So, in last weeks BMC post, I showed a number of probability distributions, one of which was the Fermi distribution. As you saw, at low temperatures, the Fermi distribution approached a step function. This itself was a bit weird, as it implied that there are particles in non-zero energy levels even when the temperature is zero. This distribution’s shape produces a number of bizarre properties, the great fun pop-science kind of things that makes physics popular among you, the unwashed masses (I kid, I love this stuff too).
Pauli wanna cracker?
So, the strangeness of the Fermi distribution arises from a strange property of fermions (most of the particles we know and love, like protons, neutrons, and electrons). This property is the Pauli Exclusion Principle. This is the property that gives the Fermi distribution its “boxiness”: It prevents fermions from occupying the same energy level in a state. This has important implications, as it affects the properties of matter at macroscopic scales: chemistry would break if electrons were bosons. This property is due to their half-integer spin, which makes two identical fermions in the same state have exactly opposite wavefunctions. This causes destructive interference, excluding the system from existence. But all that is just details, the key thing to remember is that you can’t put two identical fermions in the same quantum state. This exclusion results in a number of very strange properties.
Strange Fermion Properties
The piling up of Fermions into higher energy levels means that, even at absolute zero, some fermions in a
system have (relatively) high energies. The highest level (essentially, the level at which you run out of fermions) is called the Fermi Energy (this can also be expressed as a temperature using Boltzmann’s constant, even though we are technically at absolute zero). A fermi gas (a collection of unbounded fermions) acts extremely strangely because of this “extra” energy. For one thing, because some particles are forced into higher energy levels, their momentum and velocity are non-zero even at absolute zero. Put enough fermions together, and you can’t bring them all to a stop without separating them. This is why the common definition of temperature (the average kinetic energy of the particles) is incorrect, because even at the lowest possible temperature (zero kelvin), a fermi gas has non-zero average speed, momentum, and kinetic energy. Fermi gasses become uncompressible when all of the available energy states fill up. In otherwords, even at absolute zero, a fermi gas has some non-zero pressure (the Fermi Pressure). The electrons in white dwarves and the neutrons in neutron stars form a fermi gas, and the outward fermi pressure is what counteracts the crushing inward pressure of gravity (as opposed to the energy released in nuclear fusion). Just for an idea as to how intense the fermi pressure can be, the surface gravity on a neutron star tends to be ~100 billion times more intense than the surface gravity on Earth. The fermi pressure is just as intense when the star is in equilibrium (if it was weaker, the star would collapse further, and if it was stronger, the star would expand). At really high densities and really low temperatures, the fermi energy becomes an insurmountable obstacle. In fact, when it seems to be overcome, it is actually by sidestepping the problem. A neutron star is denser than a white dwarf, but both are sustained by the fermi pressure. How does a star go from the density of a white dwarf to that of a neutron star (a 10 million-fold increase) when the fermi pressure pushes out on the white dwarf? By reducing the number of particles, of course! The fermi pressure occurs because there are an equal number of particles and available states. Drop the number of particles, and you drop the pressure. This happens in the electron-degenerate to neutron-degenerate transition by two processes: beta capture (which turns a proton and electron into a neutron), and a transition from fermi gas to fermi liquid. This state transition alters the magnitude of the fermi energy by “bundling up” the particles, so that many fermions behave as if they are a single particle. Other phenomena can drop the fermi energy even further, collapsing a neutron star into an even denser object called a quark star. These almost-black-holes are composed of an exotic type of matter called quark-gluon plasma, where the protons and neutrons are broken down into their constituent components, fermions called quarks, and bosons called gluons. These very dead stars are still somewhat controversial, as there have only been a handful of candidate supernovae that produce the light curves expected from a so-called “quark nova“. If you are more interested in these fascinating objects, check out that some of the smart dudes at the University of Calgary are working on at the Quark Nova Project.