How many possible states can 10 floating marbles occupy in a 10x10x10 box?

[Unteroffizier]

Greetings!

I do not know whether or not this is the correct location for my post. I also apologize in advance if my question seems too simple, nonsensical, or downright idiotic. I am only a high school 1st year, therefore I do not have much knowledge of anything significant.

So, while watching some educational videos on thermodynamics, I had a thought. Say we have a box filled with 10 magical floating marbles that can pass through each other (to avoid inconsistencies). They all start out in one corner of the box, where they are "born". Each sets out in a different direction, bouncing off walls as they go.

Sooner or later, all the marbles will end up, even if only for a very short time, in the same spot as they started, correct? Given enough time, surely they must go through every possible combination of locations and end up in point 0, no?

Well, let's say this is true. For the sake of simplicity, let's say that the box has a volume of 10x10x10. So that means there are, in total, 1000 places a marble may occupy. Every second, each marble moves to a different area.

How would I calculate the number of possible states of the group of marbles? 10^1000? I'm truly clueless. I've never been proficient in mathematics, so I have no idea.

Also, how would I calculate the time it would take for all the marbles to go back to their original locations? Or is that even possible to measure at least somewhat accurately?

Again, forgive me for the overall dumb question.

Thanks.

 

[BvU]

Hello there, and

Good questions. Thought experiments like this form a good basis for thermodynamics. But to get further you need some more mathematics.
And you need to specify a few more things that make difference: the marbles can pass through each other, so i suppose several marbles can take up the same position. That settles one issue. The other is: can the marbles be distinguished from each other ? I.e.: is a state which differs from another state in that marbles 1 and 2 are interchanged a different state, or not ?

Counting goes like this if the marbles can be distinguished:
marble 1 can go in 1000 places
marble 2 idem
together that makes 1000² states already
So with 10 distinguishable marbles you end up with 1000¹⁰ = 10³⁰ states, a 'little' less than the 10¹⁰⁰⁰ you envisioned...

 

[mfb]

And here the case of distinguishable marbles, which is a bit more complicated:

Imagine the 1000 possible spaces written as a list, with marbles among them, e.g. with 1 marble at place 2 and 2 marbles at place 4:
1, 2, M, 3, 4, M, M, 5, ...
In total our list has 1010 entries, but the first one cannot be a marble, so we have 1009 entries where 10 of those are marbles. Every arrangement is a different state. How many options are there to pick 10 places out of 1009? This can be calculated with the binomial coefficients: (1009 choose 10) = 2.88*10²³ (rounded).

As you can see, the number is significantly smaller in this case.

Add different velocities to the possible states of the particles, assume that every state (for a given energy) has the same probability, and you get thermodynamics.

 

[Unteroffizier]

Thank you both for your answers! I appreciate your assistance very much!

Indeed, I require more mathematics to progress in this field. I won't pretend to understand everything you guys said (I probably would if I gave it more effort, but I'm quite exhausted after another day of vectors and intervals). In fact, I haven't taken a class on thermodynamics yet (still have to finish irregular motion), so I suppose I can only wait to truly understand everything I've been told.

Also, to add to my question, I'd like to ask about the entropy of the system I described above. I don't really understand entropy, but from what I can gather, a higher entropy equals a more chaotic system (the marbles moving faster, located in separate locations). Let us shrink the system then, making the marbles into molecules and the box significantly smaller. As I implied; sooner or later, the system will return to its original state, the state where all marbles (particles) are in point 0, the starting point (surely that's inevitable if they can pass through each other). Does that mean that the entropy of the system will be lower at that point?

One way I could possibly argue against it would be the fact that the molecules are in fact still in motion when they meet in point 0. Another counter-argument could be that the closer together the molecules are pressed, the higher the heat, the higher their velocities, therefore ultimately more entropy? But that only applies if they cannot pass through each other, right?

Am I thinking about this right? Thermodynamics is a fascinating topic, and I quite simply cannot wait to learn it properly.

 

[mfb]

Does that mean that the entropy of the system will be lower at that point?

"Yes". In quotation marks because it is "unfair" to pick this specific point in time. Most of the time they are not at the same place, and thermodynamics is about the average over large samples (in particles or in time).

 

[Unteroffizier]

I see! So entropy isn't measured like, say, instantaneous velocity. Instead it's the overall average entropy in a certain timespan? Interesting.

Thanks for the quick response! Would you have any tips for studying thermodynamics when the time comes?

 

[BvU]

If you can find a copy (perhaps in the library), P.W. Atkins: The 2nd law (Scientific American Library Paperback) is old but very good reading.

 

[Unteroffizier]

There's a small library in my school, so perhaps I'll find it. Thanks for the tip! Should be a good read!