• A system in thermal contact with a heat reservoir.

• A system thermally and molecularly open to its environment.

• Let the closed system \(\psi_i\) have the energy of \(E_i\), and the system is within the environment with the energy of \(E_{\rm env}\). The total energy is \(E_i + E_{\rm env} = E\).
• \(P(\psi_i)\) is proportional to the number of microstates of environment \(\Psi_I\), consistent with \(\psi_i\).
  • \[ P(\psi_i) \propto \Omega(E_{\rm env}) = \Omega(E - E_i) \]
• Taylor expand at \(E\), then truncate it under the assumption \(E,E_{\rm env} \gg E_i\) \[ \ln(\Omega(E-E_i)) \rightsquigarrow \ln(\Omega(E)) - \frac{\partial \ln(\Omega(E))}{\partial E}E_i \]
  • \(\frac{\displaystyle\partial f(E)}{\displaystyle \partial E}\) represents the derivative evaluated at \(E\).
  • Therefore, \[ P(\psi_i) \propto \exp\left(-\frac{\partial \ln(\Omega(E))}{\partial E}E_i\right) \]
  • Introduce \[ \beta = \frac{1}{k_B T} = \frac{\partial \ln(\Omega(E))}{\partial E} \]
  • Finally, \[ P(\psi_i) = \frac{1}{Z}\exp(-\beta E_i) \] where \(Z\) is the normalizing factor which is the sum of Boltzmann weight of every state.
  • \(\psi_i\) is in the canonical ensemble by definition.

• \[ P(\psi_i) \propto \Omega(E-E_i, N-N_i) \]
  • The Taylor expansion then becomes \[ \ln(\Omega(E-E_i,N-N_i)) \approx \ln(\Omega(E,N)) - \frac{\partial \ln(\Omega(E,N))}{\partial E}E_i - \frac{\partial \ln(\Omega(E,N))}{\partial N}N_i. \]
  • Introduce the chemical potential \[ \beta\mu = - \frac{\partial \ln(\Omega(E,N))}{\partial N}. \]
    • Negative sign is included to make the particles flow from the higher potential area to the lower potential area.
    • It is more negative when the entropy is more variable per molecule transfer.
  • Finally, \[ P(\psi_i) = \frac{1}{Z}\exp(-\beta E_i + \beta\mu N_i) \]

• \[ P(\psi_i) = \frac{1}{Z}\exp(-\beta E_i + \beta\sum_j\mu_j N_i^{(j)}) \] where \(N_i^{(j)}\) each represent a unique chemical.

\[ \frac{1}{T} = \frac{\partial S(E)}{\partial E} \]

\[ Z = \sum_i \exp(-\beta E_i) \]

It contains the information of all the microstates available at thermal equillibrium.

\[ Z = \frac{\Omega(E)}{\Omega_{\rm env}(E)} \]

\begin{align*} N =& \sum_i P(\psi_i)N_i\\ =& \frac{1}{\beta}\sum_i \frac{1}{Z} \frac{\partial}{\partial \mu}\exp(-\beta E_i + \beta\mu N_i) \\ =& \frac{1}{\beta}\frac{\partial}{\partial \mu} \ln Z \end{align*}

\begin{align*} U =& \frac{1}{Z}\sum_i -\left( \frac{\partial}{\partial \beta} \exp(-\beta E_i + \beta \mu N_i)\right)_{\mu} + \mu N\\ =& -\left(\frac{\partial}{\partial \beta} \ln Z\right)_{\mu} + \mu N \end{align*}

\begin{align*} P =& \sum_i P(\psi_i) \left(-\frac{\partial E_i}{\partial V}\right)\\ =& \frac{1}{Z}\left( \frac{1}{\beta}\frac{\partial}{\partial V}\right) \sum_i \exp(-\beta E_i + \beta\mu N_i)\\ =& \frac{1}{\beta}\frac{\partial}{\partial V} \ln Z \end{align*}

• When we do not care about \(N\), we can set \(\mu = 0\).

\begin{align*} E =& \sum_i P(\psi_i)E_i\\ =& \frac{1}{Z}\sum_i -\frac{\partial}{\partial \beta} \exp(-\beta E_i)\\ =& -\frac{\partial}{\partial \beta} \ln Z \end{align*}

• \[ \delta E^2 = \langle E_i^2\rangle - E^2 \]
• \[ \delta E^2 = -\frac{\partial^2}{\partial\beta^2}\ln Z \]

• Energy flows from the higher temperature to the lower temperature.
  • It is due to the entropy in the lower temperature part increases faster than it decreases in the higher temperature part.

• \[ dE = TdS - PdV + \mu dN \]

• \[dS \ge 0\]

• \[ T = 0 \implies S = 0 \]