Statistical Physics - Equilibrium

[SchroedingersLion]

Good evening,

I have a question to a short introduction to statistical mechanics in a book about molecular dynamics simulation.
It introduces the fundamental assumption: Every microscopic state with a fix total energy E is equally probable.

I attached the section. I understand it all, except for the very last sentence.

Why does the author claim that, if we start at some random energy for E₁, we will have energy exchange until eq. (2.1.3) holds? As he explained it, "equilibrium" is only the energy state with the largest number of corresponding micro configurations - and, since all of them are equally likely according to the assumption from the beginning, the equilibrium is the most probable energy state. But this does not explain why any other state would change towards equilibrium through energy exchange on its own.

The authors did say that their way of introducing statistical physics is somewhat dirty. Is there missing a point?

Regards,
SL

 

[DrClaude]

Why does the author claim that, if we start at some random energy for E₁, we will have energy exchange until eq. (2.1.3) holds? As he explained it, "equilibrium" is only the energy state with the largest number of corresponding micro configurations - and, since all of them are equally likely according to the assumption from the beginning, the equilibrium is the most probable energy state. But this does not explain why any other state would change towards equilibrium through energy exchange on its own.

There are a few unspecified assumptions in the text. The one that is important here is that the systems you are considering are big enough that fluctuations in energy will be too small to be measurable once equilibrium is reached, where the probability of energy going to from system 1 to system 2 will be equal to the reverse probability.

The two systems are always exchanging energy, but the number of microstates corresponding to the equilibrium condition is so huge that the flow energy from one system to the other is not observable anymore. The function Ω is so narrowly peaked that its width is not measurable.

 

[SchroedingersLion]

The two systems are always exchanging energy, but the number of microstates corresponding to the equilibrium condition is so huge that the flow energy from one system to the other is not observable anymore. The function Ω is so narrowly peaked that its width is not measurable.

Ok, to put it in my own words:
The systems are always exchanging energy. If we are out of equilibrium, the probability for energy going from one system to the other is not the same as the reverse probability, that's why we will approach the equilibrium energy distribution sooner or later.

But from the introduced terms in the extract, this is not really obvious, because there is not a single term that describes energy flow from system 1 to system 2.
From my point of view: Just because equilibrium has the greatest probability does not mean that any other state will reach equilibrium.
Greatest probability to me would just imply: If I prepared an ensemble of these systems in random micro configurations, out of all energy distributions, the equilibrium distribution is the most probable.
Only if I introduce the additive assumptions that the probability to transfer energy from one system to the other is only equal to the reverse probability in equilibrium, I find that the system will always equilibrium.

 

[vanhees71]

Well, on average every state will end up in thermal equilibrium. To understand this really, in my opinion you have to study non-equilibrium statistical physics, particularly the Boltzmann equation. The best introduction I know about this topic is Landau&Lifshitz vol. X (btw. also vol. V for equilibrium statistical physics is excellent).

 

[DrClaude]

From my point of view: Just because equilibrium has the greatest probability does not mean that any other state will reach equilibrium.

Indeed, there are some other considerations here. That's another unwritten assumption: that the systems under consideration can reach equilibrium in a reasonable time. Some systems are metastable, but then again it is not the point of equilibrium thermodynamics to study such systems. For instance, the Sun is not in equilibrium with the universe, but there are some other ways to model it thermodynamically.

Greatest probability to me would just imply: If I prepared an ensemble of these systems in random micro configurations, out of all energy distributions, the equilibrium distribution is the most probable.

I should probably add another unwritten assumption: ergodicity. In such cases, it is assumed that ensemble averages, like you are describing, are the same as time averages. That means that the system will explore enough microstates in a relative short amount of time such that it will explore macrostates of increasing probability as time goes by (again, this would not be the case for a metastable system).

 

[SchroedingersLion]

Thank you guys!