This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
I'll go over all of these equations.
Boltzmann formulation (microcanonical or for large \(N\)): \begin{align} S(E) &= k_B \ln g(E) \end{align} where \(g\) is the number of microstates (or energy eigenstates). We spent little time on the Boltzmann formulation, but it is helpful to know that at sufficiently high temperatures the entropy approaches \(k_B\) times the logarithm of the number of energy eigenstates.
Gibbs formulation (always true): \begin{align} S(E) &= -k_B \sum_{i}^{\text{all states}} P_i \ln P_i \end{align}
The average value of any quantity is given by the weighted average \begin{align} \left<X\right> &= \sum_i^{\text{all states}} P_i X_i \end{align} In particular, the internal energy is given by \begin{align} U &= \sum_i^{\text{all states}} P_i E_i \end{align}
The following is the old review that I wrote.
\begin{align} dA &= \left(\frac{\partial {A}}{\partial {B}}\right)_{C}dB + \left(\frac{\partial {A}}{\partial {C}}\right)_{B}dC \end{align} You can:
- Do algebra
- Interpret coefficients as partial derivatives
- Integrate
\begin{align} \left(\frac{\partial {\left(\frac{\partial {A}}{\partial {B}}\right)_{C}}}{\partial {C}}\right)_{B} = \left(\frac{\partial {\left(\frac{\partial {A}}{\partial {C}}\right)_{B}}}{\partial {B}}\right)_{C} \end{align}
\begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} &= \frac{1}{\left(\frac{\partial {B}}{\partial {A}}\right)_{C}} \\ \left(\frac{\partial {A}}{\partial {B}}\right)_{D} &= \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \\ \left(\frac{\partial {A}}{\partial {B}}\right)_{C} &= -\frac{\left(\frac{\partial {A}}{\partial {C}}\right)_{B}}{\left(\frac{\partial {B}}{\partial {C}}\right)_{A}} \end{align}
\begin{align} \Delta S &= \int {\mathit{\unicode{273}}} \frac{Q_{\text{quasistatic}}}{T} \\ {\mathit{\unicode{273}}} Q &= TdS \\ C_\alpha &= T \left(\frac{\partial {S}}{\partial {T}}\right)_{\alpha} \end{align}
\begin{align} \Delta U &= Q + W \\ dU &= {\mathit{\unicode{273}}} Q + {\mathit{\unicode{273}}} W \\ dU &= TdS - pdV \end{align}
\begin{align} \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \ge 0 \end{align}
You can subtract \(TS\) from \(U\) or add \(pV\) to \(U\) to create new thermodynamic potentials that are convenient when \(T\) or \(p\) are held fixed or controlled.
From any thermodynamic potential you can use the equality of mixed partial derivatives to create a relationship between two different partial derivatives.
\begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} \\ Z &= \sum_i^{\text{all states}} \frac{e^{-\beta E_i}}{Z} \\ \beta &= \frac{1}{k_BT} \\ F &= -k_BT \ln Z \\ U &= \sum_i^{\text{all states}} P_i E_i \\ S &= -k_B\sum_i^{\text{all states}} P_i \ln P_i \end{align}
assignment Homework
Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.
assignment Homework
A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.
In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).
Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.
If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?
face Lecture
120 min.
Gibbs entropy information theory probability statistical mechanics
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 Lecture
120 min.
chemical potential Gibbs distribution grand canonical ensemble statistical mechanics
These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.group Small Group Activity
60 min.
face Lecture
120 min.
phase transformation Clausius-Clapeyron mean field theory thermodynamics
These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.assignment Homework
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 \(\varepsilon\).
From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.
Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.
Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework