Chemical Potential Demon
Previously we used the demon algorithm as an ideal thermometer, where the demon was an extra degree of freedom that could exchange energy with the system. A plot of the log of the demon energy distribution $\ln P(E_d)$ versus the demon energy $E_d$ is a straight line with a slope equal to $1/kT$. We now also allow the demon to exchange particles with the system, so that a plot of the log of the demon particle number distribution $\ln P(N_d)$ versus the demon particle number $N_d$ for a particular value of $E_d$ is a straight line with a slope equal to $\mu/kT$.
The program simulates a simple model in one dimension where the particles have discrete values of momentum and position. A single particle state is defined by its location in a twodimensional momentumposition phase space. To represent a dilute classical system only one particle per phase space point is allowed. Attractive interactions between neighboring particles can be added and a hard core repulsion can be added to prevent more than one particle at the same position.
The goal of this simulation is to explore how $\mu$ depends on particle density and the effect of the interactions.
Problem: The chemical potential of a onedimensional ideal gasProgram ChemicalPotentialDemon implements this algorithm for the onedimensional ideal gas and computes $P(E_d,N_d)$, the probability that the demon has energy $E_d$ and $N_d$ particles.
 Use the default parameters $N=100$, $E=200$, and $L=100$, where $N$ is the total number of particles, $L$ is the length of the box, and $E$ is the maximum total energy. The program uses units such that $\Delta x \Delta p=1$, and $m=1/2$ so that the kinetic energy of a particle is $\epsilon=p^2$. These choices imply that the momentum and energy are integers. The $N$ particles are initially randomly placed on the sites of the phase space lattice with no more than one particle on a site such that the total energy does not exceed $E$. The initial demon energy $E_d=0$ and particle number $N_d=0$. Run the simulation and discuss the qualitative features of the positions of the particles in phase space. Do some particles have the same position? Run the simulation for different values of $E$, but the same values of the other parameters. Where in phase space are most of the particles located?
 Explain why the simulation in part (a) is identical to a simulation of an ideal gas in the semiclassical limit. Show that the chemical potential is given by \begin{equation} \mu = T \ln[(L/N)(\pi T)^{1/2}]. \label{eq:usingthermo/mudemon} \end{equation} (Remember that $\Delta x \Delta p=1$.)
 Use the same parameters as in part (a) and run for about 200 Monte Carlo steps per particle (mcs) to equilibrate the system. Then click Zero Averages and average over at least $1000\,$mcs. (The simulation will run faster if you increase the steps per display to 10 or 100.) After the simulation is stopped, click the Plot Demon Distributions button to display the demon energy and particle number distributions. The plot of $\ln P(E_d, N_d = 1)$ versus $E_d$ should be approximately linear for small $E_d$. Use the plots to estimate the values of $T$ and $\mu$ and compare your Monte Carlo results with the exact result that you found in part (b).
 How do you expect $\mu$ to change if you lower the particle density? Run the simulation for $N= 50$, $E = 100$, and $L= 100$ and compare the temperature and chemical potential obtained from the demon distributions with the results found in part (c).
What are the effects of including interactions between the particles in a dilute classical gas?
 If the interaction has a hard core, no two particles can have the same position. How do you expect $\mu$ to change compared to the ideal gas? Use Program ChemicalPotentialDemon and determine the effect of including a hard core on $\mu$. For this comparison set $L = 200$ and $N = 100$. Why must $L$ be much larger than $N$?
 If we include an attractive interaction between particles that are nearest neighbors in position, the energy of the system will be lowered for some particle additions, thus giving the demon more energy. How do you expect $\mu$ to change compared to the hard core system?
Resources

Problems 7.41 and 7.42 in Statistical and Thermal Physics: With Computer Applications, 2nd ed.,
Harvey Gould and Jan Tobochnik, Princeton University Press (2021).