Statistical Mechanics

• 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 \(E,E_{\rm env} \gg E_i\) \[ \ln\Omega(E-E_i)\sim \ln\Omega(E) - \frac{\partial \ln\Omega}{\partial E}E_i. \] Therefore, \[ P(\psi_i) \propto \exp\left(-\frac{\partial \ln\Omega}{\partial E}E_i\right). \] We here introduce: \[ \beta = \frac{1}{k_{\rm B} T} = \frac{\partial \ln\Omega}{\partial E}. \]

Finally, \[ P(\psi_i) = \frac{1}{Z}\exp(-\beta E_i) \] where \(Z\) is the normalizing factor called partition function, which is the sum of Boltzmann weight of every state.

• \[ P(\psi_i) \propto \Omega(E-E_i, N-N_i) \]
  • 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.

• \[ dE = TdS - PdV + \mu dN \]

• \[dS \ge 0\]

• \[ T = 0 \implies S = 0 \]

Let us define the "radial momentum" \( G \), from the "central moment of inertia" \( I \):

\begin{align*} \frac{1}{2} \dv{I}{t} &= \sum_k m_k \dv{\vb{r}_k}{t} \vdot \vb{r}_k \\ &= \sum_k \vb{p}_k \vdot \vb{r}_k := G. \end{align*} \begin{align*} \dv{G}{t} = 2T + \sum_k \vb{F}_k \vdot \vb{r}_k. \end{align*}

When averaged under the assumption \[ \expval{\dv{G}{t}}_{\tau} := \frac{1}{\tau}\int_0^{\tau} \dv{G}{t}\dd{t} = \frac{G(\tau) - G(0)}{\tau}= 0 \] which is the case for bounded system, the virial theorem is obtained: \[ 2\expval{T} = -\sum_k \expval{\mathbf{F}_k\vdot \mathbf{r}_k}. \]

Using the Newton's third law of motion, after decomposing the force into force on \( k \)th particle by \( j \)th particle \( \vb{F}_{jk} \): \[ \dv{G}{t} = 2T + \sum_{k > j} \vb{F}_{jk} \vdot (\mathbf{r}_k - \mathbf{r}_{j}). \]

If we assume that the force \( \mathbf{F}_{jk} \) can be given by the potential energy \( V_{jk} \): \[ \mathbf{F}_{jk} = - \nabla_{\mathbf{r}_k}V_{jk} = -\dv{V_{jk}}{r_{jk}} \left( \frac{\mathbf{r}_k - \mathbf{r}_j}{r_{jk}} \right), \] it gives

\begin{align*} \dv{G}{t} = 2T - \sum_{k > j} \dv{V_{jk}}{r_{jk}}r_{jk}. \end{align*}

• In astrophysics, it is used to tell whether a star will expand or shrink.

\[ \pdv{\eta}{t} = - M \frac{\delta E}{\delta \eta} \] where \( \eta \) is the scalar-valued state variable of fluid with \( \pm 1 \) indicating both domains, \( M \) is the mobility representing the rate of change of domain, and \( E \) is the total energy of the system. The total energy is given by: \[ E[\eta] = \int_{\Omega} F(\eta) + \frac{\varepsilon^2}{2} \| \nabla \eta \|^2\dd{\Omega}, \] where \( \varepsilon \) controlling the relative contribution of potential energy and interfacial energy, \( F \) is the potential energy, often given by \( (\eta^2 - 1)^2/4 \).

\[ \pdv{\eta}{t} = - D \nabla^2 \frac{\delta E}{\delta \eta} \]

The total volume of each phase is conserved due to the conservation law \[ \pdv{\eta}{t} = - \nabla \cdot \mathbf{j} \] where \( \mathbf{j} = -D \nabla \mu \) in which \( \delta E / \delta \eta \) is identified with chemical potential \( \mu \).