Then we can find the equilibrium state of the system by looking for the minima of F=E-TS,
where \( S= k ln(\Omega)\). Does that make sense?
Here is a suggestion. I think what we really want is just a graph of \( kT ln\Omega\) as a function of \(N_{\uparrow} - N_{\downarrow}\) and then to maybe add that to the e-e energy term and graph that to see which one controls the equilibrium. (or really, how they both influence it...) So instead of getting all caught up in formal math, and Stirling's formula etc. (which is fine, but might take too much time) maybe someone could just do that numerically for a particular case like N=1000 or so, and for a particular value of \(e^2/a\), like say 1 or 2 eV. Then graph the Free Energy as a function of \(N_{\uparrow} - N_{\downarrow}\) for a few values of T. Does that make any sense?
By whatever method, either analytically or numerically, what we need to keep moving forward is an approximate expression for S that includes terms up to 4th order in \(N_{\uparrow} - N_{\downarrow}\). I think S has a maximum at \(N_{\uparrow} - N_{\downarrow}=0\) and one can do a Taylor series expansion near there including the \((N_{\uparrow} - N_{\downarrow})^2\) and \((N_{\uparrow} - N_{\downarrow})^4\) terms. One can also fit a numerically generated S in that way.
This might help you get started.
Plot [ln[100!/(n!(100-n)!)]] and [66.78-(n-50)^2/50] from 2 to 98
PS. Does this ring a bell?
ReplyDelete"...multiplicity function for a two state system, W(n,N), is the number of spin states such that n of the N spins point in the z-direction. This function is given by the combinatoric function C(N,n)..." wikipedia
PS. I think it is the 2nd law that leads to the principle that F is minimized for a system in thermal equilibrium (with a well-defined T).
ReplyDelete"In Thermodynamics the second law applies to processes in isolated systems, but most systems are not isolated. For a system in contact with a constant temperature environment that undergoes a process at constant volume, the second law tells us the system will evolve towards a state in which the potential U-TS reaches a minimum. U-TS is called F, the Helmholtz free energy."
(from a guy named Campov more or less)
So if you want to understand more about why we minimize F, you can read about how to connect this to the 2nd law somewhere.