Gibbs sum for a two level system

• Gibbs sum Microstate Thermal average energy
• face Gibbs entropy approach

face Lecture

120 min.

Gibbs entropy approach
Thermal and Statistical Physics 2020

These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.
• face Chemical potential and Gibbs distribution

face Lecture

120 min.

Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
• face Review of Thermal Physics

face Lecture

30 min.

Review of Thermal Physics
Thermal and Statistical Physics 2020

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
• assignment Boltzmann probabilities

assignment Homework

Boltzmann probabilities
Energy Entropy Boltzmann probabilities Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies $(-\epsilon,0,\epsilon)$.
1. At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy $U$? What is the entropy $S$?
2. At very low temperature, what are the three probabilities?
3. What are the three probabilities at zero temperature? What is the internal energy $U$? What is the entropy $S$?
4. What happens to the probabilities if you allow the temperature to be negative?
• 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 2020
1. Show 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 $\frac1{V^{\frac23}}$.

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

• group Hydrogen emission

group Small Group Activity

30 min.

Hydrogen emission
Contemporary Challenges 2022 (5 years)

In this activity students work out energy level transitions in hydrogen that lead to visible light.
• assignment Approximating a Delta Function with Isoceles Triangles

assignment Homework

Approximating a Delta Function with Isoceles Triangles
Static Fields 2022 (5 years)

Remember that the delta function is defined so that $\delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases}$

Also: $\int_{-\infty}^{\infty} \delta(x-a)\, dx =1$.

1. Find a set of functions that approximate the delta function $\delta(x-a)$ with a sequence of isosceles triangles $\delta_{\epsilon}(x-a)$, centered at $a$, that get narrower and taller as the parameter $\epsilon$ approaches zero.
2. Using the test function $f(x)=3x^2$, find the value of $\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx$ Then, show that the integral approaches $f(a)$ in the limit that $\epsilon \rightarrow 0$.

• 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 Ideal gas calculations

assignment Homework

Ideal gas calculations
Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.

1. How much heat (in joules) is added to the gas in each of these two processes?

2. What is the temperature at the end of the second process?

3. Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?

• assignment Free energy of a two state system

assignment Homework

Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

• Thermal and Statistical Physics 2020
1. Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy $\varepsilon$. Find the Gibbs sum for this system is in terms of the activity $\lambda\equiv e^{\beta\mu}$. Note that the system can hold a maximum of one particle.

2. Solve for the thermal average occupancy of the system in terms of $\lambda$.

3. Show that the thermal average occupancy of the state at energy $\varepsilon$ is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

4. Find an expression for the thermal average energy of the system.

5. Allow the possibility that the orbitals at $0$ and at $\varepsilon$ may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because $\mathcal{Z}$ can be factored as shown, we have in effect two independent systems.