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I'm reading through Statistical Physics by F. Mandl and in the chapter about the 2nd law of thermodynamics he states that:

The basic distinction between the initial and final states in such an irreversible process is that in the final state we have a less complete knowledge of the state of the system.

Why is this true? Let's say our gas exists in the left side of a container separated by a partition. We know that all the molecules are definitely on the left side.

Now we remove the partition. The gas rushes into the vacuum and eventually spreads itself throughout the container. The book mentions that macroscopic fluctuations of the gas moving around would not be observable unless we waited for a period of time around the order of the age of the universe. So then if we assume that 50% of the gas is on the left side and 50% is on the right side (assume that the partition is infinitesimally thin and unbreakable/unbendable), then how is it that we know less information of the system in this state?

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  • $\begingroup$ Do you know how entropy is defined? $\endgroup$
    – ClassicStyle
    Commented Feb 27, 2016 at 2:43
  • $\begingroup$ @TylerHG Yes, it can be defined in a general sense as the amount of disorder of a system. But the thing is that when the gas is on the left side only, it's still technically disordered because we don't know the exact position of every molecule in space. We only know in the macroscopic sense that the gas is on the left side. When we remove the partition, we know that the gas is now on both the left and right sides, but we still don't know the exact positions of every molecules in space. $\endgroup$
    – whatwhatwhat
    Commented Feb 27, 2016 at 2:46
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    $\begingroup$ It depends on how you define "information". Physicists have a somewhat quirky way of defining it. What they mean is that there are more places where the gas molecules can be after the volume expands, so there are more possible microscopic configurations. From an information theoretical perspective the volume/number of states is "potential information" or absence of information, if you like. You can look at it the other way round... it's an increase in the amount of information you would have to gather to completely describe an actual state of the gas. $\endgroup$
    – CuriousOne
    Commented Feb 27, 2016 at 2:49
  • $\begingroup$ @CuriousOne so the extra information is simply observing that the volume changed from $V_1$ to $V_2$? $\endgroup$
    – whatwhatwhat
    Commented Feb 27, 2016 at 3:00
  • $\begingroup$ Quite the opposite. Look at it this one: one molecule and a tiny compartment in which it can't move. How many states that the system have? 1, right? The entropy of that is 0. Now double the volume. Now the molecule can be on one of two places. How much information can you store in that? 1 bit. Physicists call that an entropy of 1 (with a funny factor on top :-)). Now double the volume, again. Now the particle can be in one of four states. That's 2 bits and physicists assign an entropy of 2 (units). Add more particles, more volume, the numbers get bigger, but the principle stays the same. $\endgroup$
    – CuriousOne
    Commented Feb 27, 2016 at 4:02

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Partially answered already in the question and comments. CuriousOne answered it.

Entropy is a measure of the possible microstates of the system, i.e., the different positions and velocities of each of the molecules. When you double the volume each of the molecules doubles the number of possible x, y, and z's. The possible number of states has increased. S has gone up.

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