Two (sorta meta) questions related to entropy

[hideelo]

I'm taking thermal physics this semester and I have two questions that have been floating around my head,

1. Time's arrow and second law:

If I have system (box of gas) and at t_i I know all the positions and momenta of all the particles, let's assume it's not in a state with maximum entropy. If I wanted to know the state of the system at some other time t_j I could say that given that the particles in my system have momenum, given that they are in a box with walls and the particles inevitably collide with those walls, I know that that the system must evolve into some other state and the particles will in general have different momenta and positions. I could now use statistics and try and predict what that other state might be. Turns out, that in general I should expect that at any other time t_j I should expect the entropy to go up simply based on the fact that a state with more entropy has a greater multiplicity, and is therefore more likely. However no where so far have I placed any constraints on if t_i < t_j or t_i > t_j . So why can't I argue that given any system, I could expect that in the past it should have had a higher entropy? In other words it seems to me that I should assume that however and whenever I find a system, it is most probably in the state of lowest entropy it ever has or will be.

Now, this must be false because that gives predictions about nature that strongly depend on when I choose to measure the system, and worse this argument goes against the second law. So why is this false?

2. Entropy and symmetry

If I have some crystalline solid, its symmetry group will be S_k for some k depending on the exact structure of the crystal. I know that as a liquid it has higher entropy and as a gas even higher entropy. However in addition to the increase in entropy, the symmetry of this system also just increased, it's symmetry group is now S^2 which has continuous symmetry under arbitrary rotation. Is increase in entropy somehow related to increase in symmetry? Is there some generalization under which this holds?

TIA

 

[DrClaude]

So why can't I argue that given any system, I could expect that in the past it should have had a higher entropy?

First, let me note that the second law is a statement on probability: we observe what is more probable. There is more chance that we go from a low probability (low entropy) state to a high probability (high entropy) one than the other way around. For a system reasonably big (macroscopic), the probability of observing it in a low entropy state is so incredibly low that in practice it never occurs.

That leads me to the second answer: because that's not what we observe. The arrow of time can be seen as stemming from the fact that the universe started out in a state with extremely low entropy. Check out for instance http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers.../lebowitz_370.pdf.

If I have some crystalline solid, its symmetry group will be S_k for some k depending on the exact structure of the crystal. I know that as a liquid it has higher entropy and as a gas even higher entropy. However in addition to the increase in entropy, the symmetry of this system also just increased, it's symmetry group is now S^2 which has continuous symmetry under arbitrary rotation. Is increase in entropy somehow related to increase in symmetry? Is there some generalization under which this holds?

I don't understand why you say that liquids and gases have symmetry.

 

[hideelo]

First, let me note that the second law is a statement on probability: we observe what is more probable. There is more chance that we go from a low probability (low entropy) state to a high probability (high entropy) one than the other way around. For a system reasonably big (macroscopic), the probability of observing it in a low entropy state is so incredibly low that in practice it never occurs.

That leads me to the second answer: because that's not what we observe. The arrow of time can be seen as stemming from the fact that the universe started out in a state with extremely low entropy. Check out for instance http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers.../lebowitz_370.pdf.

I don't understand why you say that liquids and gases have symmetry.

The low entropy of the big bang is a good point, I'll think about it.

Can you double check the link you added though, I couldn't access it.

As to the second question, rotational symmetry. Gas looks the sme no matter which way you look at it, the same doesn't hold for a solid crystal.

 

[DrClaude]

For the first question, I keep hearing the claim that the/an origin of the arrow of time is the second law of thermodynamics. I don't see it however, I see no reason why entropy should increase in one direction of time and not the other.

Take a box with 10 molecules and follow their trajectories. They will sometime occupy a small volume of the box, some fill the box uniformly. If I were to show you a movie of the molecules moving around, you couldn't tell if I was showing it as it was filmed or in reverse. There is no arrow of time.

If you take a box with Avogadro's number of molecules, and start them in one corner, they will soon fill the entire volume, and then you could wait for the age of the universe, and nothing would change. If I showed you a film of the molecules and you saw them all clump together in a corner of the box, you would know that I was showing the film backwards: there is an arrow of time. If I were to show you a film taken at equilibrium, you would again not know if I was showing it in the normal way or in reverse. So you see the arrow of time appear (generally speaking) when systems go from low entropy to high entropy.

As to the second question, rotational symmetry. Gas looks the sme no matter which way you look at it, the same doesn't hold for a solid crystal.

From that point of view, a solid sphere would also have very high symmetry. You have to look at where the individual atoms/molecules are: in a crystal, you have order and symmetry, in a liquid and in a gas, you don't.

 

[hideelo]

Take a box with 10 molecules and follow their trajectories. They will sometime occupy a small volume of the box, some fill the box uniformly. If I were to show you a movie of the molecules moving around, you couldn't tell if I was showing it as it was filmed or in reverse. There is no arrow of time.

If you take a box with Avogadro's number of molecules, and start them in one corner, they will soon fill the entire volume, and then you could wait for the age of the universe, and nothing would change. If I showed you a film of the molecules and you saw them all clump together in a corner of the box, you would know that I was showing the film backwards: there is an arrow of time. If I were to show you a film taken at equilibrium, you would again not know if I was showing it in the normal way or in reverse. So you see the arrow of time appear (generally speaking) when systems go from low entropy to high entropy.From that point of view, a solid sphere would also have very high symmetry. You have to look at where the individual atoms/molecules are: in a crystal, you have order and symmetry, in a liquid and in a gas, you don't.

Thanks, food for thought :-)