is anyone of you learning Statistical Mechanics ? come and let's learn together!!!


by Twukwuw
Tags: learn, learning, mechanics, statistical
Twukwuw
Twukwuw is offline
#1
Feb3-07, 07:11 AM
P: 53
haha, I am currently taking statistical mechanics in my undergraduate 2nd year. This subject is so interesting! I would like to invite you all to discuss and debate about the our understanding of subject!!!

Well, I would start the rolling ball. Below is my understanding of the basic ideas underlying statistical mechanics. "Attacks" and Critics to my understanding is well well welcome!!!

For a system (or assembly) of weakly-interacting particles, the detail of interaction (be it collision, Coulomb force, gravitational force and etc.) is not a concern of statistical mechanics as long as the particles can exchange energy with each other yet do not alter the energy levels of each other’s. In fact, statistical mechanics is not able to tell the details of the interaction or what the interaction is. In this sense the complexity of atomic dynamics is veiled.

In Statistical mechanics’ picture, the existence of thermodynamic equilibrium, at first, is just a statistical consequence of

The system is composed of a very large number of particles

This is because; the large number of particles will create an overwhelming number of microstates that give the same macrostate M. Statistical mechanics identifies this macrostate M as “equilibrium-macrostate”. In other words, at thermodynamic equilibrium the system is always in one of these microstates or, in macrostate M. Now, let’s re-examine the arguments and see what the underlying principles are.

1) There is an overwhelming number of microstate giving the same macrostate M
Implication: For the overwhelming number of microstate to exist, the must be a large number of particles. This implies the number of particles has to be conserved at thermo-dynamic equilibrium. This is in turn related to the assumption that the number of charges is conserved. This assumption reveals gauge symmetry.

2) This macrostate M is the equilibrium-macrostate
Implication: For the overwhelming microstates to give overwhelming probability (or most of the time the assembly is in that region), we are assuming “all the accessible microstate are equally probable”. This reveals the symmetry in time’s direction (reversibility in time) as pointed out by Callen (pg 468).

3) The accessible microstates (accessible region in -space) doesn’t change with time
Implication: If the accessible microstates change with time, the probability will keep re-distributing in new sets of macrostates as time goes on. This makes the probability distribution over energy/other parameters changes with time, and so are the statistical results. Here we are assuming conservation of energy, momentum and angular momentum. This reveals the symmetry of physical laws in space-time translation and rotation.

4) At thermodynamic equilibrium/equilibrium-macrostate there will be some steady thermodynamic variables about the system
Implication: Some thermodynamic variable show steady values at equilibrium as a result of the steady equilibrium-macrostate. The steady value in this case is the mean value given by the equilibrium-macrostate. However, some thermodynamic variables show steady values simply because they are the ones survive from the temporal averaging process during measurement. This kind of thermodynamic variable has zero changing-rate, hence revealing broken symmetry. An example is the volume of a solid.
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baryon
baryon is offline
#2
Feb14-07, 12:49 AM
P: 35
Can you summarize that?
StatMechGuy
StatMechGuy is offline
#3
Feb14-07, 12:14 PM
P: 223
This sounds like as good a thread as any to post in...

So, a few things for your consideration:

(1) I'm not sure how gauge symmetries with conserved charges really becomes a player in statistical mechanics. I have not seen it in normal equilibrium statistical mechanics, at least. In what context have you seen this, I'm curious?

(2) If thermodynamics obeys time reversal symmetry, then why does there seem to be an arrow of time due to entropy? The concept of a reversible process is a useful approximation, but there are also irreversible processes. Callen discusses them in his book (since you mentioned it).

(3) There are a "large" number of microstates, but what about statistical fluctuations? Fluctuations do exist, and in some physical systems they are the driving factor (I'm thinking of phase transitions here). How can you suppress discussion of fluctuations in the context of equilibrium statistical mechanics?

(4) On a related note, how good is the average of a statistical ensemble when we talk about thermodynamics. Okay, volume is a pretty good average, but what about pressure? How do we know that the averages we take in statistical mechanics are a good indication of how the system actually behaves?

I leave you to work these out, since you seem to love the subject so much.

baryon
baryon is offline
#4
Feb14-07, 07:21 PM
P: 35

is anyone of you learning Statistical Mechanics ? come and let's learn together!!!


I think you mean symmetry, Twukwuw, not gauge symmetry.


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