Exercise: One-dimensional Ising model

1. Choose $N=20$, $T=1.0$, and all the spins initially pointing ``up'' as the initial state. Calculate the energy after each step, and estimate the number of steps for the system to reach equilibrium.

2. Pick all the spins initially poiting randomly. Estimate the time that takes for the system to reach equilibrium.

3. Choose $N=20$ and equilibrate the system for 100 MC steps. Use at least 200 MC steps to determine the mean energy $\langle E \rangle$ and magnetization $\langle M \rangle$ as a function of $T$ in the range $T=0.5$ to $5.0$. Plot $\langle E \rangle$ as a function of $T$ ans discuss its qualitative features. Compare your computed results for the mean energy to the exact values:
   
   $$\langle E \rangle = -N \tanh{\left(\frac{J}{k_BT}\right)}.$$
   
   
   What are yout results for $\langle M \rangle$? Do they depend on the initial configuration?

4. Is the acceptance ratio and increasing or decreasing function of $T$? Does the Metropolis algorithm become more or less efficient at low temperatures?