The Enigma of Entropy

— Deepak Dhar

 

One of the most mysterious of laws of nature is the second law of thermodynamics. There are several equivalent formulations of this law. For our present discussion, it is enough to take the Clausius formulation that says that for an isolated system evolving in time, the entropy cannot decrease. Many of you have encountered this in your B.Sc. textbooks already. Let me explain my reasons for calling it enigmatic. I do not have any answers. I just want to say why I think that it is an interesting question.

Let me explain myself using a parable. A foreign scientist is visiting a laboratory in China. In the day time, he discusses his work with his hosts, and in the evenings, he has to find some place to eat. Unfortunately, he knows no Chinese.

So, he goes to a restaurant, sits down on a table, and the waiter brings in the menu. It has a very long menu. All kinds of dishes listed, about one thousand in all, and the price of each in understandable numerals. Knowing what he is willing to spend, the man selects one of the entries in the correct price range at random, and points to it. Eventually, the waiter brings him the food, he eats and leaves. Next day, it is the same story. The food is satisfactory and the man has no complaints.

But, after a few days, he discovers a new restaurant in the neighbourhood. It is the same price range, and same quality of food. He tells his friends that he prefers this restaurant to the other. The friend knows how he places his order, and wants to know what is special about the second restaurant. The man says that there the menu is bigger, ten thousand entries, and “I like to have more choices”.

Of course, one would think that the man is quite foolish. What difference does it make that the menu has one thousand or one million entries, after all, in one dinner, you are only going to eat one meal! The problem is that systems with no brains at all do exactly this. Forgetting the deliberate anthropomorphic flavour added to the story, I find it infinitely amazing that “having more choices” should be such an important fundamental principle of Nature.

Consider a known mass of a dilute gas, say hydrogen, kept in a box at a fixed temperature. Then, given this specification of the macroscopic system, we are able to determine other properties of the system, like the pressure, or the viscosity of the gas etc… However, given the macroscopic parameters characterizing the system, we cannot really tell the precise position and velocity of each individual molecule: there are very many “microstates” corresponding to a fixed macrostate. According to famous equation of Boltzmann, the entropy S is equal to k log Ω, where Ω is the number of microstates available to the system. So, the entropy is not determined by the current state (the actual microstate), but by what others it could have been in. It seems to be not a function of what is, but of what could have been!

Often one explains the notion of entropy by saying that it is a measure of disorder. The system is prepared in a macrostate, but it could have been in any of the microstates of the system. When the system is observed after some time, we can only say that each microstate occurs with probability 1/Ω. To any such probability distribution, in which microstate i occurs with probability pi, we can define a quantity, the Shannon entropy SSh= Σipilogpi. But this measures the uncertainty the observer feels, given the limited information. It seems to be a measure of disorder in the head of the observer, not in the system.

Continuing our story, when our hero is asked to explain what is the point of having more choices, he says, “Well, my boss in the lab is a very nosey person. I think he wants to know exactly I eat, even though it is none of his business. I choose the second restaurant, just to spite him.” This reason seems not really plausible. In fact, most of the time, the boss (the external observer) does not want to know the details of what the person ate, and maybe the hero is not really so full of negative emotions, like spitefulness.

In fact, for systems that undergo deterministic evolution, the microstate at later time is fully determined by the present, and there is no need to invoke any probability ideas. In this scenario, the hero of our story has a simpler method of making the selection : he selects the item with the right price that comes next in the menu to what he chose on the previous day. Here he has no choice whatsoever. In that case, why does he still prefer the second restaurant over the first?

That is the mystery.

 

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