Thermal equilibrium in statistical physics

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Thermal equilibrium in statistical physics is defined as the state with the maximum number of possible realizations, as explained in Nolting's textbook. In systems with two subsystems that can exchange energy, equilibrium occurs when the product of their phase space volumes is maximized, leading to equal temperatures. The discussion highlights confusion about the reasoning behind maximizing phase space volume and its connection to statistical entropy. There is a concern about whether one needs to assume that maximizing entropy indicates equilibrium or if this can be derived independently. The conversation reflects a desire to clarify these concepts while acknowledging the intuitive nature of the relationships involved.
jonas_nilsson
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I am mixed up about thermal equilibrium in statistical physics. And I hope you excuse me if I use unconventional words, I am from Sweden, my book is in german and I try to express myself in english.

In my book (Noltings "Grundkurs theoretische Physik, Band 6") thermal equilibrium is defined as the state which is characterized by the maximum number of possible realizations. Further, he says that in a system concisting of two subsystems (that can exchange energy) this is exactly when omega_1 * omega_2 has a maximum, where the omegas are the phase space volumes. He goes on to show that this means that the temperature in the two systems is equal.

So far so good, and the main argument, that the phase space volume should be maximized, seems plausible, but I can't fully understand it. Could someone give me a hint? Or is this reasoning OK while it leads to results we know are right from experience? I find the book really nice, but this arguing isn't really developed by the author, or is it just way to obvious?
 
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It's highly intuitive if one takes into account Boltzmann's formula.Then u realize that the quantity which is really maximized is the statistical entropy.

Daniel.
 
Hi, and thanks for the quick reply!

Well I agree, you can see it in that formula, but it's sort of a "hen/chicken and the egg"-problem for me; what comes first? If you with Boltzmann's formula mean the distribution for the canonic ensemble (or the reduced distribution of Maxwell-Boltzmann?) I can see it in the logical steps. But doesn't your arguing need us to allready know that maximization of entropy means finding equilibrium? I am trying to sort of forget that right now, and then derive the results that are known from thermodynamics. Or is that wrong/impossible? Should we already assume that we from the thermodynamics know that the entropy is maximized?

I'm going to have a look at it all again tomorrow, right now it's late and I'm confused, but thanks anyway! I'll probably see some light tomorrow, along with new problems, and I'll write again =)

/Jonas
 

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