. But fun to think about.

However, one thing is sure: entropy changes can only occur in "the neighbourhood" of a special (initial or final) state. If not, you are in equilibrium, which is by far the state that prevails during the longest time. At least in a purely classical system.

 

[Tsunami]

Ok, that makes sense.

I still want to solve one question though (just trying to get my fundamentals straight ):


If, in principle, we can move to a state of lower entropy by reversing all velocities, why don't we?


I can see how this can be tackled by saying that we lack sufficient information to know the exact reversal of motion, but that doesn't satisfy me. Surely, the second law must be stronger than that?

 

[Andrew Mason]

Does entropy decrease with the action of forces? I mean, does the entropy in a given amount of lava decrease, when the lava crystallizes?

Entropy of any system can increase or decrease in a process. But the entropy of the universe will always increase in the process.

Also, thermal energy is the vibration of atoms/molecules hitting each other (correct me if I'm wrong). So, if we can calculate how two billiard balls hit each other (and the change in their momentums), we should be able to calculate how two atoms/molecules hit each other as well in the limits of the uncertainty principle. Of course, we will need some freakin-powerful supercomputers and stuff, not to mention taking the London dispersion force into account, but doesn't that mean that entropy isn't an increase in disorder?

Entropy is not really about disorder.


The second law of thermodynamics simply says that the energy of the universe flows in a way that makes it more disperse.

Entropy is really about energy dispersion and probability. Consider the break in game of pool (assume a frictionless surface, perfectly elastic collisions and rebounds from the cushions). The relatively high energy of the cue ball is dispersed into all 16 balls, with each ball having a portion of that original energy. The probability that the balls will all collide with the cue ball in such a way that they all stop and the cue ball regains its original energy is very low. It would not violate the laws of physics. It would just be extremely improbable.

AM

 

[Andrew Mason]

Hay, physics isn't the only science (although it is the basic natural science), if memory serves, philosophy can be called a science as well.

"In science there is only physics; all the rest is stamp collecting."

Ernest Rutherford (before he won the Nobel Prize for Chemistry)​

AM

 

[Andrew Mason]

Poincaré Recurrence Theorem - Zeno's Paradox in reverse

The Poincaré Recurrence theorem states that any possible event, no matter how improbable, will occur in a time interval $\Delta t$ if we make $\Delta t$ arbitrarily large.

Zeno's paradox says that two objects separated by a distance s and approaching at a fixed speed v will never hit each other in an infinite number of time intervals $\Delta t_i$ if we make the time intervals arbitrarily small (so that $\sum \Delta t_i$ < s/v).

It seems to me that both are correct mathematically. But both have nothing to do with the real world.

Similarly Quantum Mechanics has nothing to do with the lunar orbit. There is a finite probability that the moon will stop obeying Newton's laws of motion and pass through the Earth and out the otherside. This would violate Newton's laws of motion but would not violate QM. It is just that the probability of this happening is very small - so small that it will not likely happen in 10^10^10^10^10^10^10^10^10 lifetimes of the universe. This does not invalidate Newton's laws of motion.

So to the suggestion that the Poincaré Recurrence theorem shows that entropy of the universe can and will decrease spontaneously, I say: 'not in this universe'.

AM

 

[Tsunami]

However, you can make an adjusted Zeno's paradox that suits the real world: you can place a finite lower boundary on the amount of divisions you make, something like "the turtle cannot walk half a distance if this half is smaller than one of his feet"

Anyway, doing the same for Poincare probably means saying : the finite upper boundary we can think of is the duration of conscious life in the universe, so it doesn't matter.

So considering practically everything can be explained using an infinite duration of time and any improbable probability : I get your point.

 

[vanesch]

So to the suggestion that the Poincaré Recurrence theorem shows that entropy of the universe can and will decrease spontaneously, I say: 'not in this universe'.

As we don't know the laws of this universe (but only some aspects of it), we can't tell but you're probably right. The point was not this, the point was that even in a classical MODEL where Poincare recurrence holds, and where the microdynamics is time-reversible, a second law can hold during a time lapse which is "shortly" after a special initial condition.

And now when you look at where, in this world, we DO get our second law from, then this clearly points out to a similar reason: we get our entropy increase potential essentially from the influx of low-entropy radiation from the sun, and the outflux of high-entropy radiation into the blackness of space. Both are related to the state of the early universe and its expansion. If the universe were in thermodynamic equilibrium, then we wouldn't observe a second law (we wouldn't be there either).

Now whether there is some cyclic equivalent of the Poincare recurrence theorem or not for the "real" laws of this universe I don't know of course, but that was not the point. The point was that the apparent paradox between reversible microlaws and the second law can be solved by considering that you are evolving away from a special initial state. It might turn out that microlaws are, in the end, not time-reversible, who knows. But this is not necessary to explain the second law.

 

[Andrew Mason]

The point was not this, the point was that even in a classical MODEL where Poincare recurrence holds, and where the microdynamics is time-reversible, a second law can hold during a time lapse which is "shortly" after a special initial condition.

I don't follow you there. What do you mean by "where the microdynamics is time-reversible"?

If the universe were in thermodynamic equilibrium, then we wouldn't observe a second law (we wouldn't be there either).

But the universe would only move into thermodynamic equilibrium because of the second law: the state of equilibrium being the most probable state.

The point was that the apparent paradox between reversible microlaws and the second law can be solved by considering that you are evolving away from a special initial state.

As a great politician once observed, there are many ways to change but only one way to remain the same. That, in a nutshell, sums up the second law.

AM

 

[vanesch]

I don't follow you there. What do you mean by "where the microdynamics is time-reversible"?

That the dynamical laws are time-symmetric (that there is a symmetry when replacing t -> -t in the dynamical laws of classical mechanics).

But the universe would only move into thermodynamic equilibrium because of the second law: the state of equilibrium being the most probable state.

Well, the point was, that this "evolution towards a more probable state" (= second law) comes about only, because we come from a state which is NOT very probable (probable in the sense of Boltzmann entropy, where the available phase space is chunked up in boxes of "equivalent macrostates", where we don't care for high-order correlations), and that there is no specific conservation law which keeps us in the small, =unprobable, box. So the time evolution of the classical state will wander from small boxes into big boxes, most of the time, because there is no specific relationship between the delimitation of the boxes, and the time evolution of the microstate. Given this a priori independence, a stochastical argument can do. It can happen that it wanders from a big box into a small box, but that is a highly peculiar happening (which happens nevertheless, due to Poincare recurrence, at the end of a "cycle"), because the size of the boxes, and the time evolution, are not really related. All this in a classical MODEL universe.
The point is simply that the timescales on which this could happen (Poincare recurrence time) is vastly longer than the time since a fixed initial condition, so the "slope of increase of box size" is still very steep, hence a clearly defined second law.

As a great politician once observed, there are many ways to change but only one way to remain the same. That, in a nutshell, sums up the second law.

That's a nice quote :-).

However, "staying the same" is microscopically as involved as "changing". It is only that the low-order correlation functions (like matter densities and so on) do not change anymore in what we call 'equilibrium'. The high-order correlation functions change as much during "equilibrium" as during "irreversible evolution" in a classical universe.

 

[Farsight]

Doublepost deleted.

 

[Farsight]

Time is to do with motion over a distance. The direction doesn't matter, so changing the direction won't reverse the arrow of time. What you need for that, is a negative distance. There's one in here, thankyou google. But apologies if it's bunk, I haven't read it through.

http://www.comcity.com/distance-time/Photon%20Kinematics.html

 

[lalbatros]

Tsunami,

Already in 70's some people made computer simulations showing that entropy can decrease (close to its original value) after a long time. This is just unavoidable when the microscopic laws are reversible. Such simulations of course need to consider a small number of particles. The "recurrence time" that can simulated in this way increases very fast with the number of particles. This recurrence time also increases -of course- with the precision of the recurrence, a perfect recurrence requires generically an infinite time.

Some models can make things easier to visualise, like particles in a box with periodicity conditions: then there may even be no need to calculate the trajectories since they are analytically simple.

More obvious simulations were dealing with velocity-reversal. In such simulations, the entropy evolves exactly in the opposite directions than in the forward simulation (it decreases) until the initial condition is recovered, and thereafter the entropy increases again as usual.

Many people consider that these simulations don't represent physical reality, and that therefore something is missing in the physics to account for irreversibility. The interpretation is simple however. Preparing a system that could exhibit anti-thermodynamics behaviour is theoretically possible from the microscopic law of physics. But this preparation is extremely difficult and such systems are "fragile": they loose their exceptional properties by any small perturbation.

That's an interresting and fundamental topic. But I have not seen that it delivered any real progress in physics, till now.