Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas

For a given canonical ensemble $\mathcal{C}$ at temperature $T$, the **mathematical Helmholtz free energy** is defined as the product of the negative Boltzmann constant $k_B$, the temperature $T$, and the natural logarithm of the mathematical partition function $Z_{\text{math}}(T)$: $$F_{\text{math}}(T) = -k_B T \ln(Z_{\text{math}}(T))$$ This quantity serves as an intermediate thermodynamic potential that relates the microscopic statistical properties (via the partition function) to macroscopic thermodynamics, prior to any semi-classical corrections.

For a canonical ensemble $\mathcal{C}$ at temperature $T$, assuming that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite and the base measure $\mu$ is non-zero, the physical Helmholtz free energy $F(T)$ is related to the mathematical Helmholtz free energy $F_{\text{math}}(T)$ by the following identity: $$F(T) = F_{\text{math}}(T) + k_B T d \ln(h)$$ where $k_B$ is the Boltzmann constant, $d$ is the number of degrees of freedom of the system, and $h$ is the phase space unit.

For a canonical ensemble $\mathcal{C}$ at temperature $T$, let the Boltzmann measure $\mu_{\text{Bolt}}$ be finite, the base measure $\mu$ be non-zero, and the energy function $E$ be integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$. Then the differential entropy $S_{\text{diff}}(T)$ is related to the mean energy $U$ and the mathematical partition function $Z_{\text{math}}(T)$ by the identity: $$S_{\text{diff}}(T) = k_B \beta U + k_B \ln(Z_{\text{math}}(T))$$ where $k_B$ is the Boltzmann constant and $\beta = \frac{1}{k_B T}$ is the inverse temperature.

For a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with energy function $E$, let $T$ be the temperature with inverse temperature $\beta = \frac{1}{k_B T}$. If the Boltzmann measure $\mu_{\text{Bolt}}$ is finite and the base measure $\mu$ is non-zero (ensuring the mathematical partition function $Z_{\text{math}}(T)$ is strictly positive), then the natural logarithm of the probability density $\rho(i)$ for a microstate $i \in \iota$ is given by: $$\log(\rho(i)) = -\beta E(i) - \log(Z_{\text{math}}(T))$$

For any temperature $T > 0$, the product of the Boltzmann constant $k_B$ and the inverse temperature $\beta$ (defined as $\beta = \frac{1}{k_B T}$) is equal to the reciprocal of the temperature: $$k_B \beta = \frac{1}{T}$$

For a canonical ensemble $\mathcal{C}$ at temperature $T$, suppose the energy function $E$ is integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$, the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, and the base measure $\mu$ is non-zero. Then the thermodynamic entropy $S(T)$ is related to the differential entropy $S_{\text{diff}}(T)$ by the following identity: $$S(T) = S_{\text{diff}}(T) - k_B d \ln(h)$$ where $k_B$ is the Boltzmann constant, $d$ is the number of degrees of freedom, and $h$ is the fundamental phase space unit. This identity represents the semi-classical correction required to relate the mathematical differential entropy to the absolute thermodynamic entropy.

For a canonical ensemble $\mathcal{C}$ at temperature $T$, suppose that the energy function $E$ is integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$, the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, and the base measure $\mu$ is non-zero. If the number of degrees of freedom is $d = 0$, then the absolute thermodynamic entropy $S(T)$ is equal to the differential entropy $S_{\text{diff}}(T)$: $$S(T) = S_{\text{diff}}(T)$$ This result implies that there is no semi-classical correction required when the system has zero degrees of freedom.

For a canonical ensemble $\mathcal{C}$ at temperature $T$, let $S(T)$ be the absolute thermodynamic entropy and $S_{\text{diff}}(T)$ be the differential entropy. Suppose that the energy function $E$ is integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$, the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, and the base measure $\mu$ is non-zero. If the fundamental phase space unit $h$ is equal to 1, then the thermodynamic entropy and the differential entropy are equal: $$S(T) = S_{\text{diff}}(T)$$ This result reflects that the semi-classical correction term $k_B d \ln(h)$ vanishes when the phase space unit is unity.

For a canonical ensemble $\mathcal{C}$ at temperature $T > 0$, suppose that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, the base measure $\mu$ is non-zero, and the energy function $E$ is integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$. Then the Helmholtz free energy $F(T)$, the mean energy $U(T)$, and the absolute thermodynamic entropy $S(T)$ satisfy the fundamental thermodynamic identity: $$F(T) = U(T) - T S(T)$$

For a canonical ensemble $\mathcal{C}$ with $d$ degrees of freedom and phase space unit $h$, at a temperature $T > 0$, assume that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, the base measure $\mu$ is non-zero, and the energy function $E$ is integrable with respect to the canonical probability measure. Then the differential entropy $S_{\text{diff}}(T)$ is related to the mean energy $U$ and the Helmholtz free energy $F$ by the identity: $$S_{\text{diff}}(T) = \frac{U - F}{T} + k_B d \ln h$$ where $k_B$ is the Boltzmann constant. This relation accounts for the semi-classical correction term $k_B d \ln h$ required to connect the mathematical differential entropy to the physical thermodynamic quantities.

For a canonical ensemble $\mathcal{C}$ at temperature $T > 0$, assume that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite, the base measure $\mu$ is non-zero, and the energy function $E$ is integrable with respect to the canonical probability measure $\mu_{\text{Prod}}$. If the system either has zero degrees of freedom ($d = 0$) or the phase space unit is normalized to unity ($h = 1$), the semi-classical correction term $k_B d \ln h$ vanishes. In this case, the differential entropy $S_{\text{diff}}$ is equal to the difference between the mean energy $U$ and the Helmholtz free energy $F$, divided by the temperature $T$: $$S_{\text{diff}} = \frac{U - F}{T}$$

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function and $s \subseteq \mathbb{R}$ be a set. If $f$ has a derivative $f'$ at a point $x$ within the set $s$, and $f(x) \neq 0$, then the composition $\log(f(t))$ also has a derivative at $x$ within $s$, which is given by $\frac{1}{f(x)} \cdot f'$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function and $s \subseteq \mathbb{R}$ be a set. If $f$ has a derivative $f'$ at a point $x$ within the set $s$, and $f(x) \neq 0$, then the composition $\log(f(t))$ also has a derivative at $x$ within $s$, which is given by $\frac{1}{f(x)} \cdot f'$.

Let $\mathcal{C}$ be a canonical ensemble on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$. For a given temperature $T$, let $\beta = \frac{1}{k_B T}$ be the inverse temperature and $\mu_{\text{Bolt}}$ be the corresponding Boltzmann measure. Then, for any real-valued function $\phi: \iota \to \mathbb{R}$, the integral of $\phi$ with respect to the Boltzmann measure $\mu_{\text{Bolt}}$ is equal to the integral of the product of $\phi$ and the Boltzmann factor $e^{-\beta E}$ with respect to the base measure $\mu$: $$\int_\iota \phi(x) \, d\mu_{\text{Bolt}}(x) = \int_\iota \phi(x) e^{-\beta E(x)} \, d\mu(x)$$

Let $\mathcal{C}$ be a canonical ensemble on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$. For a given temperature $T$, let $\beta = \frac{1}{k_B T}$ be the inverse temperature and $\mu_{\text{Bolt}}$ be the corresponding Boltzmann measure. The integral of the energy function $E$ with respect to the Boltzmann measure is equal to the integral of the product of the energy and the Boltzmann factor $e^{-\beta E}$ with respect to the base measure $\mu$: $$\int_\iota E(x) \, d\mu_{\text{Bolt}}(x) = \int_\iota E(x) e^{-\beta E(x)} \, d\mu(x)$$

Let $\mathcal{C}$ be a canonical ensemble on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$. For a given temperature $T$ with corresponding inverse temperature $\beta = \frac{1}{k_B T}$, the mean energy $U$ is equal to the ratio of the integral of the energy weighted by the Boltzmann factor $e^{-\beta E}$ to the integral of the Boltzmann factor itself: $$U = \frac{\int_{\iota} E(i) e^{-\beta E(i)} \, d\mu(i)}{\int_{\iota} e^{-\beta E(i)} \, d\mu(i)}$$ where the denominator corresponds to the mathematical partition function $Z_{\text{math}}(T)$.

Let $\mathcal{C}$ be a canonical ensemble on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$. For a given temperature $T > 0$ with inverse temperature $\beta = \frac{1}{k_B T}$, assume that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite and the base measure $\mu$ is non-zero. If the mathematical partition function, defined as $Z_{\text{math}}(\beta) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$, is differentiable at $\beta$ (within the domain $\beta > 0$) with derivative $$\frac{d Z_{\text{math}}}{d\beta} = -\int_{\iota} E(i) e^{-\beta E(i)} \, d\mu(i),$$ then the mean energy $U$ of the ensemble at temperature $T$ is given by the negative derivative of the logarithm of the mathematical partition function with respect to $\beta$: $$U = -\frac{d}{d\beta} \log Z_{\text{math}}(\beta)$$

Consider a canonical ensemble $\mathcal{C}$ with a non-zero base measure $\mu$, energy function $E$, $d$ degrees of freedom, and phase space unit $h$. Provided that the Boltzmann measure is finite for all inverse temperatures $\beta > 0$, then for any $\beta \in (0, \infty)$, the logarithm of the physical partition function $Z$ is related to the logarithm of the mathematical partition function $Z_{\text{math}}$ by the following identity: $$\log Z(\beta) = \log Z_{\text{math}}(\beta) - d \log h$$ where $Z_{\text{math}}(\beta) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$.

Consider a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E$. Let $T > 0$ be the temperature and $\beta = \frac{1}{k_B T}$ be the inverse temperature. Suppose that the Boltzmann measure $\mu_{\text{Bolt}}$ is finite for all inverse temperatures $\beta > 0$ and the base measure $\mu$ is non-zero. Then the derivative of the logarithm of the physical partition function $Z$ with respect to $\beta$ is equal to the derivative of the logarithm of the mathematical partition function $Z_{\text{math}}$ with respect to $\beta$: $$\frac{d}{d\beta} \log Z(\beta) = \frac{d}{d\beta} \log Z_{\text{math}}(\beta)$$ where the mathematical partition function is defined as $Z_{\text{math}}(\beta) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$. The derivatives are taken within the domain $\beta \in (0, \infty)$.

Consider a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E$. Let $T > 0$ be the temperature and $\beta = \frac{1}{k_B T}$ be the inverse temperature. Suppose that the base measure $\mu$ is non-zero and the Boltzmann measure $\mu_{\text{Bolt}}$ is finite for all inverse temperatures $\beta > 0$. If the mathematical partition function $Z_{\text{math}}(\beta) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$ is differentiable at $\beta$ with derivative $$\frac{d Z_{\text{math}}}{d\beta} = -\int_{\iota} E(i) e^{-\beta E(i)} \, d\mu(i),$$ then the mean energy $U$ at temperature $T$ is equal to the negative derivative of the logarithm of the physical partition function $Z$ with respect to $\beta$: $$U = -\frac{d}{d\beta} \log Z(\beta)$$ where the derivative is taken within the domain $\beta \in (0, \infty)$.

For a canonical ensemble $\mathcal{C}$ over a space of microstates $\iota$ at temperature $T$, if the canonical probability measure $\mu_{\text{Prod}}$ is a probability measure, and the microstate energy function $E$ and its square $E^2$ are integrable with respect to $\mu_{\text{Prod}}$, then the energy variance $\text{Var}(E)$ is equal to the mean square energy $\langle E^2 \rangle$ minus the square of the mean energy $\langle E \rangle$: $$\text{Var}(E) = \langle E^2 \rangle - \langle E \rangle^2$$

For a canonical ensemble $\mathcal{C}$ and a real number $t$, this function returns the mean energy $U$ of the system at temperature $t$. It is defined by converting the real value $t$ into a non-negative temperature and calculating the expectation value of the energy function $E$ with respect to the canonical probability measure $\mu_{\text{Prod}}$: $$U(t) = \int_{\iota} E(i) \, d\mu_{\text{Prod}}(T)$$ where $T$ is the temperature corresponding to the real value $t$.

For a canonical ensemble $\mathcal{C}$, this function calculates the mean energy $U$ as a function of the real-valued inverse temperature $\beta$. It is defined as the mean energy $U(T)$ evaluated at the temperature $T = \frac{1}{k_B \beta}$, where $k_B$ is the Boltzmann constant and $\beta$ is the non-negative part of the input $b \in \mathbb{R}$.

For a canonical ensemble $\mathcal{C}$ and a temperature $T$, the heat capacity at constant volume $C_V$ is the derivative of the mean energy $U(T)$ with respect to temperature $T$. It is defined as the derivative of the function $U(t)$ (representing `meanEnergy_T`) within the set of positive real numbers $(0, \infty)$, evaluated at the value of the temperature $T$: $$C_V(T) = \frac{dU}{dT}$$ where $U(t)$ is the mean energy of the ensemble at temperature $t$.

For a canonical ensemble $\mathcal{C}$ and a temperature $T > 0$, let $U(\beta)$ be the mean energy expressed as a function of the inverse temperature $\beta = \frac{1}{k_B T}$. If $U(\beta)$ is differentiable at $\beta$ (within the domain $(0, \infty)$) with derivative $\frac{dU}{d\beta}$, then the heat capacity at constant volume $C_V$ at temperature $T$ is given by: $$C_V = \frac{dU}{d\beta} \cdot \left( -\frac{1}{k_B T^2} \right)$$ where $k_B$ is the Boltzmann constant and the derivative $\frac{dU}{d\beta}$ is evaluated at the inverse temperature corresponding to $T$.

For a canonical ensemble $\mathcal{C}$ at temperature $T > 0$, let $U(\beta)$ be the mean energy expressed as a function of the inverse temperature $\beta = \frac{1}{k_B T}$, where $k_B$ is the Boltzmann constant. Suppose that $U(\beta)$ is differentiable at $\beta$ (within the interval $(0, \infty)$) and that the energy variance $\text{Var}(E)$ satisfies the identity $\text{Var}(E) = -\frac{dU}{d\beta}$. Then the heat capacity at constant volume $C_V$ is given by: $$C_V = \frac{\text{Var}(E)}{k_B T^2}$$