Small deviations from equilibrium and Lagrange multipliers

[Ted Ali]

Homework Statement

Consider the grand canonical ensemble, for a system A with N identical particles. This system can exchange particles with a reservoir A'. We consider only small deviations from equilibrium. What is the necessary and sufficient condition that the Lagrange multiplier $\alpha$ of N, must satisfy?

Relevant Equations

The Lagrange multiplier $\alpha$ is: $$\alpha =\left( \frac{\partial \ln \Omega}{\partial N'} \right)$$.

According to the book "Principles of Statistical Mechanics" by Amnon Katz, page 123, $\alpha$ must be such that $\exp ( -\alpha N )$ can be expanded in powers of $\alpha$ with only the first order term kept. Is this the necessary and sufficient condition for small deviations from equilibrium?

Thank you in advance,
Ted.

 

[Juan Carlos]

Hello Ted.

Looking through your expressions, I guess that you are using the same notation of the book ("Principles of Statistical Mechanics" by Amnon Katz).

Two points:

What is the expression of your statistical functions $f$'s?
I assume that it should be something like this for system $A$,

$$f=\exp(\Omega\ +\ \alpha N )\ ,$$
right? (please correct me if I am wrong).

May you elaborate on how you obtained the Lagrange multiplayer $\alpha$? If your derivation is correct, then you have the necessary condition.