Intuitive statistical mechanical explanation for thermodynamic 2nd law

1. Apr 1, 2012

vinovinovino

I don't have the intuitive picture of thermodynamic 2nd law, in terms of statistical mechanics. That is, why should the number of microstates be maximized in equilibrium? Anyone gives an intuitive explanation? Thanks a lot.

2. Apr 1, 2012

Rap

Its not that the number of microstates is maximized in equilibrium. You have a fixed number of microstates, and most of those microstates are equilibrium microstates. The system bounces around among those microstates, and since almost all of them are equilibrium microstates, the system is almost always in an equilibrium microstate.

The real question is, why are almost all microstates equilibrium microstates? You can get an intuitive feel for this by considering a deck of cards - why, after I shuffle a deck of cards, are the red suits and the black suits more or less evenly distributed throughout the deck? The more or less even distribution corresponds to equilibrium.

The number of ways you can arrange the cards in a deck (the microstate) is a fixed number. Almost all of them will have the red and black suits more or less evenly distributed (equilibrium macrostate). Only a small number will have a lot of black cards in the top half and a lot of red cards in the bottom half. (non-equilibrium macrostate).

3. Apr 1, 2012

nonequilibrium

Well that's one way to go about it, but it seems rather backward. IMO it would make more sense to define equilibrium as the macrostate with the most number of microstates corresponding to it. Then questions like "The real question is, why are almost all microstates equilibrium microstates" get tautological answers, and it becomes really evident to see why a system is most likely in its equilibrium state, as it follows directly from the definition of equilibrium and the plausible guess that every microstate is equally likely.

4. Apr 1, 2012

Rap

I don't like to look at it that way. As I see it, classical thermodynamics and the four laws are a concise statement of experimental results. Equilibrium is defined in classical thermodynamics. Statistical mechanics then comes along and explains classical thermodynamics. Stat. mech. says, look, the system can be in any of a huge number of microstates, and the overwhelming majority of those states correspond to the macrostate that classical thermodynamics calls equilibrium - the equilibrium macrostate. Questions like "why are almost all microstates equilibrium microstates" is then a statistics problem, and a good question. The fact that systems at equilibrium almost always stay in equilibrium suggests that each microstate is equally likely.

A finer point is the difference between the idea that 1) the number of microstates corresponding to the equilibrium macrostate is larger than the number of microstates for any other macrostate and 2) the number of microstates corresponding to the equilibrium macrostate is OVERWHELMINGLY larger than the number of microstates for any other macrostate. This is the difference between saying that the system will be in an equilibrium macrostate most often, and that it will be in an equilibrium macrostate almost always.

The real question is why is it OVERWHELMINGLY larger, not just larger. Why is the equilibrium microstate not simply the most common, but almost the only one? Thats a good statistical question, and answering it will address the OP, I think.

5. Apr 1, 2012

nonequilibrium

Well your first paragraph seems to be a matter of taste. I like to give precedence to the concepts of statistical mechanics, as they apply more broadly. Analogously, I don't like to see the Boltzmann entropy as an explanation for thermodynamic entropy, rather I like to see the latter as a specific application of the former.

As for your other paragraphs, I agree!

6. Apr 1, 2012

Rap

Well, you are probably right about it being a matter of taste, but I like to keep clear in my mind the difference between a phenomenological theory and an explanatory theory. I think classical thermodynamics and Maxwell's equations, for example, are phenomenological - they are a condensed description of experimental results which make very little use of untestable abstractions - they are expressed in terms of measurements. The explanatory theories, like statistical mechanics, or quantum mechanics make use of untestable or unmeasureable abstractions like the microstate, or the wave function. I don't like to confuse the two, and I take the phenomenological theories as the bottom line, while the explanatory theories are not to be believed in a religious sense.

I also take issue with the idea that classical thermodynamic entropy is a specific application of Boltzmann (or information) entropy. Classical thermodynamic entropy is defined in terms of concrete macroscopic measurements, making no reference to microstates or probabilities or statistics. It does not speak the language of statistical mechanics, while statistical mechanics DOES speak the language of classical thermodynamics. Statistical mechanics explains thermodynamic entropy, but thermodynamic entropy is not an application of any statistical concept.