One-Dimensional Classical Ideal Gas

Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state.Ludwig Boltzmann

For an ideal gas, the energy of a configuration is independent of the positions of the particles, and the total energy of the system is the sum of the kinetic energies of the individual particles. Since kinetic energy is proportional to the square of the velocity, for an ideal gas the only coordinates of interest are the velocity coordinates. To change a configuration, we simply choose a particle at random and change its velocity by a small but random amount.

This program implements a microcanonical Monte Carlo simulation of a classical ideal gas in one dimension. The variable MCS, the number of Monte Carlo steps per particle, plays an important role in Monte Carlo simulations. The output of K_avg and v_avg—the mean kinetic energy and mean velocity of the gas, respectively—is in arbitrary units.

Monte Carlo Simulation of an Ideal Gas

This program investigates some of the equilibrium properties of a one-dimensional classical ideal gas.

Notes

gfortran -g -std=f2008 -Wall -Wextra -O2 -fall-intrinsics -o boltzmann boltzmann.f08

The method employed here is straightforward. First, select a particle at random: i_count = INT(FLOAT(N) * random) + 1. Then, change the velocity of the particle by a small but random amount: v_new(i_count) = v_old(i_count) + (1.0 - 2.0 * random) * delta. Now, find the change in kinetic energy of the particle, dE = 0.5 * v_new(i_count)**2 - 0.5 * v_old(i_count)**2, taking the mass of each particle as unity for simplicity. An intermediate state of the particle, referred to as a microstate, is accepted if dE ≤ 0.0, as all systems naturally seek a global minimum-energy state. If this is not the case, then particle i_count could be in a microstate dictated by the temperature T of the thermal reservior the ideal gas is in contact with. For a canonical distribution, the probability that particle i_count is in a microstate as result of an energy exchange with the surrounding environment is given by boltz = EXP(-dE * beta), where beta = 1.0 / T (Boltzmann's constant $k$ is also taken as unity). Based on this interaction, an acceptable microstate would be determined by boltz ≥ random. This process is repeated over an MCS number of steps for each particle, keeping a running average of some equilibrium properties as the algorithm proceeds.