[NOTES/SM-03002] Application of Micro canonical Ensemble to Two Level System

Consider an isolated system of \(N\) particles each of which can exist in two levels \(E_0, E_1\). We take \(E_1=0, E_2=\epsilon\). The microstates of this system are specified by giving occupation number \(n\) of level \(E_2\), for then the occupation number of state \(E_1\) is \(N-n\) of the two levels. If the total number of particles is \(N\) then \(N=n_1+n_2\). The total energy of the system is \(E=n_1E_2+n_2E_2= n\epsilon\).

The number of microstate of a system for a given \(n\), hence also for a fixed energy \(E=n\epsilon\), is equal to the number of ways \(n\) particles can be selected from a \(N\) particles. Thus we have \begin{equation} \Omega(n) = \frac{N!}{n!(N-n)!}. \end{equation} Therefore the entropy is given by \begin{equation} S =k \ln \Omega(n) = k(\ln N! -\ln n! -\ln (N-n)!. \end{equation} Next we use the Stirling approximation \begin{equation} \ln N! = n\ln n - n, \end{equation} which is valid for \(n>>1\). This gives \begin{equation} S(n) = k\big[N\ln N - N - (n \ln (n- n) \big)- \big((N-n)\ln (N-n) - (N-n)\big) \big] \end{equation} We are interested in finding the number of particles in the second level at temperature. The temperature is brought in by recalling the result \begin{equation} \frac{1}{T} = \pp[S]{E}. \end{equation} For the present case, the energy dependence is hiding in\(n\), \(E=n\epsilon\). Therefore, we get \begin{eqnarray} \frac{1}{T} &=& \pp[S(n)]{E} = \frac{1}{\epsilon}\pp[S(n)]{n}\\ &=& - \ln n + \ln (N-n). \end{eqnarray} Thus we get \begin{equation} \frac{1}{T} = \frac{k}{\epsilon}\ln \frac{N-n}{n}. \end{equation} Taking exponential and solving for \(n/N\) we arrive at \begin{equation} \frac{n}{N} = \frac{1}{e^{\epsilon/kT} +1}. \end{equation} In the two limits \(\epsilon << kT\) and \(\epsilon>> kT\), the number of particles in upper level approaches \begin{equation} n = \begin{cases} \frac{1}{2}, & \quad \text{if } \epsilon << kT,\\ N e^{-\epsilon/kT},& \quad \text{if } \epsilon >> kT. \end{cases} \end{equation}