Physlib.StatisticalMechanics.CanonicalEnsemble.Finite

Let $\mathcal{C}$ and $\mathcal{C}_1$ be canonical ensembles. If both $\mathcal{C}$ and $\mathcal{C}_1$ satisfy the finiteness condition `IsFinite` (which implies that their respective spaces of microstates are finite and their associated measures are counting measures), then their additive composition $\mathcal{C} + \mathcal{C}_1$ also satisfies the `IsFinite` condition.

Let $\mathcal{C}$ be a canonical ensemble. If $\mathcal{C}$ satisfies the finiteness condition `IsFinite` (which implies that its microstate space $\iota$ is a finite type and its associated measure is the counting measure), then for any measurable equivalence $e : \iota_1 \simeq^m \iota$, the ensemble $\text{congr}(\mathcal{C}, e)$ (the ensemble $\mathcal{C}$ re-indexed over the state space $\iota_1$ via $e$) also satisfies the `IsFinite` condition.

Let $\mathcal{C}$ be a canonical ensemble. If $\mathcal{C}$ satisfies the finiteness condition `IsFinite` (which implies that its microstate space is finite and its associated measure is the counting measure), then for any natural number $n \in \mathbb{N}$, the $n$-fold composition of the ensemble $n \cdot \mathcal{C}$ (denoted by `nsmul n 𝓒`) also satisfies the `IsFinite` condition.

In a canonical ensemble $\mathcal{C}$ that satisfies the finiteness condition `IsFinite` (which implies the microstate space is finite and the measure $\mu$ is the counting measure), the associated measure $\mu$ is a finite measure.

In a canonical ensemble $\mathcal{C}$ where the microstate space $\iota$ is finite and the measure $\mu$ is the counting measure (as specified by the `IsFinite` condition), if $\iota$ is non-empty, then the measure $\mu$ is non-zero ($\mu \neq 0$).

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates, the Shannon entropy at a given temperature $T$ is defined by the formula \[ S = -k_B \sum_{i} p_i \ln p_i \] where $k_B$ is the Boltzmann constant and $p_i$ is the probability of the system being in microstate $i$ at temperature $T$.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$, the mathematical partition function at temperature $T$ is equal to the sum of the Boltzmann factors over all microstates: \[ Z(T) = \sum_{i \in \iota} e^{-\beta E_i} \] where $E_i$ is the energy of microstate $i$ and $\beta = \frac{1}{k_B T}$ is the inverse temperature.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$, the partition function $Z(T)$ at temperature $T$ is given by the sum of the Boltzmann factors over all microstates: \[ Z(T) = \sum_{i \in \iota} e^{-\beta E_i} \] where $E_i$ is the energy of microstate $i$ and $\beta = \frac{1}{k_B T}$ is the inverse temperature.

For a finite canonical ensemble $\mathcal{C}$ at temperature $T$, the unnormalized Boltzmann measure of a microstate $i$ is given by $\exp(-\beta E_i)$, where $\beta = \frac{1}{k_B T}$ is the inverse temperature and $E_i$ is the energy associated with the state $i$.

Consider a canonical ensemble $\mathcal{C}$ with a finite number of microstates. For any temperature $T$, the Boltzmann measure $\mu_{\text{Bolt}}(T)$ (the measure where the weight of a state $i$ is given by $e^{-\beta E_i}$, with $\beta = \frac{1}{k_B T}$) is a finite measure.

For a finite canonical ensemble $\mathcal{C}$ at temperature $T$, the value of the normalized probability measure $\mu_{\text{Prod}}(T)$ for a single microstate $i \in \iota$ is equal to the probability $p_i(T)$ of the system being in that state.

For a canonical ensemble $\mathcal{C}$ with a finite number of microstates at temperature $T$, the mean energy $\langle E \rangle(T)$ is equal to the sum over all microstates $i$ of the energy $E_i$ of the microstate multiplied by its probability $p_i(T)$ of occurrence: \[ \langle E \rangle(T) = \sum_{i} E_i p_i(T) \]

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates at temperature $T$, the Shannon entropy $S(T)$ is given by \[ S(T) = -k_B \sum_{i} p_i \ln p_i \] where $k_B$ is the Boltzmann constant and $p_i$ is the probability of the system being in microstate $i$ at temperature $T$.

For a canonical ensemble $\mathcal{C}$ with a finite and non-empty set of microstates $\iota$, the probability $p_i$ of the system being in microstate $i \in \iota$ at temperature $T$ is less than or equal to 1.

For a canonical ensemble $\mathcal{C}$ where the set of microstates $\iota$ is finite and non-empty, the mathematical partition function $Z_{\text{math}}(T)$ is strictly positive for any temperature $T$. That is, \[ Z_{\text{math}}(T) > 0. \] The mathematical partition function $Z_{\text{math}}(T)$ corresponds to the integral of the Boltzmann factor $e^{-\beta E}$ over the microstate space, which reduces to the sum $\sum_{i \in \iota} e^{-\beta E_i}$ in the finite case.

For a canonical ensemble $\mathcal{C}$ where the set of microstates $\iota$ is finite and non-empty, the physical partition function $Z(T)$ is strictly positive for any temperature $T$. That is, \[ Z(T) > 0. \] In this finite discrete case, the partition function $Z(T)$ reduces to the sum of Boltzmann factors $\sum_{i \in \iota} e^{-\beta E_i}$, where $\beta = (k_B T)^{-1}$. Since each exponential term is positive and the set of microstates is non-empty, the sum is strictly positive.

For a canonical ensemble $\mathcal{C}$ where the set of microstates $\iota$ is finite and non-empty, the probability $P(i; T)$ of the system being in a microstate $i \in \iota$ at temperature $T$ is non-negative. That is, \[ P(i; T) \ge 0. \] In this finite context, the probability is given by the Boltzmann distribution $P(i; T) = \frac{1}{Z(T)} e^{-\beta E_i}$, where $E_i$ is the energy of the state, $\beta = \frac{1}{k_B T}$ is the inverse temperature, and $Z(T)$ is the partition function.

For a canonical ensemble $\mathcal{C}$ with a finite and non-empty set of microstates $\iota$, the sum of the probabilities of all microstates $i \in \iota$ at a given temperature $T$ is equal to 1: \[ \sum_{i \in \iota} P(i; T) = 1 \] where $P(i; T)$ denotes the probability of the system being in microstate $i$ at temperature $T$.

For a canonical ensemble $\mathcal{C}$ where the set of microstates $\iota$ is finite and non-empty, the Shannon entropy $S$ at temperature $T$ is non-negative. That is, \[ S \ge 0, \] where the Shannon entropy is defined as $S = -k_B \sum_{i \in \iota} p_i \ln p_i$, $k_B$ is the Boltzmann constant, and $p_i$ is the probability of the system being in microstate $i$.

For a finite canonical ensemble $\mathcal{C}$ at temperature $T$ with a discrete set of microstates $\iota$, the Shannon entropy $S_{\text{Shannon}}(T)$, defined by \[ S_{\text{Shannon}}(T) = -k_B \sum_{i \in \iota} p_i \ln p_i \] (where $k_B$ is the Boltzmann constant and $p_i$ is the probability of the system being in microstate $i$), is equal to the differential entropy $S_{\text{diff}}(T)$ of the system.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$ (satisfying the `IsFinite` condition), the thermodynamic entropy $S_{\text{thermo}}(T)$ at temperature $T$ is equal to the discrete Shannon entropy $S_{\text{Shannon}}(T)$: \[ S_{\text{thermo}}(T) = S_{\text{Shannon}}(T) \] where the Shannon entropy is defined as $S_{\text{Shannon}}(T) = -k_B \sum_{i \in \iota} p_i \ln p_i$, $k_B$ is the Boltzmann constant, and $p_i$ is the probability of the system being in microstate $i$ at temperature $T$. In this finite case, all semi-classical correction factors vanish because the degrees of freedom $dof = 0$ and the phase space unit is $1$.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $i \in \iota$ at temperature $T$, the mean square energy $\langle E^2 \rangle$ is equal to the sum over all microstates of the squared energy of each state weighted by its probability: \[ \langle E^2 \rangle = \sum_{i \in \iota} E_i^2 p_i(T) \] where $E_i$ is the energy of microstate $i$ and $p_i(T)$ is the probability of the system being in that state.

For a canonical ensemble $\mathcal{C}$ with a finite and non-empty set of microstates $i \in \iota$ at temperature $T$, the energy variance $\text{Var}(E)$ is equal to the mean of the squared energies weighted by their probabilities minus the square of the mean energy: \[ \text{Var}(E) = \left( \sum_{i \in \iota} E_i^2 p_i(T) \right) - \langle E \rangle^2 \] where $E_i$ represents the energy of microstate $i$, $p_i(T)$ is the probability of the system being in state $i$ at temperature $T$, and $\langle E \rangle$ is the mean energy of the system.

For a canonical ensemble with a finite set of microstates $i \in \iota$, the partition function $Z$ is defined as a function of the inverse temperature $\beta \in \mathbb{R}$ by the sum: \[ Z(\beta) = \sum_{i \in \iota} e^{-\beta E_i} \] where $E_i$ represents the energy of microstate $i$.

For a finite canonical ensemble where the set of microstates $\iota$ is non-empty, the partition function $Z(\beta) = \sum_{i \in \iota} e^{-\beta E_i}$ is strictly positive for any inverse temperature parameter $\beta \in \mathbb{R}$.

For a canonical ensemble with a finite set of microstates $\iota$, the Boltzmann probability $p_i(\beta)$ of a microstate $i \in \iota$ at inverse temperature $\beta \in \mathbb{R}$ is defined as the ratio: \[ p_i(\beta) = \frac{e^{-\beta E_i}}{Z(\beta)} \] where $E_i$ represents the energy of microstate $i$ and $Z(\beta) = \sum_{j \in \iota} e^{-\beta E_j}$ is the partition function of the system.

For a canonical ensemble with a finite set of microstates $\iota$, the mean energy $U(\beta)$ as a function of the inverse temperature parameter $\beta \in \mathbb{R}$ is defined as the expected value of the energy over all microstates: \[ U(\beta) = \sum_{i \in \iota} E_i p_i(\beta) \] where $E_i$ is the energy of microstate $i$, and $p_i(\beta)$ is the Boltzmann probability of microstate $i$, given by: \[ p_i(\beta) = \frac{e^{-\beta E_i}}{Z(\beta)} \] where $Z(\beta) = \sum_{j \in \iota} e^{-\beta E_j}$ is the partition function of the system.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$, the general measure-theoretic mean energy $\langle E \rangle_\beta$ at a positive inverse temperature $\beta > 0$ is equal to the finite sum definition of mean energy $U(\beta)$, which is given by the expected value: \[ U(\beta) = \sum_{i \in \iota} E_i p_i(\beta) \] where $E_i$ is the energy of microstate $i$ and $p_i(\beta) = \frac{e^{-\beta E_i}}{\sum_{j \in \iota} e^{-\beta E_j}}$ is the Boltzmann probability of that microstate.

For a canonical ensemble with a finite and non-empty set of microstates $\iota$, the mean energy $U(\beta)$ is a differentiable function of the inverse temperature parameter $\beta \in \mathbb{R}$. The mean energy is defined as: \[ U(\beta) = \sum_{i \in \iota} E_i p_i(\beta) \] where $E_i$ is the energy of microstate $i$, and $p_i(\beta) = \frac{e^{-\beta E_i}}{\sum_{j \in \iota} e^{-\beta E_j}}$ is the Boltzmann probability of that state.

For a finite canonical ensemble $\mathcal{C}$, the partition function $Z(\beta) = \sum_{i \in \iota} e^{-\beta E_i}$ is differentiable with respect to the inverse temperature parameter $\beta \in \mathbb{R}$.

For a finite canonical ensemble $\mathcal{C}$ with microstates indexed by $i$ and corresponding energies $E_i$, this function maps the inverse temperature parameter $\beta \in \mathbb{R}$ to the sum $\sum_i E_i e^{-\beta E_i}$. This sum represents the numerator used in the definition of the mean energy $U(\beta) = \frac{\sum_i E_i e^{-\beta E_i}}{Z(\beta)}$, where $Z(\beta)$ is the partition function.

For a finite canonical ensemble $\mathcal{C}$ with microstates indexed by $i$ and corresponding energies $E_i$, the numerator of the mean energy, defined as the function $f(\beta) = \sum_i E_i e^{-\beta E_i}$ for $\beta \in \mathbb{R}$, is differentiable with respect to $\beta$.

For a finite canonical ensemble with microstates indexed by $i$ and corresponding energies $E_i$, let $Z(\beta) = \sum_i e^{-\beta E_i}$ be the partition function and $N(\beta) = \sum_i E_i e^{-\beta E_i}$ be the mean energy numerator. For any inverse temperature $\beta \in \mathbb{R}$, the derivative of the partition function with respect to $\beta$ is equal to the negative of the mean energy numerator: $$\frac{d}{d\beta} Z(\beta) = -N(\beta).$$

For a finite canonical ensemble with microstates indexed by $i$ and corresponding energies $E_i$, let $N(\beta) = \sum_i E_i e^{-\beta E_i}$ denote the mean energy numerator. For any inverse temperature $\beta \in \mathbb{R}$, the derivative of this numerator with respect to $\beta$ is given by $$\frac{d}{d\beta} N(\beta) = -\sum_i E_i^2 e^{-\beta E_i}.$$

For a finite canonical ensemble with microstates indexed by $i \in \iota$ and corresponding energies $E_i$, let $U(\beta)$ be the mean energy and $p_i(\beta)$ be the Boltzmann probability of microstate $i$ at inverse temperature $\beta \in \mathbb{R}$. The derivative of the mean energy with respect to $\beta$ is given by the expression: $$\frac{d}{d\beta} U(\beta) = U(\beta)^2 - \sum_{i \in \iota} E_i^2 p_i(\beta),$$ where $U(\beta)^2$ is the square of the mean energy and the sum represents the expected value of the squared energy $\langle E^2 \rangle_\beta$.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$, let $U(\beta)$ be the mean energy as a function of the inverse temperature $\beta \in (0, \infty)$. For any positive temperature $T > 0$ with corresponding inverse temperature $\beta = \frac{1}{k_B T}$, the derivative of the mean energy with respect to $\beta$ is equal to the negative of the energy variance $\text{Var}(E)$ at that temperature: $$\frac{dU}{d\beta} = -\text{Var}(E)$$ where the energy variance is defined as $\text{Var}(E) = \langle E^2 \rangle - \langle E \rangle^2$.

For a canonical ensemble $\mathcal{C}$ with a finite set of microstates $\iota$, let $T > 0$ be a strictly positive temperature. The heat capacity $C_V$ of the system is related to the variance of the energy fluctuations $\text{Var}(E)$ by the expression: $$C_V = \frac{\text{Var}(E)}{k_B T^2}$$ where $k_B$ is the Boltzmann constant and $\text{Var}(E) = \langle E^2 \rangle - \langle E \rangle^2$.