Fermi-Dirac Statistics
On the quantization of the perfect monoatomic gas (1926)
The power of mathematical physics allows us to describe numerous states of matter: from the most mundane cup of water to the blinding streams of plasma on the sun. In the most extreme conditions, however, the behavior of matter must be modeled in a way that is uniquely quantum mechanical.
In 1926, Enrico Fermi published a paper that laid the foundation for what would later be known as Fermi-Dirac statistics. This work, entitled On Quantizing an Ideal Monatomic Gas, was a critical step in understanding the quantum behavior of particles that obey the Pauli exclusion principle such as electrons, neutrinos, and certain types of atomic gases.
Fermi’s study was motivated by the need to reconcile classical thermodynamics with quantum mechanics in describing an ideal monatomic gas at extremely low temperatures—monotonic gas meaning that it is a gas made up of individual atoms that do not bond with one another like helium or neon. This framework also becomes invaluable in an analysis of the high-density environments of many stellar cores.
Limits of the Ideal Gas Law
Before quantum mechanics, the behavior of gases was understood using classical thermodynamics and statistical mechanics. According to classical theory, an ideal gas obeys the equation of state:
where P is pressure, V is volume, N is the number of molecules, k is Boltzmann’s constant, and T is temperature.
However, this classical model fails at very low temperatures. Experimental observations suggested that the specific heat of gases, a measure of the amount of heat required to raise the temperature of a substance by one degree, deviated from classical predictions as temperatures approached absolute zero.
This problem becomes even more pressing with respect to Nernst’s heat theorem (also known as the third law of thermodynamics), which states that the entropy of a system should approach zero as the temperature approaches absolute zero. Classical physics, which predicted a constant heat capacity at extremely low temperatures, was clearly incomplete.
It thus became the guiding light of the scientific community to develop a statistical model based on quantum processes that could explain these deviations and provide a more complete description of ideal gases.