# "Where’s your second law of thermodynamics now?"

• etotheipi

#### etotheipi

Prof. Tong used the Liouville theorem to prove the Poincaré recurrence theorem in his notes, that given an initial point ##P## in phase space, for any neighbourhood ##D_0## of ##P## there exists a point ##P' \in D_0## that will return to ##D_0## in a finite time.

To illustrate the theorem, he says that if you put a bunch of molecules of gas in a corner of a room, then eventually they'll end up back where they started [granted, the recurrence time exceeds the universe lifetime!].

The question at the end of the section is "Where's your second law of thermodynamics now?!". I can't figure out how its consistent, so I wondered if someone could explain? Thanks!

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Demystifier, dRic2, vanhees71 and 1 other person
I wouldn't bet on the room still being there in a trillion years or so.

swampwiz, vanhees71, russ_watters and 2 others
To illustrate the theorem, he says that if you put a bunch of molecules of gas in a corner of a room, then eventually they'll end up back where they started [granted, the recurrence time exceeds the universe lifetime!].

Yes, this is known to be theoretically possible. But laws of nature (including the laws of thermodynamics) are generalisations of experimental observations and any reproducible experiment will show you that it doesn't happen.

binis and etotheipi
Dr. Poincare was found dead in his library yesterday evening. The coroner's report indicates the cause of death as asphyxiation but the detectives report no evidence of foul play, no ligature marks on the doctor's neck, and no evidence of break in. According to Head Detective Tong, the working theory is that all of the oxygen molecules coalesced in the upper corner of the room, and by the time they diffused back into a normal configuration the doctor had suffocated...

binis, swampwiz, Delta2 and 7 others
The second law is not a fundamental law. It is due to the special initial conditions of the universe, and the limited time that has passed since the beginning of the universe.

etotheipi
Are you bothered by this with two molecules? Ten? A hundred? Where does this start to be a problem for you?

swampwiz, vanhees71, russ_watters and 1 other person
Are you bothered by this with two molecules? Ten? A hundred? Where does this start to be a problem for you?
A debate about that might end up as a viscous circle.

The second law arises from the statistics of large numbers. While it may be possible to observe configurations of molecules that appear to violate the second law, those configurations are so rare that you will practically never see them in a large system with many particles. You can quantify these notions using large deviation theory.

binis, vanhees71 and etotheipi
Are we yet at the point where the electrons in the logic gates of my computer may be all bunched up on one side, because it seems to be throwing bugs a lot more often than the last one. Is it still more likely to be bad programming, or could it be the laws of thermodynamics beginning to break down on such tiny processor junctions?

Are we yet at the point where the electrons in the logic gates of my computer may be all bunched up on one side, because it seems to be throwing bugs a lot more often than the last one. Is it still more likely to be bad programming, or could it be the laws of thermodynamics beginning to break down on such tiny processor junctions?
Do a little research on your own and you can calculate that. Estimate the volume of your logic gate. Then estimate the number of electrons or holes in a N type or P type region. Google searches should bring you to all those numbers. What is the answer? Is it of the order 10 electrons or is it much larger?

Also remember, that electrons in a semiconductor or a conductor don't wander aimlessly like a gas in a room. They react to electric and magnetic fields.

vanhees71
Are we yet at the point where the electrons in the logic gates of my computer may be all bunched up on one side, because it seems to be throwing bugs a lot more often than the last one. Is it still more likely to be bad programming, or could it be the laws of thermodynamics beginning to break down on such tiny processor junctions?
Most computer chips are still at 22 nanometer features, and fastest processors at 7 nM. Quantum effects start at 2-3 nM. Flash memory used in storage and also controls computer motherboard, tunnels electrons into insulator. The electrons eventually tunnel back out. The storage vendors says your data is safe for 150 years. The companies making the flash chips say the data will leak out within 10 years. Thanks to Apple, we have no error corection on DRAM. Using E and C we find DRAM stores data as 1/3 -1/4 of an electron. High energy cosmic rays still give ~3 counts per second per square inch in a tall building basement. Think of the silicon junctions of solar cells being degraded by cascade radiation from lower power cosmic rays and Solar protons.

Phase space, not physical space...

What you are talking about is probability. Yes it is possible for 100 coins to flip to all heads, given enough tosses. But generally they will tend toward half heads & half tails, and that's your second law of thermodynamics, that everything tends toward equilibrium.
Its the difference between looking for an improbable instance, versus the probability of an average outcome.

binis, vanhees71 and Delta2
Prof. Tong used the Liouville theorem to prove the Poincaré recurrence theorem in his notes, that given an initial point ##P## in phase space, for any neighbourhood ##D_0## of ##P## there exists a point ##P' \in D_0## that will return to ##D_0## in a finite time.

To illustrate the theorem, he says that if you put a bunch of molecules of gas in a corner of a room, then eventually they'll end up back where they started [granted, the recurrence time exceeds the universe lifetime!].

The question at the end of the section is "Where's your second law of thermodynamics now?!". I can't figure out how its consistent, so I wondered if someone could explain? Thanks!
The point is that, for a system with many (say ##10^{23}##) degrees of freedom the typical Poincare recurrence time is much much larger than the age of the Universe. So during times which are relevant to human observations, Poincare recurrence cannot be observed. For all practical purposes the Poincare recurrence is irrelevant. Below I attach a table of typical time scales in physics, at the end you will see that Poincare recurrence times are the biggest (the last time scale is a joke).

vanhees71 and etotheipi