In Part 6 I introduced the notion of entropy in the context of heat engines, defining it as dS = dQ / T. The great Ludwig Boltzmann gave us an equivalent, statistical notion of entropy, establishing it as a measure of disorder.
Imagine a gas in a chamber (labelled A in the left part of the figure below), separated by a partition from another chamber (B) of the same volume. This second chamber is empty to start with. If the partition disappears, the molecules of the gas start moving into the right half of the enlarged chamber, and soon the gas occupies the entire (doubled) volume uniformly.
Next let us ask the question: What is the probability that all the n molecules of the gas will ever occupy the left half of the chamber again? This probability is the same as that of flipping a coin n times, and finding ‘heads’ in each case, namely ½n.
Considering the fact that usually n is a very large number (typically of the order of the Avogadro number, i.e. ~1023), the answer is very close to zero. In other words, the free expansion of the gas is practically an irreversible process. On removal of the partition, the gas has spontaneously gone into a state of greater disorder, and it cannot spontaneously go back to the initial state of order (or rather less disorder).
Why do we say 'greater disorder'? Because the probability of finding any specified molecule at any location in the left half of the chamber is now only half its earlier value. Here 'order' means that there is a 100% chance that a thing is where we expect it to be, so 50% chance means a state of less order, or greater disorder.
So, intuitively we have no trouble agreeing that, left to themselves (with no inputs from the outside), things are more likely to tend towards a state of greater disorder. This is all that the second law of thermodynamics says. It says that if we have an ISOLATED system, then, with the passage of time, it can only go towards a state of greater disorder on its own, and not a state of lesser disorder. This happens because, as illustrated by the free-expansion-of-gas example above, a more disordered state is more probable.
How much has the disorder of the gas increased on free expansion to twice the volume? Consider any molecule of the gas. After the expansion, there are twice as many positions at which the molecule may be found. And at any instant of time, for any such position, there are twice as many positions at which a second molecule may be found, so that the total number of possibilities for the two molecules is now 22. Thus, for n molecules there are 2n more ways in which the gas can fill the chamber after the free expansion. We say that, in the double-sized chamber, the gas has 2n more 'accessible states', or 'microstates'. [This is identical to the missing-information idea I explained in Part 21.]
The symbol W is normally used for the number of microstates accessible to a system under consideration. This number doubled when the gas expanded to twice the volume. So, one way of quantifying the degree of disorder is to say that entropy S, a measure of disorder, is proportional to W; i.e., S ~ W.
Boltzmann did something even better than that for quantifying disorder. He defined entropy S as proportional to the logarithm of W; i.e. S ~ log W.
In his honour, the constant of proportionality (kB) is now called the Boltzmann constant. Thus entropy is defined by the famous equation S = kB log2W (or just S = k log W).
To see the merit of introducing the logarithm in the definition of entropy, let us apply it to calculate the increase of entropy when the gas expands to twice the volume. Since W = 2n, we get S ~ n. This makes sense. Introduction of the logarithm in the definition of entropy makes it, like energy or mass, a property proportional to the number of molecules in the system. Such properties are described as having the additivity feature. If the number of molecules in the gas is, say, doubled from n to 2n, it makes sense that the defined entropy also doubles in a linearly proportionate fashion.
The introduction of log W, instead of W, in the definition of entropy was done for the same reasons as those for defining missing information I (cf. Part 21). In fact,
I = S.
Also, the original (thermodynamic) and the later (statistical mechanics) formulations of entropy are equivalent. In the former, entropy of an isolated system increases because a system can increase its stability by obliterating thermal gradients. In the latter, entropy increases (and information is lost) because spontaneous obliteration of concentration gradients (and the ensuing more stable state) is the most likely thing to happen. Concentration gradients get obliterated spontaneously because that takes the system towards a state of equilibrium and stability.
For an isolated system, maximum stability, maximum entropy, and maximum probability all go together.