The Ground State Energy of a Two-Dimensional Bose Gas

Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model since it pictured the atom as a miniature “solar system” with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton as...

Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.

The internal energy of a thermodynamic system is the sum of the kinetic and potential energies of all the molecules or entities in the system. The kinetic energy of an individual molecule includes contributions due to its rotation and vibration, as well as its translational energy. The potential energy is associated only with the interactions between one molecule and the other molecules of the system. Neither the system's location nor its motion is of any consequence as far as the internal...

A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.

The word "gas" comes from the Flemish word meaning "chaos," first used to describe vapors by the chemist J. B. van Helmont. Consider a container filled with gas, with a continuous and random motion of molecules. During collisions, the velocity component parallel to the wall is unchanged, and the component perpendicular to the wall reverses direction but does not change in magnitude. If the molecule’s velocity changes in the x-direction, then its momentum is changed.

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...