## Chemical Potential MC

For a classical many-particle system the chemical potential $\mu$ can be divided into two parts, one equal to the chemical potential of an ideal gas, and the other associated with interactions of the particles, which is denoated as $\mu_{\rm excess}$. We can compute $\mu_{\rm excess}$ in a Monte Carlo simulation by using the Widom insertion method such that $$\mu_{\rm excess} = -kT\langle e^{-\beta \Delta U} \rangle,$$ where $\Delta U$ is the change in the potential energy of the system if an imaginary particle is placed at a random position in the system. The average is formed by doing this imaginary insertion many times throughout the simulation.

The program does a standard Metropolis simulation of particles interacting through the Lennard-Jones potential, and computes $\mu_{\rm excess}$.

**Problem: The excess chemical potential of a Lennard-Jones fluid**

Program `ChemicalPotentialMC` implements the Widom insertion method to estimate the chemical potential
for a system of particles interacting with the Lennard-Jones potential.
Determine the density dependence of the excess chemical potential of a Lennard-Jones fluid.
Are your results for $\mu_{\rm excess}$ consistent with qualitative arguments given in the text?

## Resource

Problem 7.6 in*Statistical and Thermal Physics: With Computer Applications*, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).