Statistical Mechanics: Maximum Entropy

[Niles]

Homework Statement

Hi all.

I've read some about the MaxEnt method (maximum entropy), and one question of mine is still unanswered: Why is it that we wish to maximize the entropy?

When looking at a gas with average energy U, the higher the entropy, the "less" we know about the gas because the multiplicity is high. So how can this "high-entropy" distribution tell us anything about the gas? Wouldn't we want the entropy to be low, so we know which state the particles of the gas is in?

Thanks in advance.


Niles.

 

[marcusl]

You can never know anything about each individual molecule because there are simply too many to follow (about 10^24 in one mole) and their motions and dynamics are rapid and complex. (10^30 coordinates to track when you consider (3 coordinates of position and 3 of velocity). Hence statistical mechanics literally deals with the average behavior of the ensemble of microstates. The individuality making up a microstate is ignored.

Any low entropy state implies a high degree of order that isn't justified by the limited information available to you. Imagine the classical example: why are air molecules evenly distributed in the room you are sitting in instead of being slightly more concentrated in one side? The answer is that there are something like 10^100 more ways to arrange the molecules in the high entropy macrostate, compared to a macrostate that has 1 microjoule lower entropy. (I forget the exact number, but you can look it up or calculate it. It's astronomical.)

From pure combinatorics and probability, you'd be very foolish to assume that the system occupies anything but the maximum entropy state. Thus you always solve for the maxent macrostate.

 

[Niles]

Ahh, I see. A nice explanation; thanks!