Expert: Daniel Mazur - 8/27/2009

Question
Hi Daniel,

This is the first time I've started studying the topic of thermodynamics in any detail. It's the only topic that I didn't know much about so far.

So far it's quite comprehensible. But then I came across the concept of entropy. I had read somewhere, I think in stephen hawking's book sometime back that entropy is a measure of chaos and randomness and this randomness always increases in a system.

Now when I studied from my book, it said entropy that we "define" entropy

as dS = dQ/T
where S, Q and T stand for entropy, heat given to the system and temperature of the system in some equilibrium state.

Now who defined this, how was it defined and why was it defined that way???

Also if it is somehow related to the chaos and randomness(and how exactly), is the law that the chaos always increases, a fundamental law of nature???

If it is ..... can't we state it in another way as follows??
Suppose we have a sample of a gas with some molecules having a greater kinetic energy than the others.
So the molecules with the greater K.E. have more chances of coming near the slower molecules than the slower molecules going near the faster ones. So faster molecules will tranfer their energy to the slower molecules at a faster rate than the slower molecules will to the faster ones.

In the case of other laws like newton's first law, we can't understand that law in terms of some other thing....i.e. we don't know why F=ma....
But in the case of this law...the law of incresing entropy....it isn't so..
Can't we understand this law on the basis of some molecules having a greater probability than others????

Thanks ,
Shikhin

Answer
Dear Shikhin,

entropy is indeed a measure of randomness in a system, but you cannot start understanding it from the point of thermodynamics. Thermodynamics is based on chaos and randomness, but it only uses it, it does not study any properties of chaos, nor how chaos arise from order. The equation you've written is a (a guy named Clausius found it) thermodynamic definition of entropy. Chaos, and therefore entropy, is studied at the level of statistical mechanics. At the level of statistical mechanics entropy is defined as S = -k_B*Sum(w_i*ln(w_i)), where w_i is the "number of realizations of a physical system, given some set of values of macroscopic state variables". The k_B is the Boltzmann constant, which connects purely combinatorial number w_i to the physical Universe. Let me give you an example:

If you take 100 ideally elastic balls, put them in a box and (by shaking the box for example) give the balls a total energy E=2kJ. The balls (no gravitation, no friction, no losses) will continue moving in the box, bouncing off the walls and off one another. As you can imagine, the energies {E1,E2,...,E100} are all time-dependent. However, if you treat the balls as indistinguishable, generate a histogram and observe the time development of the histogram, you will find, that NO MATTER what the initial energy distribution (and associated histogram) looked like, the system always evolves in time into a state characterized by the same histogram - one with majority of the balls having their kinetic energy near the "average energy" = 2kJ/100 = 20J. Do you know why? It is because such a situation has "maximum number of realizations" of all situations, consequently it has maximum entropy and it represents the "thermodynamic equilibrium" of the system. It can be proven, why the "maximum number of realizations" happens this way, but please consult a textbook of Stat. Mechanics for it. It is not too difficult.
So, now you know, what entropy is. Entropy as such was found in studying evolution (time dependence) of statistical ensembles of indistinguishable objects. It simply fell out of the equations that a quantity exists, which for any given system left alone (!) can only grow (or stay constant, when maximum is reached) with time. Yes, it IS a fundamental law of nature. Thermodynamically it is defined by the equation you wrote, although I a better form is:

\Delta S = S_2 - S_1 = \int_1^2 (dQ/T)

Here I used LaTEX notation, look at Wikipedia http://en.wikipedia.org/wiki/Boltzmann_constant for its intelligible form. I should stress out that you MUST pay a close attention to the assumptions, under which this equation is valid, namely the part about dQ being transferred to (from) the system in a reversible (!) process. You may also find http://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics) enlightening. As to why entropy was defined this way I cannot answer very well - do you have a better definition? I think it's simply that dQ and T are macroscopic observables such that using them the definition has the most concise mathematical form.

[Q]If it is ..... can't we state it in another way as follows??
Suppose we have a sample of a gas with some molecules having a greater kinetic energy than the others. So the molecules with the greater K.E. have more chances of coming near the slower molecules than the slower molecules going near the faster ones. So faster molecules will tranfer their energy to the slower molecules at a faster rate than the slower molecules will to the faster ones.[/Q]
I think that you meant to say that any faster molecule experiences higher average frequency of collisions with other molecules than any slower molecule. Yes, this is a fact, but it is an illuminating example, not an equivalent form of writing the law. The law of non-decreasing entropy IS explained by the "maximum number of realizations" and therefore via the chaos notion. There is no more fundamental way of explaining it.

Cheers,
Daniel