Nature is made of two totally different kinds of fundamental particles, bosons and fermions. Fermions are conserved in number and obey the Pauli Exclusion Principle... no two identical fermions in the same region of space can be in a state with the same quantum numbers. Fermions all have intrinsic spin 1/2 ℏ. Complex systems of fermions which happen to have a total angular momentum of 1/2 ℏ also behave like fundamental point fermions by obeying the exclusion principle. This behavior is referred to as Fermi-Dirac statistics. On the other hand, pointlike fundamental particles with an intrinsic spin of zero or of one ℏ unit are called Bosons. Any number of bosons in the same region of space can occupy precisely the same state. Fundamental pointlike bosons are not conserved in number, but any complex system which happens to have a total angular momentum of zero can behave like a boson. This behavior is called Bose-Einstein statistics.

Satyendra Nath Bose (1894 – 1974 )

Albert Einstein (1879 – 1955)

Paul Dirac (1902 – 1984)

Enrico Fermi (1901 – 1954)

Statistical physics began with the great late 19th Century physicists Maxwell and Boltzmann asking, in an ideal gas consisting of non-interacting particles which have only a kinetic energy, what is the probability that a given particle has a given speed v? The key factor that appeared in their solution was exp(-E/kT) where E is the kinetic energy, and k is Boltzmann's constant, which is 1.38 × 10-23 Joules per Kelvin, or 8.62 × 10-5 eV/K. Remember that in physics, kT is equal to (2/3) of the average kinetic energy of a constituent particle in a system. The Kelvin is the ONLY unit of T for which this statement is correct!

In quantum physics, what we need is the probability that a given state in a quantum system is occupied by one or more particles. The Bose-Einstein distribution gives the answer for bosons, while the Fermi-Dirac distribution supplies the answer for fermions. Again, the absolute temperature T of the system is a key parameter.

The meaning of the Fermi energy E_F and of kT for a Fermi-Dirac distribution.

BOSE-EINSTEIN CONDENSATION!  Bose–Einstein condensates were first predicted in 1924–1925 by Albert Einstein, crediting a pioneering paper by Satyendra Nath Bose establishing a new field now known as quantum statistics. It was not until 1995 that the Bose–Einstein condensate was actually created in the lab by Eric Cornell and Carl Wieman of the University of Colorado, Boulder, using rubidium atoms; later that year, Wolfgang Ketterle of MIT produced a BEC using sodium atoms. In 2001 Cornell, Wieman, and Ketterle shared the Nobel Prize in Physics “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates.” [Composite particles, such as atoms, also will behave as bosons or fermions. Depending on the number of electrons, protons and neutrons, an atom can have integer or half-integer total spin and, therefore, be a boson or fermion.]

It is remarkable that when physicists came to investigate atomically flat systems, with effectively only two space dimensions, they discovered that in these systems the distinction between fermions and bosons breaks down, and the particles behave like “anyons,” because in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles' state function after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their state function, leaving us with only two possibilities: bosons, whose state function remains the same after a single exchange, and fermions, whose exchange only changes the overall sign of their state function.

Anyons are just one example of the discoveries being made in what is probably the hottest single topic of research currently ongoing,  in all of condensed matter physics--- namely, topological materials. We will have more to say about them later in the course.