5:49 Contrary to what he says, the equally distributed state is more likely with 2 than with 4 particles: 1/2 versus 3/8. Equally distributed states get less likely as the number of particles increases.
Good point on the math Laureano. I guess that the macroscopic result is what he had in mind, with the fluctuations away from that 50/50 state being smaller and smaller as N increases. Thanks for your comment.
Hey dont compare 1/2 (two particles) with 3/8(4 particles).In case of two particles from the initial all on one side (1/4) to one on each side (2/4) is 100% more .While with 4 particle from the initial all on one side(1/16) to two on each side (6/16) is 500% more. That's what he means when he says it becomes ever more likely to get the equally distributed state with increasing number of particles.
Yes, the equally distributed case is the most probable case, but as the the number of particles increases, the equally distributed case becomes less and less probable., though remaining the most probable of the possible cases.
Thanks, it's really clear. Still I don't understand the concept of macrostates: in all of these examples, aren't we talking about the same macrostate?I mean the box always has the same temperature, pressure and volume, no matter how the particles are distributed, right?
I have trouble understanding some things regarding entropy being described with probability. In the two particle example, you have two possibilities where you have 1 particle on the left and 1 particle on the right. The two separate instances of this are described as equal. So why are they counted as two separate possibilities and not one single possibility? Where you would then have a total of just 3 possibilities of equal probability. I'd be a fool to say its incorrect, I just don't understand it Edit: Additionally, why are the two possibilities where the particles are on the same side as each other not considered to be the same configuration? Leaving you with only 2 actual possibilities. The structure of the two particles is either 1. They are at the same side, or 2. They are on opposite sides.
Label the particles: Particle 'a' Particle 'b' Imagine two possibilities: Possibility 1: particle a on the left and particle b on the right Possibility 2: particle b on the left and particle a on the right Read carefully the difference. They are both swapped around. Both possibilities lead to the same configuration, i.e. a particle on the left and a particle on the right. It's just that particle a is on the left in one possibility and particle b is on the left in the other, but they lead to the same thing. So there are 4 possibilities, but there are three configurations. It's just that two possibilities can lead to the exact same configuration. So those configurations aren't equally likely. The first one is 1/4 chance, the second (which has two possibilities that lead to it) is 2/4 and the last is 1/4. So there is a 25% chance of getting the first configuration, 50% of getting the second one and 25% of getting the last. The one with the highest likelihood always seems to be the one where there are an equal amount in either side, as there are more possibilities that lead to those configurations.
beautifully explained
Great explanation Sir
5:49 Contrary to what he says, the equally distributed state is more likely with 2 than with 4 particles: 1/2 versus 3/8. Equally distributed states get less likely as the number of particles increases.
Good point on the math Laureano. I guess that the macroscopic result is what he had in mind, with the fluctuations away from that 50/50 state being smaller and smaller as N increases. Thanks for your comment.
It is not the fraction of the states having equal distribution but the absolute number of such states that matters.
I think what he want to say actually is the "probabilistic density" increases as the number of particles increases.
Hey dont compare 1/2 (two particles) with 3/8(4 particles).In case of two particles from the initial all on one side (1/4) to one on each side (2/4) is 100% more .While with 4 particle from the initial all on one side(1/16) to two on each side (6/16) is 500% more. That's what he means when he says it becomes ever more likely to get the equally distributed state with increasing number of particles.
Yes, the equally distributed case is the most probable case, but as the the number of particles increases, the equally distributed case becomes less and less probable., though remaining the most probable of the possible cases.
Great explanation,
Thanks, it's really clear. Still I don't understand the concept of macrostates: in all of these examples, aren't we talking about the same macrostate?I mean the box always has the same temperature, pressure and volume, no matter how the particles are distributed, right?
The volume would be the same but not for the other two because the density would not be the same. You can also refer to the old PV=nRT to see that.
Thank you!
Awesome
If the particles are indistinguishable how we can more than 1 configuration for each microstate?
The states are equivalent, but not the same. If two particles swap position, this is counted as another microstate, but an equivalent one
During a large shock waveyou have a lot of upset molecules on 1 side of the box
Thanks 🙏
I have trouble understanding some things regarding entropy being described with probability.
In the two particle example, you have two possibilities where you have 1 particle on the left and 1 particle on the right. The two separate instances of this are described as equal. So why are they counted as two separate possibilities and not one single possibility? Where you would then have a total of just 3 possibilities of equal probability.
I'd be a fool to say its incorrect, I just don't understand it
Edit: Additionally, why are the two possibilities where the particles are on the same side as each other not considered to be the same configuration? Leaving you with only 2 actual possibilities. The structure of the two particles is either 1. They are at the same side, or 2. They are on opposite sides.
Label the particles:
Particle 'a'
Particle 'b'
Imagine two possibilities:
Possibility 1: particle a on the left and particle b on the right
Possibility 2: particle b on the left and particle a on the right
Read carefully the difference. They are both swapped around.
Both possibilities lead to the same configuration, i.e. a particle on the left and a particle on the right. It's just that particle a is on the left in one possibility and particle b is on the left in the other, but they lead to the same thing.
So there are 4 possibilities, but there are three configurations. It's just that two possibilities can lead to the exact same configuration.
So those configurations aren't equally likely. The first one is 1/4 chance, the second (which has two possibilities that lead to it) is 2/4 and the last is 1/4.
So there is a 25% chance of getting the first configuration, 50% of getting the second one and 25% of getting the last. The one with the highest likelihood always seems to be the one where there are an equal amount in either side, as there are more possibilities that lead to those configurations.