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I heard the definitions, but I have much trouble understanding them. My teacher told me that in an example with two chambers and gas particles (look below) entropy increases, but I cannot understand why.

Let’s say that we have $2$ empty, identical chambers, separated by a wall. Inside one of these chambers we have a gas particle with non-zero kinetic energy. Let’s say that if we had more than one particle, theirs collisions would be elastic and they wouldn’t lose energy while hitting the walls.

Now, let’s remove the wall and let the particle fly freely. After some time it can come back to the initial position, so entropy didn’t change at all. So the process isn’t irreversible. If we have $2$ particles, then after some time they can come back to the initial state too. The same with $3, 4, 5, \dots, n$ particles. Even if for large $n$ it is less and less possible, it is still possible, so it still shouldn’t be an irreversible process, but from what I heard, it is.

Where is my misunderstanding? It seems nonsensical than for some $n$ the process wasn’t irreversible, but for $n+1$ it was, so... was the process irreversible from the very beginning? If yes, how does entropy increase, if the particle goes back to its initial state?

Now, let’s look at the problem from a different side. I heard that entropy is a measure of the number of ways in which a system may be arranged. Let’s say that we have $1$ particle and $2$ chambers. It can happen that the particle will be just flying from one chamber to the other and back, with probability of being in a specific one of them equal to $\frac{1}{2}$. So, entropy shouldn’t change then, should it?

Now, if we have $2$ particles, the probability changes, but there is still very likely for all particles to be either in first or second chamber, so entropy cannot change. The same with $3$ particles, $4$ particles… $n$ particles.

The chances for all particles being in the same chamber are getting lower and lower, but are always positive, so entropy shouldn’t really increase, as in the “long run”, the system will be going back to its initial state with probability equal to $1$.

If I am wrong and it is irreversible for many particles, then there should be a certain number $n$ for which the process is reversible, but for $n+1$ it isn't. I don't see why should this happen, especially because probability for particles to be in a certain chamber decreases gradually. Thus there should be no reason for such drastic changes.

And if it is irreversible, then it should be irreversible for the single particle too. But this particle can go back perfectly to it's initial state, why then does the entropy increase?

How can this process even be irreversible?

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    $\begingroup$ you should see this part of lecture by Dr. Richard Feynman youtube.com/watch?v=2OU_f7vYDys $\endgroup$ Jan 26, 2018 at 17:19
  • $\begingroup$ These are excellent questions... $\endgroup$
    – The Quark
    Nov 2, 2023 at 13:46
  • $\begingroup$ @aaaaasaysreinstateMonica The video is not available anymore. Would you know another link to the recording? $\endgroup$
    – The Quark
    Nov 2, 2023 at 13:46

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Your mistake is that by "state" you mean following: particle either in section A or section B. In reality what you use for computing entropy is number of ways for particle to be in some point of phase space, that is having particle coordinate and impulse (all 3D).

But you can omit these details. Consider box with single particle and 2 sections divided by barrier. Number of ways for this system to be is 1: particle is always on one side. Remove the divider and now you have two states: particle on the left or particle on the right (each with probability 1/2). Entropy increased, because:

Before: $$\begin{align}S&=-k_b\sum_i p_i \ln(p_i)\\&=-k_b\times 1\times \ln(1)\\&=0\end{align}$$

After: $$\begin{align}S &= -k_b\sum_i p_i \ln(p_i)\\ &=-k_b(0.5\ln(0.5) + 0.5\ln(0.5))\\ &=-k_b\times (-0.7)\\&=0.7k_b>0\end{align}$$

Now, consider reversibility. It means that after entropy increased you will have to wait very long time for system to come back to initial state. When your body dies, you need to wait really long to become alive again.

For number of particles it is much more probable that they will be distributed randomly across box's sections, rather than concentrated on one side. For 3 particles there are 2 options: either all on right, or all on left. But there are many more options (if all particles are distinguishable) for them to be around the box.

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