Physlib.StatisticalMechanics.CanonicalEnsemble.Basic

For any canonical ensemble $\mathcal{C}$, the reference measure $\mu$ associated with the microstate space is $\sigma$-finite.

Two canonical ensembles $\mathcal{C}$ and $\mathcal{C}'$ defined over the same microstate space $\iota$ are identical if their defining components are equal. Specifically, if they have the same energy function ($E = E'$), the same number of degrees of freedom ($d = d'$), the same phase space unit ($h = h'$), and the same underlying measure ($\mu = \mu'$), then $\mathcal{C} = \mathcal{C}'$.

For any canonical ensemble $\mathcal{C}$, the energy function $E$ (mapping microstates to real values) is a measurable function with respect to the underlying measurable space of the microstates.

Given two canonical ensembles $\mathcal{C}_1$ and $\mathcal{C}_2$ on microstate spaces $\iota_1$ and $\iota_2$ respectively, their sum $\mathcal{C}_1 + \mathcal{C}_2$ is a canonical ensemble on the product space $\iota_1 \times \iota_2$ defined by: - The energy function $E: \iota_1 \times \iota_2 \to \mathbb{R}$ is the sum of the individual energy functions: $E(i_1, i_2) = E_1(i_1) + E_2(i_2)$. - The total number of degrees of freedom $d$ is the sum of the individual degrees of freedom: $d = d_1 + d_2$. - The underlying measure $\mu$ on $\iota_1 \times \iota_2$ is the product measure $\mu_1 \times \mu_2$. - The phase space unit $h$ (the semi-classical normalization constant) is inherited from the first ensemble $\mathcal{C}_1$. Note that this composition is physically meaningful primarily when both systems share the same phase space unit.

The canonical ensemble where the set of microstates is the empty set $\emptyset$. For this ensemble, the energy function is the unique function from the empty set, the number of degrees of freedom is $0$, and the microstate measure $\mu$ is the zero measure.

Given a canonical ensemble $\mathcal{C}$ on a state space $\iota$ and a measurable equivalence $e : \iota_1 \simeq \iota$, the function constructs a new canonical ensemble on $\iota_1$. - The energy function $E_1$ on $\iota_1$ is defined by the composition $E_1 = E \circ e$, where $E$ is the energy function of $\mathcal{C}$. - The measure $\mu_1$ on $\iota_1$ is the push-forward of the measure $\mu$ of $\mathcal{C}$ under the inverse equivalence $e^{-1}$ (i.e., $\mu_1 = (e^{-1})_* \mu$). - The physical constants, including the number of degrees of freedom $d$ and the phase space unit $h$, remain identical to those in $\mathcal{C}$.

Let $\mathcal{C}$ be a canonical ensemble on a state space $\iota$ with energy function $E : \iota \to \mathbb{R}$. Given a measurable equivalence $e : \iota_1 \simeq \iota$, let $E_1$ be the energy function of the canonical ensemble transported to the state space $\iota_1$ via $e$ (denoted as `𝓒.congr e`). Then the composition of $E_1$ with the inverse equivalence $e^{-1} : \iota \simeq \iota_1$ satisfies $E_1 \circ e^{-1} = E$.

Given a natural number $n$ and a canonical ensemble $\mathcal{C}$ defined on a state space $\iota$, the function constructs a new canonical ensemble representing $n$ non-interacting, distinguishable copies of $\mathcal{C}$. The state space of the resulting ensemble is the product space $\iota^n$, modeled as the type of functions $\text{Fin } n \to \iota$. - The energy function $E$ for the composite system is the sum of the energies of the individual components: $E(f) = \sum_{i=0}^{n-1} E_{\mathcal{C}}(f_i)$ for any configuration $f$. - The total number of degrees of freedom is $n \cdot \text{dof}(\mathcal{C})$. - The reference measure is the product measure $\mu^{\otimes n}$ (the product of $n$ copies of the measure $\mu$ of $\mathcal{C}$). - The phase space unit $h$ remains the same as in the original ensemble $\mathcal{C}$.

For a canonical ensemble $\mathcal{C}$ defined over a type $\iota$, the microstates are the set of all possible configurations or states of the physical system, represented by the type $\iota$.

For any two canonical ensembles $\mathcal{C}_1$ and $\mathcal{C}_2$, the number of degrees of freedom of the composite ensemble $\mathcal{C}_1 + \mathcal{C}_2$ is equal to the sum of the degrees of freedom of the individual ensembles: $$\text{dof}(\mathcal{C}_1 + \mathcal{C}_2) = \text{dof}(\mathcal{C}_1) + \text{dof}(\mathcal{C}_2)$$

Given two canonical ensembles $\mathcal{C}_1$ and $\mathcal{C}_2$, the phase space unit of their sum $\mathcal{C}_1 + \mathcal{C}_2$ is equal to the phase space unit of the first ensemble $\mathcal{C}_1$.

For a canonical ensemble $\mathcal{C}$ and a natural number $n$, let $n \cdot \mathcal{C}$ denote the ensemble representing $n$ non-interacting, distinguishable copies of $\mathcal{C}$. Then the number of degrees of freedom of this composite ensemble is equal to $n$ times the number of degrees of freedom of the individual ensemble $\mathcal{C}$, that is, $\text{dof}(n \cdot \mathcal{C}) = n \cdot \text{dof}(\mathcal{C})$.

For any natural number $n$ and any canonical ensemble $\mathcal{C}$, the phase space unit of the ensemble formed by $n$ distinguishable, non-interacting copies of $\mathcal{C}$ is equal to the phase space unit of the original ensemble $\mathcal{C}$.

For a canonical ensemble $\mathcal{C}$ and a measurable equivalence $e : \iota_1 \simeq \iota$ between state spaces, the number of degrees of freedom of the ensemble constructed by transporting $\mathcal{C}$ along $e$ (denoted as $\mathcal{C}.\text{congr } e$) is equal to the number of degrees of freedom of the original ensemble $\mathcal{C}$.

Let $\mathcal{C}$ be a canonical ensemble. For any measurable equivalence $e : \iota_1 \simeq \iota$ between state spaces, the phase space unit $h$ of the ensemble transported along $e$ is equal to the phase space unit of the original ensemble $\mathcal{C}$.

For two canonical ensembles $\mathcal{C}$ and $\mathcal{C}_1$ with respective measures $\mu$ and $\mu_1$, the measure of their composite ensemble $\mathcal{C} + \mathcal{C}_1$ is given by the product measure $\mu \times \mu_1$.

Let $\mathcal{C}$ be a canonical ensemble with reference measure $\mu$. For any natural number $n$, the reference measure of the ensemble consisting of $n$ distinguishable, non-interacting copies of $\mathcal{C}$ is equal to the $n$-fold product measure $\prod_{i=1}^n \mu$ (also denoted $\mu^{\otimes n}$).

For a canonical ensemble $\mathcal{C}$, the reference measure $\mu$ of the composite ensemble consisting of $0$ copies of $\mathcal{C}$ is equal to the product measure over the empty index set $\text{Fin } 0$.

For any two canonical ensembles $\mathcal{C}$ and $\mathcal{C}_1$, the energy of a microstate $i = (i_1, i_2)$ in the composite ensemble $\mathcal{C} + \mathcal{C}_1$ is the sum of the energies of the individual microstates: $E_{\mathcal{C} + \mathcal{C}_1}(i) = E_{\mathcal{C}}(i_1) + E_{\mathcal{C}_1}(i_2)$.

For a canonical ensemble $\mathcal{C}$ and a natural number $n$, let $n\mathcal{C}$ represent the composite canonical ensemble consisting of $n$ non-interacting, distinguishable copies of $\mathcal{C}$. For any microstate $f$ of this composite system—which is a collection of microstates $(f_0, f_1, \dots, f_{n-1})$ where each $f_i$ belongs to the original ensemble $\mathcal{C}$—the total energy $E_{n\mathcal{C}}(f)$ is the sum of the energies of the individual components: $$E_{n\mathcal{C}}(f) = \sum_{i=0}^{n-1} E_{\mathcal{C}}(f_i)$$

Let $\mathcal{C}$ be a canonical ensemble defined on a state space $\iota$ with an energy function $E$. Given a measurable equivalence $e : \iota_1 \simeq \iota$, the energy of a microstate $i \in \iota_1$ in the transported canonical ensemble $\mathcal{C}.\text{congr } e$ is equal to the energy of $e(i)$ in the original ensemble. That is, $E_{\text{congr}(e)}(i) = E_{\mathcal{C}}(e(i))$.

Let $\mathcal{C}$ be a canonical ensemble on a state space $\iota$ with a $\sigma$-finite reference measure $\mu$. For any natural number $n$, the ensemble $(n+1)\mathcal{C}$, representing $n+1$ non-interacting and distinguishable copies of $\mathcal{C}$, is equal to the ensemble formed by the sum $\mathcal{C} + n\mathcal{C}$ transported along the measurable equivalence $e: \iota^{n+1} \simeq \iota \times \iota^n$. This equivalence $e$ is defined by the mapping $f \mapsto (f(0), f \circ \text{succ})$, which identifies a sequence of $n+1$ microstates with a pair consisting of the first microstate and the remaining $n$ microstates.

Consider two canonical ensembles $\mathcal{C}$ and $\mathcal{C}_1$ with associated reference measures $\mu$ and $\mu_1$ respectively. If both $\mu$ and $\mu_1$ are non-zero measures, then the measure associated with the composite canonical ensemble $\mathcal{C} + \mathcal{C}_1$ is also non-zero.

Let $\mathcal{C}$ be a canonical ensemble on a state space $\iota$ with an associated measure $\mu$. If $\mu$ is not the zero measure, then for any measurable equivalence $e : \iota_1 \simeq \iota$, the measure of the transported canonical ensemble $\mathcal{C}.\text{congr}(e)$ on $\iota_1$ is also not the zero measure.

Consider a canonical ensemble $\mathcal{C}$ with an associated reference measure $\mu$. If $\mu$ is a non-zero measure, then for any natural number $n$, the reference measure of the composite canonical ensemble representing $n$ non-interacting, distinguishable copies of $\mathcal{C}$ is also non-zero.

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$, and given a temperature $T$ with corresponding inverse temperature $\beta = \frac{1}{k_B T}$, the Boltzmann measure $\mu_{\text{Bolt}}$ is the measure on $\iota$ defined by the density function $f(i) = e^{-\beta E(i)}$ with respect to $\mu$. Mathematically, for any measurable set $A \subseteq \iota$, the measure is given by: $$\mu_{\text{Bolt}}(A) = \int_A e^{-\beta E(i)} \, d\mu(i)$$

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E$, the Boltzmann measure $\mu_{\text{Bolt}}$ at temperature $T$ (defined by the density $e^{-\beta E}$ with respect to $\mu$, where $\beta = (k_B T)^{-1}$) is a $\sigma$-finite measure.

For a given temperature $T$, let $\mathcal{C}$ and $\mathcal{C}_1$ be two canonical ensembles. The Boltzmann measure $\mu_{\text{Bolt}}$ of the composite ensemble $\mathcal{C} + \mathcal{C}_1$ (defined on the product of their microstate spaces) is equal to the product measure of the individual Boltzmann measures of $\mathcal{C}$ and $\mathcal{C}_1$ at temperature $T$. Mathematically, this is expressed as: $$(\mathcal{C} + \mathcal{C}_1).\mu_{\text{Bolt}}(T) = \mathcal{C}.\mu_{\text{Bolt}}(T) \times \mathcal{C}_1.\mu_{\text{Bolt}}(T)$$ where the Boltzmann measure for an ensemble with energy $E$ and base measure $\mu$ is defined by the density $e^{-\beta E}$ with respect to $\mu$, and $\beta = (k_B T)^{-1}$.

Let $\mathcal{C}$ be a canonical ensemble on a state space $\iota$ with energy function $E$ and base measure $\mu$. Given a measurable equivalence $e : \iota_1 \simeq \iota$, let $\mathcal{C}_{\text{congr}(e)}$ be the transported canonical ensemble on $\iota_1$, which has energy function $E \circ e$ and base measure $(e^{-1})_* \mu$. For any temperature $T$ with inverse temperature $\beta = (k_B T)^{-1}$, the Boltzmann measure $\mu_{\text{Bolt}}$ of the transported ensemble is equal to the push-forward of the Boltzmann measure of the original ensemble under the inverse equivalence $e^{-1}$: $$\mu_{\text{Bolt}}(\mathcal{C}_{\text{congr}(e)}) = (e^{-1})_* \mu_{\text{Bolt}}(\mathcal{C})$$ where $(e^{-1})_* \nu$ denotes the push-forward of a measure $\nu$ along the map $e^{-1}$.

Let $\mathcal{C}$ be a canonical ensemble on a state space $\iota$ with a $\sigma$-finite reference measure $\mu$. For any natural number $n$, let $n\mathcal{C}$ denote the canonical ensemble representing $n$ non-interacting, distinguishable copies of $\mathcal{C}$ defined on the product state space $\iota^n$ (the space of functions $\{0, \dots, n-1\} \to \iota$). For a given temperature $T$, the Boltzmann measure $\mu_{\text{Bolt}}$ of the composite ensemble $n\mathcal{C}$ is equal to the $n$-fold product measure of the individual Boltzmann measure of $\mathcal{C}$ at temperature $T$. Mathematically, this is expressed as: $$(n\mathcal{C}).\mu_{\text{Bolt}}(T) = \bigotimes_{i=0}^{n-1} \mathcal{C}.\mu_{\text{Bolt}}(T)$$ where the Boltzmann measure for an ensemble with energy $E$ and base measure $\mu$ is defined by the density $e^{-\beta E}$ with respect to $\mu$, and $\beta = (k_B T)^{-1}$.

For a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E : \iota \to \mathbb{R}$, if the base measure $\mu$ is not the zero measure, then for any temperature $T$ with corresponding inverse temperature $\beta = \frac{1}{k_B T}$, the Boltzmann measure $\mu_{\text{Bolt}}$ is also not the zero measure.

Given a canonical ensemble $\mathcal{C}$ with base measure $\mu$ on the space of microstates, if $\mu$ is not the zero measure, then for any temperature $T$, the resulting Boltzmann measure $\mu_{\text{Bolt}}$ is also not the zero measure.

For two canonical ensembles $\mathcal{C}$ and $\mathcal{C}_1$ at a temperature $T$, if the Boltzmann measures $\mu_{\text{Bolt}, \mathcal{C}}(T)$ and $\mu_{\text{Bolt}, \mathcal{C}_1}(T)$ are both finite, then the Boltzmann measure of the composite ensemble $\mathcal{C} + \mathcal{C}_1$ (defined on the product of their microstate spaces) is also finite. This is equivalent to stating that if the individual mathematical partition functions $Z_{\text{math}, \mathcal{C}}(T)$ and $Z_{\text{math}, \mathcal{C}_1}(T)$ are finite, then the mathematical partition function of the combined system is also finite.

Let $\mathcal{C}$ be a canonical ensemble. If the Boltzmann measure $\mu_{\text{Bolt}}$ at temperature $T$ is a finite measure, then for any natural number $n$, the Boltzmann measure of the ensemble consisting of $n$ distinguishable, non-interacting copies of $\mathcal{C}$ is also a finite measure. This is equivalent to stating that if the mathematical partition function $Z_{\text{math}}(T)$ of the original system is finite, then the mathematical partition function of the $n$-copy system is also finite.

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ with base measure $\mu$ and energy function $E: \iota \to \mathbb{R}$, and given a temperature $T$ with corresponding inverse temperature $\beta = \frac{1}{k_B T}$, the **mathematical partition function** is defined as the total mass of the microstate space under the Boltzmann measure $\mu_{\text{Bolt}}$. Mathematically, it is expressed as the integral of the Boltzmann factor over the entire space $\iota$: $$Z_{\text{math}}(T) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$$ This quantity represents the raw normalization factor for the probability density and may carry physical dimensions.

For a canonical ensemble $\mathcal{C}$ defined on a space of microstates $\iota$ with base measure $\mu$ and energy function $E$, the mathematical partition function $Z_{\text{math}}$ at temperature $T$ is equal to the integral of the Boltzmann factor over the microstate space: $$Z_{\text{math}}(T) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$$ where $\beta$ is the inverse temperature associated with $T$.

For any temperature $T$, the mathematical partition function $Z_{\text{math}}$ of the composite canonical ensemble $\mathcal{C} + \mathcal{C}_1$ (defined on the product space of microstates with additive energy) is equal to the product of the mathematical partition functions of the individual ensembles $\mathcal{C}$ and $\mathcal{C}_1$. Mathematically: $$Z_{\text{math}, \mathcal{C} + \mathcal{C}_1}(T) = Z_{\text{math}, \mathcal{C}}(T) \cdot Z_{\text{math}, \mathcal{C}_1}(T)$$ where $Z_{\text{math}}(T) = \int e^{-\beta E} \, d\mu$.

For any canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ and a measurable equivalence $e : \iota_1 \simeq \iota$, the mathematical partition function $Z_{\text{math}}$ of the ensemble transported along $e$ is equal to the mathematical partition function of the original ensemble $\mathcal{C}$ for any temperature $T$: $$Z_{\text{math}, \mathcal{C} \circ e}(T) = Z_{\text{math}, \mathcal{C}}(T)$$ where the mathematical partition function is defined as the integral of the Boltzmann factor over the microstate space, $Z_{\text{math}}(T) = \int e^{-\beta E} \, d\mu$.

For a canonical ensemble $\mathcal{C}$ and a natural number $n$, let $n\mathcal{C}$ denote the ensemble representing $n$ distinguishable, non-interacting copies of $\mathcal{C}$ (defined on the product state space $\iota^n$). For any temperature $T$, the mathematical partition function $Z_{\text{math}}$ of the composite ensemble $n\mathcal{C}$ is equal to the $n$-th power of the mathematical partition function of the individual ensemble $\mathcal{C}$. Mathematically: $$Z_{\text{math}, n\mathcal{C}}(T) = (Z_{\text{math}, \mathcal{C}}(T))^n$$ where the mathematical partition function is defined as $Z_{\text{math}}(T) = \int e^{-\beta E} \, d\mu$.

For any canonical ensemble $\mathcal{C}$ and temperature $T$, the mathematical partition function $Z_{\text{math}}(T)$, defined as the integral of the Boltzmann factor $e^{-\beta E(i)}$ over the space of microstates, is non-negative: $$Z_{\text{math}}(T) \ge 0$$

For a canonical ensemble $\mathcal{C}$ with base measure $\mu$, energy function $E$, and temperature $T$ (with $\beta = \frac{1}{k_B T}$), assume that the Boltzmann measure $\mu_{\text{Bolt}}$, defined by the density $e^{-\beta E(i)}$ with respect to $\mu$, is a finite measure. Then, the mathematical partition function $Z_{\text{math}}(T) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$ is equal to zero if and only if the base measure $\mu$ is the zero measure.

For a canonical ensemble $\mathcal{C}$ with a space of microstates $\iota$, base measure $\mu$, and energy function $E$, let $\beta \geq 0$ be the inverse temperature. The mathematical partition function $Z_{\text{math}}$ evaluated at the temperature $T$ corresponding to $\beta$ is equal to the total mass of the microstate space under the Boltzmann measure. Mathematically, this is expressed as the integral of the Boltzmann factor over the entire space: $$Z_{\text{math}}(T(\beta)) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$$ where $T(\beta)$ is the temperature associated with inverse temperature $\beta$.

For a canonical ensemble $\mathcal{C}$ with a space of microstates $\iota$, base measure $\mu$, and energy function $E$, let $T$ be a given temperature with inverse temperature $\beta = \frac{1}{k_B T}$. If the Boltzmann measure $\mu_{\text{Bolt}}$ (defined by the density $e^{-\beta E(i)}$ with respect to $\mu$) is a finite measure and the base measure $\mu$ is non-zero, then the mathematical partition function is strictly positive: $$Z_{\text{math}}(T) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i) > 0$$

Given a canonical ensemble $\mathcal{C}$ defined over a space of microstates $\iota$ with an energy function $E: \iota \to \mathbb{R}$, and a temperature $T$, the **probability density** (or probability mass function in the discrete case) for a microstate $i \in \iota$ is defined by: $$\rho(i) = \frac{e^{-\beta E(i)}}{Z_{\text{math}}(T)}$$ where $\beta = \frac{1}{k_B T}$ is the inverse temperature and $Z_{\text{math}}(T)$ is the mathematical partition function. This value represents the density of the canonical probability measure with respect to the underlying base measure $\mu$.

For a given temperature $T$ and a microstate $i = (i_1, i_2)$ belonging to the product microstate space of the composite canonical ensemble $\mathcal{C} + \mathcal{C}_1$, the probability density $\rho$ of the composite system is the product of the probability densities of the individual systems evaluated at their respective microstates: $$\rho_{\mathcal{C} + \mathcal{C}_1}(T, (i_1, i_2)) = \rho_{\mathcal{C}}(T, i_1) \cdot \rho_{\mathcal{C}_1}(T, i_2)$$ where the probability density for an ensemble is defined as $\rho(T, i) = \frac{e^{-\beta E(i)}}{Z_{\text{math}}(T)}$.

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ and a measurable equivalence $e : \iota_1 \simeq \iota$, let $\mathcal{C} \circ e$ (formally `𝓒.congr e`) denote the canonical ensemble transported along $e$ to the space $\iota_1$. For any temperature $T$ and microstate $i \in \iota_1$, the probability density $\rho$ of the transported ensemble evaluated at $i$ is equal to the probability density of the original ensemble evaluated at $e(i)$: $$\rho_{\mathcal{C} \circ e}(T, i) = \rho_{\mathcal{C}}(T, e(i))$$ where the probability density is defined as $\rho(T, i) = \frac{e^{-\beta E(i)}}{Z_{\text{math}}(T)}$, with $\beta = \frac{1}{k_B T}$ being the inverse temperature and $Z_{\text{math}}(T)$ being the mathematical partition function.

For a canonical ensemble $\mathcal{C}$ and a natural number $n$, let $n\mathcal{C}$ (formally `nsmul n 𝓒`) denote the ensemble representing $n$ distinguishable, non-interacting copies of $\mathcal{C}$ defined on the product state space $\iota^n$. For any temperature $T$ and any configuration $f \in \iota^n$ (modeled as a function $f: \{0, \dots, n-1\} \to \iota$), the probability density $\rho$ of the composite ensemble evaluated at $f$ is the product of the probability densities of the individual systems evaluated at each component microstate $f_i$: $$\rho_{n\mathcal{C}}(T, f) = \prod_{i=0}^{n-1} \rho_{\mathcal{C}}(T, f_i)$$ where the probability density $\rho$ for an ensemble is defined as $\rho(T, s) = \frac{e^{-\beta E(s)}}{Z_{\text{math}}(T)}$, with $\beta = \frac{1}{k_B T}$ being the inverse temperature and $Z_{\text{math}}(T)$ being the mathematical partition function.

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ and a temperature $T$, the probability measure $\mu_{\text{Prod}}$ is the normalized Boltzmann measure on $\iota$. It is defined by rescaling the Boltzmann measure $\mu_{\text{Bolt}}$ by the inverse of the total mass of the system (the mathematical partition function $Z_{\text{math}}$): $$\mu_{\text{Prod}} = \frac{1}{Z_{\text{math}}} \mu_{\text{Bolt}}$$ where $Z_{\text{math}} = \mu_{\text{Bolt}}(\iota) = \int_{\iota} e^{-\beta E(i)} \, d\mu(i)$, $\beta = \frac{1}{k_B T}$ is the inverse temperature, and $E(i)$ is the energy of microstate $i$.

Given a canonical ensemble $\mathcal{C}$ on a space of microstates $\iota$ and a temperature $T$, the canonical probability measure $\mu_{\text{Prod}}$ (the normalized Boltzmann measure) is a $\sigma$-finite measure.

For a canonical ensemble $\mathcal{C}$ at temperature $T$, if the Boltzmann measure $\mu_{\text{Bolt}}$ is a finite measure and the underlying base measure $\mu$ is non-zero, then the canonical probability measure $\mu_{\text{Prod}}$ is a probability measure (i.e., its total mass is 1).

For a canonical ensemble $\mathcal{C}$ at a given temperature $T$, the canonical probability measure $\mu_{\text{Prod}}(T)$ is a finite measure.

For a given temperature $T$ and two canonical ensembles $\mathcal{C}$ and $\mathcal{C}_1$, if the Boltzmann measures $\mu_{\text{Bolt}}$ of both ensembles are finite at temperature $T$, then the canonical probability measure $\mu_{\text{Prod}}$ of the composite ensemble $\mathcal{C} + \mathcal{C}_1$ (defined on the product of their microstate spaces) is equal to the product measure of the individual probability measures of $\mathcal{C}$ and $\mathcal{C}_1$. Mathematically, this is expressed as: $$(\mathcal{C} + \mathcal{C}_1).\mu_{\text{Prod}}(T) = \mathcal{C}.\mu_{\text{Prod}}(T) \times \mathcal{C}_1.\mu_{\text{Prod}}(T)$$ where $\mu_{\text{Prod}}$ is the normalized Boltzmann measure $\frac{1}{Z_{\text{math}}} \mu_{\text{Bolt}}$.

Let $\mathcal{C}$ be a canonical ensemble on a space of microstates $\iota$ with a canonical probability measure $\mu_{\text{Prod}}$. Given a measurable equivalence $e : \iota_1 \simeq \iota$, let $\mathcal{C}_{\text{congr}(e)}$ be the transported canonical ensemble on $\iota_1$. For any temperature $T$, the canonical probability measure of the transported ensemble is equal to the push-forward of the canonical probability measure of the original ensemble under the inverse map $e^{-1}$: $$\mu_{\text{Prod}}(\mathcal{C}_{\text{congr}(e)}) = (e^{-1})_* \mu_{\text{Prod}}(\mathcal{C})$$ where $(e^{-1})_* \nu$ denotes the push-forward of a measure $\nu$ along the mapping $e^{-1} : \iota \to \iota_1$.

Let $\mathcal{C}$ be a canonical ensemble defined on a space of microstates $\iota$. If the Boltzmann measure $\mu_{\text{Bolt}}$ at temperature $T$ is finite (which ensures that the mathematical partition function $Z_{\text{math}}$ is well-defined), then the canonical probability measure $\mu_{\text{Prod}}$ of the system consisting of $n$ distinguishable, non-interacting copies of $\mathcal{C}$ (denoted $n\mathcal{C}$) is the product measure of $n$ copies of the probability measure of $\mathcal{C}$. Mathematically, on the product state space $\iota^n$, the measure is given by: $$(n\mathcal{C}).\mu_{\text{Prod}}(T) = \prod_{i=0}^{n-1} \mathcal{C}.\mu_{\text{Prod}}(T)$$ where $\mu_{\text{Prod}} = \frac{1}{Z_{\text{math}}} \mu_{\text{Bolt}}$.

Let $\mathcal{C}$ and $\mathcal{C}_1$ be two canonical ensembles at temperature $T$. Suppose that the Boltzmann measures $\mu_{\text{Bolt}}$ and $\mu_{\text{Bolt}, 1}$ for both ensembles are finite at this temperature. If the energy function $E$ of $\mathcal{C}$ is integrable with respect to its canonical probability measure $\mu_{\text{Prod}}$, and the energy function $E_1$ of $\mathcal{C}_1$ is integrable with respect to its canonical probability measure $\mu_{\text{Prod}, 1}$, then the energy function of the composite ensemble $\mathcal{C} + \mathcal{C}_1$ (defined by the sum of the individual energies) is integrable with respect to its canonical probability measure $(\mathcal{C} + \mathcal{C}_1).\mu_{\text{Prod}}(T)$.