Physlib.StatisticalMechanics.CanonicalEnsemble.TwoState

For given energy levels $E_0, E_1 \in \mathbb{R}$, the two-state canonical ensemble is a canonical ensemble defined over a discrete state space of two elements (modeled as $\{0, 1\}$). The energy function $E$ for this system assigns the value $E_0$ to the first state and $E_1$ to the second state. The ensemble is equipped with the counting measure as its underlying measure.

For any energy levels $E_0, E_1 \in \mathbb{R}$, the two-state canonical ensemble defined by these energy levels is a finite canonical ensemble.

For a two-state canonical ensemble with energy levels $E_0, E_1 \in \mathbb{R}$, the partition function $Z(T)$ at temperature $T$ is given by: $$Z(T) = e^{-\beta E_0} + e^{-\beta E_1}$$ where $\beta = \frac{1}{k_B T}$ is the inverse temperature.

For a two-state canonical ensemble with energy levels $E_0, E_1 \in \mathbb{R}$, the partition function $Z(T)$ at temperature $T$ is given by the formula: $$Z(T) = 2 e^{-\beta \frac{E_0 + E_1}{2}} \cosh\left(\beta \frac{E_1 - E_0}{2}\right)$$ where $\beta = \frac{1}{k_B T}$ is the inverse temperature.

For any energy levels $E_0, E_1 \in \mathbb{R}$, the energy of the first state (index $0$) in a two-state canonical ensemble is $E_0$.

For any energy levels $E_0, E_1 \in \mathbb{R}$, the energy of the second state (index $1$) in a two-state canonical ensemble is $E_1$.

For a two-state canonical ensemble with energy levels $E_0, E_1 \in \mathbb{R}$ and temperature $T$, the probability $P(0)$ of the system being in the first state (with energy $E_0$) is given by: $$P(0) = \frac{1}{2} \left( 1 + \tanh \left( \frac{\beta (E_1 - E_0)}{2} \right) \right)$$ where $\beta = \frac{1}{k_B T}$ is the inverse temperature and $\tanh$ is the hyperbolic tangent function.

For a two-state canonical ensemble with energy levels $E_0, E_1 \in \mathbb{R}$ at temperature $T$, let $\beta$ be the inverse temperature. The probability $\rho(1)$ of the second state (the state with energy $E_1$) is given by: $$\rho(1) = \frac{1}{2} \left( 1 - \tanh \left( \frac{\beta (E_1 - E_0)}{2} \right) \right)$$

For a two-state canonical ensemble with energy levels $E_0, E_1 \in \mathbb{R}$ at temperature $T$, the mean energy $\langle E \rangle$ is given by: $$\langle E \rangle = \frac{E_0 + E_1}{2} - \frac{E_1 - E_0}{2} \tanh \left( \frac{\beta (E_1 - E_0)}{2} \right)$$ where $\beta = \frac{1}{k_B T}$ is the inverse temperature and $\tanh$ is the hyperbolic tangent function.

The entropy of a two-state canonical ensemble with energy levels $E_0$ and $E_1$ at temperature $T$ is equal to \[ \ln \left( 2 \cosh \left( \frac{\beta (E_1 - E_0)}{2} \right) \right) - \frac{\beta (E_1 - E_0)}{2} \tanh \left( \frac{\beta (E_1 - E_0)}{2} \right) \] where $\beta$ is the inverse temperature corresponding to $T$.

The Helmholtz free energy of a two-state canonical ensemble with energy levels $E_0$ and $E_1$ at temperature $T$ is equal to \[ \frac{E_0 + E_1}{2} - \frac{1}{\beta} \ln \left( 2 \cosh \left( \frac{\beta (E_1 - E_0)}{2} \right) \right) \] where $\beta$ is the inverse temperature corresponding to $T$.