Equiprobability of Particles in an Isolated Box

[Kashmir]

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Its said that the probability of having the configuration of $n$ particles in the left side is given as $P_n=C(n)/2^N$ with $C(n)$ being the total number of ways in which $n$ particles from $N$ can be placed in the left side.

Why should $P_n=C(n)/2^N$ be the probability? It should be true only if each configuration is equiprobable, but I don't think it is equiprobable. Our initial conditions (position and velocity of particles) will determine the evolution of the states of the system, so it might be that some states occur more frequently than other?

 

[BvU]

with both parts identical

Do you mean: in equilibrium ?

 

[jbriggs444]

I have trouble with the idea that we have classical particles and that we also have finitely many states.

 

[Kashmir]

I have trouble with the idea that we have classical particles and that we also have finitely many states.

These are not energy states. The configuration here is just the number of particles in each side.

 

[Kashmir]

Do you mean: in equilibrium ?

What does equilibrium mean here? If that means the state after a long time, then then the question isn't answered.

 

[jbriggs444]

These are not energy states. The configuration here is just the number of particles in each side.

Ahhh, perfect. Then all we need is that any state is reachable from any other state after some series of transitions. Then indistinguishability leads to a symmetry among the transitions and we have a Markov process.

 

[Kashmir]

Ahhh, perfect. Then all we need is that any state is reachable from any other state after some series of transitions.

Thanks for you comment.

How do we know that any state is reachable? Suppose i start the experiment with $N$ particle with $n$ on one side each with some velocity.

Also the partition into left and right side is hypothetical, the box is sealed and has no partition, we just mentally divide it into two, as I've mentioned in the original question.

 

[jbriggs444]

How do we know that any state is reachable?

In general, you cannot. It is a handwave. Albeit a pretty reasonable one.

Maybe you had one classical particle on the left side with a purely vertical trajectory. No state transitions will ever take place. But how likely is that?

 

[Kashmir]

In general, you cannot. It is a handwave. Albeit a pretty reasonable one.

Maybe you had one classical particle on the left side with a purely vertical trajectory. No state transitions will ever take place. But how likely is that?

Yes very unlikely.
Could you help me with this:

"Why should $P_n=C(n)/2N$ be the probability? It should be true only if each configuration is equiprobable, but I don't think it is equiprobable. Our initial conditions (position and velocity of particles) will determine the evolution of the states of the system, so it might be that some states occur more frequently than other"?

 

[jbriggs444]

"Why should $P_n=C(n)/2N$ be the probability? It should be true only if each configuration is equiprobable, but I don't think it is equiprobable. Our initial conditions (position and velocity of particles) will determine the evolution of the states of the system, so it might be that some states occur more frequently than other"?

If they are classical particles, they are zero size, non-interacting and, accordingly, independent. Since the two sides identical then, each particle eventually has 50/50 chance of being on either side. Since the particles are independent, all $2^N$ microstates are [eventually] equiprobable.

 

[Kashmir]

If they are classical particles, they are zero size, non-interacting and, accordingly, independent. Since the two sides identical then, each particle eventually has 50/50 chance of being on either side. Since the particles are independent, all $2^N$ microstates are [eventually] equiprobable.

Why don't the initial conditions influence the probabilities ?