Paramagnetic system: computing number of microstates

[mondeo2015]

Homework Statement

We are given a paramagnetic system of N distinguishable particles with 1/2 spin where we use N variables
s_k each binary with possible values of ±1 where the total energy of the system is known as:
$$\epsilon(s) = -\mu H \sum_{k=1}^{N} s_k$$ where $$\mu$$ is the magnetic moment of the spin and H is the applied magnetic field. We are requested to assume that N is greatly larger than 1 and we are reminded of the definition of $$arctanh(x) = \frac{1}{2} ln(\frac{1+x}{1-x})$$. We have the following three parts:
1. We are asked to find the number of microstates for a given energy E $$\Omega(E)$$ with as many justifiable simplifications as possible.
2. We are asked find $$\beta = \frac{1}{kT}$$ for a given E and we are asked to use this information to find single-spin magnetization of the system defined as $$m= \frac{1}{N}\sum_{k=1}^{N} s_k$$ in terms of $$\beta$$ and H the applied magnetic field. We are advised that m can be treated as a continuous variable where needed.
3. We are told that two systems $$(N_1,H),(N_2,H)$$ are Temperaturs $$\beta_1 , \beta_2$$ and initial magnetizations $$m_1,m_2$$ respectively, are brought into contact where only heat can be exchanged and we are asked to use the preceding parts to show that the finial magnetization at equilibrium is given by $$m = \frac{N_1m_1 + N_2m_2}{N_1+N_2}$$, noticing the magnetic field is the same for both systems.

Homework Equations

The Stirling approximation $$N! \approx Nln(N)-N$$ as seen in Wikipedia
https://en.wikipedia.org/wiki/Stirling's_approximation

The Attempt at a Solution

I was thinking for part A the idea is to use the fact that it is the total number of N distinct solutions to an integer equation where the sum of the s' must be some integer m so this is basic combinatorics where it is finding the number of +1 valued s' (denoted l) which determines the sum which is l-(N-l) = 2l-N this is equal to m therefore l must satisfy l = (N+m)/2 so the total number of possibilities is $$N \choose \frac{N+m}{2}$$ then to get rid of factorials we probably should use the assumption that N is much greater than 1 so a form of Stirling's approximation must be valid and is to be used to simplify it. For part b. I must admit I am stumped as to how to relate this to temperature and for part c I have no clue all I think must happen is thermal equilibrium but how do we reach this nice expression given? This is where I am stuck and need help, I thank all helpers.

 

[Fred Wright]

To find the number of microstates of N distinguishable objects with spin up or spin down you need to divide the N spins into two smaller groups of n_u and n_d. There are N distinguishable states so N! possible arrangements, but there are n_u indistinguishable states so there are n_u! arrangements of these. By simple combinatory analysis the total number of microstates is,
Ω = N!/ (n_u!(N- n_u)!).
The total energy is E = n_u(-μH) + n_d(μH) and from this,
n_u = (1/2)(N-E/(μH)) = (N/2)(1-E/(NμH))
n_d = N - (1/2)(N-E/(μH)) = (N/2)(1+E/(NμH))
In order to find β and magnetization I suggest you use the basic thermodynamical relations:
(∂S/∂E)_H,N = 1/T and (∂S/∂H)_E,N = m/T where the entropy is S = k_Bln(Ω)