Exercise 13.1: Classical gas in 1D

1. simulate an ideal gas of $N$ particles in 1D. Choose $N=20$, $T=100$ and 200 MC steps. Give all the particles the same initial velocity $v_0=10$. Determine the value of the maximum velocity change $\Delta v$ so that the acceptance ratio is approximately $50\%$. What is the mean kinetic energy and mean velocity of the particles?

2. We might expect that the total energy of an ideal gas to remain constant since the particles do not interact with each other and hence they cannot exchange energy directly. What is the initial value of the energy of the system? Does it remain constant? If it does not, explain how the energy changes. Explain why the measured mean particle velocity is zero even though the initial particle velocities are not zero.

3. What is a simple criterion for ``thermal equilibrium''? Estimate the number of Monte Carlo steps per particle necessary for the system to reach thermal equilibrium. What choice of the initial velocities allows the system to reach thermal equilibrium at temperature $T$ as quickly as possible?

4. Compute the mean energy per particle for $T=10$, $100$ and $400$. In order to compute the averages after the system has reached thermal equilibrium, start measuring only after equilibrium has been achieved. Increase the number of Monte Carlo steps until the desired averages do not change appreciably. What is the approximate number of warmup steps for $N=10$ and $T=100$, and for $N=40$ and $T=100$? If the number of warmup steps is different in the two cases, explain the reason for this difference.

5. Compute the probability $P(E)dE$ for the system of $N$ particles to have a total energy between $E$ and $E+dE$. Do you expect $P(E)$ to be proportional to $e^{-\beta E}$? Plot $P(E)$ as a function of $E$ and describe the qualitative behavior of $P(E)$. Doe s the plot of $\ln{(P(E))}$ yield a straight line?

6. Compute the mean energy for $T=10$, $20$, $30$, $90$, $100$ and $110$ and estimate the heat capacity.

7. Compute the mean square energy fluctuations $\langle \Delta E^2 \rangle = \langle E^2 \rangle - \langle E \rangle ^2$ for $T=10$ and $T=40$. Compare the magnitude of the ratio $\langle \Delta E^2 \rangle/T^2$ with the heat capacity determined in the previous item.