[MUSIC] Let's consider a system of non-interacting, indistinguishable particles that can have energies epsilon alpha, alpha going from one, two, etc., associated with their quantum mechanical states. The state of the system can be specified by the number of particles at each energy level that is n alpha, is the number of particles at the energy state epsilon alpha. From this, we get the total number of particles, n, is simply a summation over n alpha. And the total system energy is a summation over n alpha times epsilon alpha. In the canonical ensemble, we can write the partitioned function as. Note that in this expression we've included the Dirac delta function for the particle and the energy constraint. The indistinguishability of the particles is properly accounted for in this representation since any given set of n alpha contributes a single term without over-counting the indistinguishable states. The grand canonical ensemble is the case where we have an open system connected to a thermostat. The governing variables in this case turns out to be the chemical potential mu, the temperature t and the volume v. Now the partition function for the grand canonical ensemble simply comprises over the canonical partition function weighted by a term that goes as the exponential of beta mu N. Note that in this notation, beta is given as 1 over the thermal energy given by the Boltzmann constant times the temperature. The relevant thermodynamic potential for the open system is known as the Landow potential. The Landow potential simply turns out to be given as minus PV. The Landow potential can be related to the logarithm of the grand canonical partition function using a familiar relation. Now here comes a really important distinction. Particles can either be what is known as bosons or what is known as fermions. Now, what are bosons? Well, if the particles are bosons, then there is no restriction on the number of particles that can be put in any given state in alpha. This leads to the result that the term inside the logarithm simply turns out to be. But what happens if the particles are fermions? If the particles are fermions, then any given state in alpha can only take value zero or one, that is occupied or unoccupied. This leads to the evaluation of the term inside the logarithm as, in a generalized notation, we can write down the expression for both the bosons and the fermions. Now in this equation, the minus sign corresponds to the bosons, and the positive sign corresponds to fermions. We will now proceed to discuss a variety of elementary fundamental particles that are important which are either bosons or fermions. We will begin our discussion with electrons, then move on to phonons and then photons. Now let's begin the discussion for electrons, let's say electrons present in a metal. How can we think about the electrons present in a metal? The electrons present in a metal are at high densities, that is there are many atoms per volume, each of them contributing conducting electrons. No two electrons can exist in the same state. This is the famous Pauli's exclusion principle. As a result, the high density system fills up many single particle energy levels. The lowest unoccupied state will still have a kinetic energy much larger than the thermal energy. And so, thermal excitations result in energetics with large kinetic energy and comparatively negligible potential energy of interaction. The large kinetic energy associated with these electrons results in the large conductivity of electrons present inside a metal. As a result we can understand the behavior of electrons in a metal using the results of the thermodynamic behavior of non-interacting fermions. Now remember the example of a particle in a box. Electrons in a metal act like non-interacting particles with quantized energy levels simply given by the energies of a particle in a box. Now note that in this notation n is a vector composing of nx, ny, and nz, the quantum numbers representing x, y, and z directions. The average number of electrons in the state n is simply given by. Here we've defined the Fermi function to be. Now the total number of electrons is simply given as an infinite sum over these quantum numbers nx, ny, and nz, weighted by a factor of 2. The factor of 2 arises since the electrons can exist in a spin up, or a spin down state. For sufficiently large volume v, the spectrum of translational wave modes is effectively a continuum. Therefore, we can convert the summation to an integral over N, resulting in. Now we invoke a change of coordinate from n to k, defined as pi over L times n. With this modified definition, the energy can be rewritten as. Now let's define the chemical potential at temperature T = 0, that is, beta equals infinity to be epsilon F, also called the Fermi energy. Before we proceed, let's consider the form of the Fermi function at 0 temperature. It turns out that the Fermi function simply is 1 for all energies epsilon less than epsilon F, and 0 for energies greater than epsilon F. At 0 temperature this integration can be carried out very easily and simply gives. The important takeaway message from this equation is that the Fermi energy scales as the two-third power of the density. Now let's take a typical metal, say for instance copper, which has a mass density of nine gram per centimeter cube. Assuming each atom donates a single electron to the conducting electron gas, the density has a Fermi temperature of about 80,000 kelvin. This verifies that the Fermi energy, epsilon F, is sufficiently large to make the ideal gas approximation valid at room temperature. At room temperature, only states with energy levels very close to epsilon F will be affected by the thermal energy given by kBT. The spread in the distribution is approximately 2kBT. The average energy of an electron gas can be found using the relation. Following a similar derivation as we did before, we can write the expression as. Now the average energy depends on the integral of the Fermi function weighted by the energy to the three-halves power. We apply integration by parts to this equation, and use the expression we derived earlier for the average number of particles. Note that the new integrand depends on the derivative of the Fermi function with respect to energy. In the limit the temperature tends to zero, the derivative function is peaked near epsilon F. We can expand the integrand about the Fermi energy to get. We know that the derivative function is even about the Fermi energy in this limit. And the odd ordered terms integrate out to zero, leaving only the even ordered terms in this expression. Therefore, the final form of the energy has to take this form. A precise calculation of this low temperature expansion gives the following expression. Therefore, the heat capacity from the electrons present in the metal in the limit of small temperature gives a linear temperature dependence in the small T limit. The limiting behavior in the small T limit suggests that the plot of cv divided by the temperature approaches a constant value as the temperature tends to 0. This proves to be the behavior of the heat capacity for metals in the limit of small temperature. Now as the temperature increases, the fluctuations in the metal nuclei also contribute to the heat capacity. To summarize, we introduced a distinction for elementary particles as fermions or bosons. We learned that electrons act as non-interacting fermions. And we derived the energetics of an electron gas.
