- Fermi-Dirac function
*assignment*Pressure and entropy of a degenerate Fermi gas*assignment*Homework##### Pressure and entropy of a degenerate Fermi gas

Fermi gas Pressure Entropy Thermal and Statistical Physics 2020Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).

Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).

*face*Fermi and Bose gases*face*Lecture120 min.

##### Fermi and Bose gases

Thermal and Statistical Physics 2020Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.*face*Ideal Gas*face*Lecture120 min.

##### Ideal Gas

Thermal and Statistical Physics 2020ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.*face*Review of Thermal Physics*face*Lecture30 min.

##### Review of Thermal Physics

Thermal and Statistical Physics 2020thermodynamics statistical mechanics

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.*assignment*Distribution function for double occupancy statistics*assignment*Homework##### Distribution function for double occupancy statistics

Orbitals Distribution function Thermal and Statistical Physics 2020Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

*assignment*Energy of a relativistic Fermi gas*assignment*Homework##### Energy of a relativistic Fermi gas

Fermi gas Relativity Thermal and Statistical Physics 2020For electrons with an energy \(\varepsilon\gg mc^2\), where \(m\) is the mass of the electron, the energy is given by \(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons in a cube of volume \(V=L^3\) the momentum takes the same values as for a non-relativistic particle in a box.

Show that in this extreme relativistic limit the Fermi energy of a gas of \(N\) electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where \(n\equiv \frac{N}{V}\) is the number density.

Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

*assignment*Symmetry of filled and vacant orbitals*assignment*Homework##### Symmetry of filled and vacant orbitals

Symmetry Orbitals Thermal and Statistical Physics 2020 Show that \begin{align} f(\mu+\delta) &= 1 - f(\mu-\delta) \end{align} This means that the probability that an orbital above the Fermi level is occupied is equal to the probability an orbital the same distance below the Fermi level*being empty*. These unoccupied orbitals are called*holes*.*assignment*Ideal gas in two dimensions*assignment*Homework##### Ideal gas in two dimensions

Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

Find an expression for the energy \(U\) of the gas.

Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

*assignment*Fluctuations in a Fermi gas*assignment*Homework##### Fluctuations in a Fermi gas

Fermi gas grand canonical ensemble statistical mechanics Thermal and Statistical Physics 2020 (K&K 7.11) Show for a single orbital of a fermion system that \begin{align} \left<(\Delta N)^2\right> = \left<N\right>(1+\left<N\right>) \end{align} if \(\left<N\right>\) is the average number of fermions in that orbital. Notice that the fluctuation vanishes for orbitals with energies far enough from the chemical potential \(\mu\) so that \(\left<N\right>=1\) or \(\left<N\right>=0\).*assignment*Energy fluctuations*assignment*Homework##### Energy fluctuations

energy Boltzmann factor statistical mechanics heat capacity Thermal and Statistical Physics 2020 Consider a system of fixed volume in thermal contact with a resevoir. Show that the mean square fluctuations in the energy of the system is \begin{equation} \left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right> = k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V} \end{equation} Here \(U\) is the conventional symbol for \(\langle\varepsilon\rangle\).*Hint:*Use the partition function \(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to the mean square fluctuation. Also, multiply out the term \((\cdots)^2\).-
Thermal and Statistical Physics 2020
**Derivative of Fermi-Dirac function**Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.