Grand Canonical Partition function

[Nikitin]

Hi.

1) Does the Grand canonical partition function $\Xi$ behave in a similar fashion to the good old canonical partition function $Z$? Do you calculate thermodynamical quantities (entropy, hemholtz free energy etc.) in similar fashions?

2) Is it possible for some kind soul here to explain the basic characteristics of the partition functions for:

Microcanonical Ensemble
Canonical Ensemble
Grand Canonical Ensemble

?

3) Is it possible to extract ALL thermodynamical information about a system if you know its partition function? You just need to use the formulas, right?

4) What's up with each state in the microcanonical ensemble having equal probability to materialize? I don't get it..

thanks.

 

[Mmm_Pasta]

1) Yes, they are essentially the same. Calculations are done in the same fashion. The grand canonical ensemble allows for the particle number of a system to change as well as energy. The canonical ensemble keeps particle number constant. In the microcanonical ensemble both particle number and energy are fixed. One can show that fluctuations are on the order of 1/N, so these values change very little. This implies that these ensembles are equivalent, however calculations can be easier in one ensemble than another.

2) See above

3 ) Yep - one may have to use a computer at times since things can get nasty. =)

4) Is there any reason why one microstate will have a higher probability of occurring? If one doesn't have any previous information about a system one show that the most likely distribution is the one where every event has an equal probability to occur. This can be done by maximizing Shannon's entropy.

 

[Nikitin]

One can show that fluctuations are on the order of 1/N, so these values change very little. This implies that these ensembles are equivalent, however calculations can be easier in one ensemble than another.

How can they be equivalent? You can't use the microcanonical ensemble on a system where the particles have different energy, or the canonical ensemble on a system with varying particle number. Do you perhaps mean that you can use the grand canonical ensemble on a system instead of the canonical and microcanonical, and you can replace the microcanonical with the canonical?

And what do you mean with "fluctuations are on the order of 1/N"?

 

[WannabeNewton]

How can they be equivalent?

They are only equivalent in the thermodynamic limit, which is when the number of degrees of freedom goes to infinity. If not in this limit then the ensembles are certainly not equivalent. The "fluctuations" referred to above amounts to the variance of the random variable of interest in a given ensemble (e.g. energy in the canonical ensemble) and it can be easily shown that the relative fluctuations go as $1/\sqrt{N}$ due to Gaussianity of the distributions for these ensembles.

 

[Nikitin]

Ok, but that is ONLY when the system is in thermodynamic equilibrium. Right?

 

[WannabeNewton]

Ok, but that is ONLY when the system is in thermodynamic equilibrium. Right?

Yes.

 

[BruceW]

Ok, but that is ONLY when the system is in thermodynamic equilibrium. Right?

any of the 3 partition functions can only be used when the system is in thermodynamic equilibrium. Also, the canonical and microcanonical ensembles become similar in the limit of very many degrees of freedom. "many degrees of freedom" is a different concept than "thermodynamic equilibrium". (I'm just trying to clarify, since these concepts maybe sound the same, but are actually different things).

 

[Nikitin]

any of the 3 partition functions can only be used when the system is in thermodynamic equilibrium. Also, the canonical and microcanonical ensembles become similar in the limit of very many degrees of freedom. "many degrees of freedom" is a different concept than "thermodynamic equilibrium". (I'm just trying to clarify, since these concepts maybe sound the same, but are actually different things).

OK so basically you just assume the energy and particle numbers are constant for the entire system at thermal equilibrium. OK. Got it.

With "very many degrees of freedom" you mean that there are extremely many micro-states a system can assume? Or that each micro-state is made up of allot of particles, and hence the standard deviation of the thermodynamical variables fall off by $1/\sqrt{N}$? I'm a bit confused on the meanings..

 

[Nikitin]

BTW, does anybody know where I can find a list of formulas for extracting thermodynamical variables from the 3 partition functions?

 

[Nikitin]

bump

 

[Mmm_Pasta]

BTW, does anybody know where I can find a list of formulas for extracting thermodynamical variables from the 3 partition functions?

There is something on Wikipedia, but not quite what you want: http://en.wikipedia.org/wiki/Table_of_thermodynamic_equations

Are you familiar with partial and total derivatives? That would be more useful to get what you want as opposed to a table.

 

[BruceW]

OK so basically you just assume the energy and particle numbers are constant for the entire system at thermal equilibrium. OK. Got it.

With "very many degrees of freedom" you mean that there are extremely many micro-states a system can assume? Or that each micro-state is made up of allot of particles, and hence the standard deviation of the thermodynamical variables fall off by $1/\sqrt{N}$? I'm a bit confused on the meanings..

ah, I forgot to reply. uh, yeah, It has been a while since I did one of these derivations. I'm fairly sure that if you have number of particles much much greater than 1, and if you have number of microstates much greater than number of particles, then you will get a limit where the microcanonical ensemble acts very similar to the canonical ensemble. But if you have a phase transition, I think this approximation may not work. possibly because at phase transition, all the particles can become strongly correlated, thus the system has low degrees of freedom... I should really learn more about this too. I typically just use the canonical ensemble.

 

[Nikitin]

thanks bruce!

There is something on Wikipedia, but not quite what you want: http://en.wikipedia.org/wiki/Table_of_thermodynamic_equations

Are you familiar with partial and total derivatives? That would be more useful to get what you want as opposed to a table.

Yes of course. I know how to extract most values, but stuff like extracting variance of something, finding entropy and so on often require a bit of thinking and time. I can't afford to waste time on deriving everything during the exam, hence why I need to write down the formulas for the most important stuff (i can bring one handwritten note with me when taking the exam).

Edit: as I have been doing more and more problems, it feels like I have gained a decent understanding of the partition functions. I don't think I need any more help. thanks