The Enigma of Entropy

0.04 MB·English

by  Deepak Dhar

The Enigma of Entropy
∗
Deepak Dhar
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India.
(Dated: January 10, 2013)
This is a light-hearted takeat thethesecond law of thermodynamics.
PACSnumbers: 01.30.Rr,05.70.-a
Oneofthemostmysteriousoflawsofnatureisthesec- deliberate anthropomorphic flavour added to the story.
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ondlawofthermodynamics. Thereareseveralequivalent The amazing fact remains that “having more choices” is
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0 formulations of this law. For our present discussion,it is such an important fundamental principle of Nature.
2 enough to take the Clausius formulation that says that Consider a known mass of a dilute gas, say hydrogen,
n for an isolatedsystem evolving in time, the entropy can- kept in a box at a fixed temperature. Then, given this
a notdecrease. Manyofyouhaveencounteredthis inyour specification of the macroscopic system, we are able to
J B.Sc. textbooks already. Let me explain my reasons for determine other properties of the system, like the pres-

1 calling it enigmatic. I do not have any answers. I just sure, or the viscosity of the gas etc However, given the

wanttosaywhyIthinkthatitisaninterestingquestion. macroscopic parameters characterizing the system, we
]
h Letmeexplainmyselfusingaparable. Aforeignscien- cannotreallytelltheprecisepositionandvelocityofeach
p tistis visitingalaboratoryinChina. Inthe daytime,he individual molecule : there are very many “microstates”
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p discusseshis workwithhishosts,andintheevenings,he corresponding to a fixed macrostate. According to fa-
o has to find some place to eat. Unfortunately, he knows mous equation of Boltzmann, the entropy S is equal to
p no Chinese. klogΩ, where Ω is the number of microstates available
.
s So, he goes to a restaurant, sits down on a table, and to the system. So, the entropy is not determined by the
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the waiter rings in the menu. It has a very long menu. currentstate(theactualmicrostate),butbywhatothers
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s All kinds of dishes listed, about one thousand in all, and it could have been in. It seems to be not a function of
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the price of each in understandable numerals. Knowing what is, but of what could have been!.
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p what he is willing to spend, the man selects one of the Oftenoneexplainsthenotionofentropybysayingthat
[ entries in the correct price range at random, and points it is a measure of disorder. The system is prepared in a

to it. Eventually,the waiterbringshimthe food, he eats macrostate, but it could have been in any of the mi-
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v and leaves. Next day, it is the same story. The food is crostates of the system. When the system is observed
2 satisfactory and the man has no complaints. after some time, we can only say that each microstate
9 But, aftera few days,he discoversa new restaurantin occurs with probability 1/Ω. To any such probability
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the neighborhood. It is the same price range, and same distribution, in which microstate ioccurs with probabil-
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. quality of food. He tells his friends that he prefers this ity pi, we can define a quantity, the Shannon entropy
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restaurantto the other. The friend knows how he places SSh = Pipilogpi. But this measures the uncertainty
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his order, and wants to know what is special about the theobserverfeels,giventhelimitedinformation. Itseems
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1 second restaurant. The man says that there the menu to be a measure of disorder in the head of the observer,
: is bigger, ten thousand entries, and I like to have more not in the system.
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i choices. Continuing our story, when our hero is asked to ex-
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Of course, one would think that the man is quite fool- plain what is the point of having more choices, he says,
r ish. Whatdifferencedoesitmakethatthe menuhasone “Well, my boss in the lab is a very nosey person. I think
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thousand or one million entries, after all, in one dinner, he wantsto knowexactly I eat,eventhoughitis none of
you are only going to eat one meal! The problem is that hisbusiness. Ichoosethesecondrestaurant,justtospite
systems with no brains at all do exactly this. him.” This explanation seems not really convincing. In
As a paranthetical remark, let me add here that de- fact, most of the time, the boss (the external observer)
scriptionofasystemofsaygasmoleculesinabox“having doesnotwanttoknowthedetailsofwhatthepersonate,
a wish to” increase their entropy is quite inappropriate, and maybe the hero is not really so full of negative emo-
even though in textbooks one often encounters such ex- tions, like spitefulness. One is reminded of A. Einstein’s
pressions as “the electron wants to go the lowest energy words: Subtle is the Lord,but maliciousHe is not.” For
state”. The electron has no brains, and in no effective the more agnostic readers,‘Lord’in the above quotation
sensethiscouldbetrue. Also,ifeverybodywouldwantto may be replaced by ‘Nature’.
lower their energy, where would the extra energy go? In In fact, for systems that undergo deterministic evolu-
myparable,thiswasdonedeliberately,asarhetoricalde- tion, the microstate at later time is fully determined by
vice. Inamoreseriousdiscussion,oneshouldignorethis the present, and there is no need to invoke any proba-
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bility ideas. In this scenario, the hero of our story has a That is the mystery.
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
∗
Electronic address: [email protected]
the second restaurant over the first?