Gibbsian Ensemble: Kerson & Huang Explained

[emob2p]

Hi,
I'm taking a course in Stat Mach using Kerson and Huang's Statistical Mechanics book. I am quite confused with their treatment of a Gibbsian Ensemble. They say imagine an infinite copies of the same system whose state can be represented by a point in phase space. Then $$\rho (p,q,t) = d^{3N}p d^{3N}q$$ is the number of representative points contained in the infinitesimal volume. So if we integrate this over allowed p's and q's, we should get infinity because we started out w/ an infinite number of total systems. Can this be correct?

 

[actionintegral]

Not if the volume of integration is finite or infinite but integrable.

 

[charng]

Hi there, thank you for sharing your thoughts and questions about the Gibbsian Ensemble and Kerson & Huang's explanation. I can understand your confusion, as the concept of an infinite number of copies of a system can be difficult to grasp. However, rest assured that their treatment of the Gibbsian Ensemble is indeed correct.

To better understand the concept, let's first define what a Gibbsian Ensemble is. It is a collection of systems that are all in thermal equilibrium with each other, but each system can have different values for its macroscopic variables (such as energy, temperature, etc.). This ensemble allows us to study the behavior of a system as a whole, rather than focusing on individual systems.

Now, let's address your concern about the infinite number of systems. It is true that we start with an infinite number of systems, but we are not considering the entire ensemble as a whole. Instead, we are looking at a small, infinitesimal volume in phase space. This volume represents the possible states that a single system can have. So, when we integrate over all possible values of momentum and position, we are essentially summing up the number of systems that can have those specific values. This number will be finite, as we are only considering a small portion of the entire ensemble.

I hope this explanation helps clarify your doubts about the Gibbsian Ensemble. Keep in mind that this concept can be a bit abstract, but with practice and further study, it will become more intuitive. Best of luck in your studies!