Exercise 12.1: MC simulation of the 1D ideal gas

1. Write a program to simulate a 1D ideal gas using the demon algorithm.

2. Use the initial condition that all particles have the same velocity $v_0$. The mass of the particles is set to unity. Choose $N=20$, total energy $E=10$, a maximum velocity change $\Delta v = 2$, and 50 steps for the simulation. Increase the number of MC steps per particle until the desired averages are constant within a desired range.

3. What is the initial mean velocity per particle? What is the equilibrium mean velocity per particle?

4. Compute the mean energy of the demon and the mean system energy per particle. Do your results in parts 3 and 4 depend on whether the particles are chosen randomly or sequentially?

5. Choose $E=20$ and find the value of the maximum velocity change $\Delta v$ that yields an acceptance rate of $50\%$. Compute the mean demon energy and mean system energy per particle after equilibrium has been established. Then consider $E=40$ and obtain an approximate relation between the mean demon energy and the mean system energy per particle.

6. The MC simulation in the microcanonical ensemble is done at a fixed total energy with no reference to temperature. Determine the temperature by the relation
   
   $$\frac{1}{2}m\langle v^2 \rangle = \frac{1}{2} k_B T.$$
   
   
   Set the Boltzamann's constant to unity. Use this relation to obtain $T$. Is $T$ related to the mean demon energy?

7. What is the meaning of the ``acceptance ratio''? What is the empirical relation between the acceptance ratio and $\Delta v/v_0$?