PhyslibAlpha.ProbabilisticTheory.LocalNet.Thermodynamic

Given a thermodynamic family $F$ indexed by $i \in \iota$, a region $X$, and a local observable $a \in A(X)$, this function returns the expectation value of $a$ with respect to the finite-volume state $\omega_i$ if the region $X$ is contained within the finite volume $\Lambda_i$ (i.e., $X \le \Lambda_i$). If $X$ is not contained in $\Lambda_i$, the value is $0$. Specifically, the value is given by $\omega_i(\iota_{X \to \Lambda_i}(a))$ when $X \le \Lambda_i$, where $\iota_{X \to \Lambda_i}$ is the inclusion of the local observable into the algebra of the larger volume.

For a thermodynamic family $F$ with states $\omega_i$ on finite volumes $\Lambda_i$, let $X$ be a region and $a \in A(X)$ be a local observable. If $X$ is contained within the finite volume $\Lambda_i$ (i.e., $X \le \Lambda_i$), then the expectation value $F.\text{val}(X, a, i)$ is given by $\omega_i(\iota_{X \to \Lambda_i}(a))$, where $\iota_{X \to \Lambda_i}$ is the inclusion map of the local observable into the algebra of the larger volume $\Lambda_i$.

A state $\sigma$ of the infinite system (represented as a `NetState`) is the thermodynamic limit of a family $F$ if, for every region $X$ and every local observable $a \in A(X)$, the expectation values $F.\text{val}(X, a, i)$ converge to the expectation value $\sigma.\omega(X, a)$ along the filter $\mathcal{U}$ associated with $F$. Specifically, $\sigma$ satisfies: $$\forall X, \forall a \in A(X), \lim_{i \to \mathcal{U}} F.\text{val}(X, a, i) = \sigma.\omega(X, a)$$ where $F.\text{val}(X, a, i)$ denotes the expectation of the observable $a$ in the finite volume $\Lambda_i$ (vanishing if $X$ is not contained in $\Lambda_i$).

Let $F$ be a thermodynamic family of states on a net of observables. If $\sigma$ and $\tau$ are two states of the infinite system (net states) that both satisfy the condition of being a thermodynamic limit of $F$, then $\sigma = \tau$. Specifically, if for every region $X$ and every local observable $a \in A(X)$, the limit of the finite-volume expectation values along the associated ultrafilter $\mathcal{U}$ converges to both $\sigma(a)$ and $\tau(a)$, then the states $\sigma$ and $\tau$ are identical.

Let $F$ be a thermodynamic family of states indexed by $i \in \iota$ with an associated filter $\mathcal{U}$ on $\iota$. For any local region $X$ in the set of regions $R$, there exists a state $\phi$ in the state space $\mathcal{S}(A(X))$ of the local algebra $A(X)$ such that for every local observable $a \in A(X)$, the finite-volume expectation values converge to $\phi(a)$ along the filter $\mathcal{U}$. That is, $$\lim_{i \to \mathcal{U}} \text{val}_F(X, a, i) = \phi(a),$$ where $\text{val}_F(X, a, i)$ denotes the expectation value of the observable $a$ in the $i$-th finite-volume state.

Let $F$ be a thermodynamic family of states $(\omega_i)_{i \in I}$ on a net of observables, and let $\mathcal{U}$ be an ultrafilter on the index set $I$. There exists a state $\sigma$ of the infinite system (a net state) such that for every local region $X$ and every local observable $a \in A(X)$, the limit of the expectation values of $a$ along the ultrafilter $\mathcal{U}$ converges to the expectation value of $a$ in the state $\sigma$. That is, $$\lim_{i \to \mathcal{U}} \omega_i(a) = \sigma(a)$$ where $\omega_i(a)$ represents the expectation value of $a$ in the finite-volume state indexed by $i$ (formally defined as the value of $a$ under $\omega_i$ when the region $X$ is contained in the finite volume $\Lambda_i$).

Let $F$ be a thermodynamic family of states $(\omega_i)_{i \in I}$ defined on a net of observables $N$, and let $\mathcal{U}$ be its associated ultrafilter. The thermodynamic limit state is a specific state $\sigma$ of the infinite system (a net state) chosen such that for every local region $X$ and every local observable $a \in A(X)$, the expectation value $\sigma(a)$ is the limit of the finite-volume expectation values along the ultrafilter $\mathcal{U}$: $$\sigma(a) = \lim_{i \to \mathcal{U}} \text{val}_F(X, a, i)$$ where $\text{val}_F(X, a, i)$ is the expectation value of $a$ in the $i$-th finite-volume state.

Let $F$ be a thermodynamic family of states $(\omega_i)_{i \in I}$ and let $\mathcal{U}$ be its associated ultrafilter. For any local region $X$ and any local observable $a \in A(X)$, the expectation values of $a$ in the finite-volume states (given by $\text{val}_F(X, a, i)$) converge along the ultrafilter $\mathcal{U}$ to the expectation value of $a$ in the thermodynamic limit state $\sigma$ (denoted `F.limitState`). That is: $$\lim_{i \to \mathcal{U}} \text{val}_F(X, a, i) = \sigma(a)$$ where $\text{val}_F(X, a, i)$ is the expectation value of $a$ in the $i$-th finite-volume state $\omega_i$ if the region $X$ is contained within the volume $\Lambda_i$, and $0$ otherwise.

Let $F$ be a thermodynamic family of states defined on a net of observables $N$ with an associated ultrafilter $\mathcal{U}$. Let $\sigma$ be the thermodynamic limit state (denoted by `limitState`) of the family $F$. Then $\sigma$ is a limit of $F$ in the sense that for every local region $X$ and every local observable $a \in A(X)$, the expectation value of $a$ in the state $\sigma$, denoted $\sigma.\omega(X, a)$, is the limit of the finite-volume expectation values $F.\text{val}(X, a, i)$ along the ultrafilter $\mathcal{U}$: $$\lim_{i \to \mathcal{U}} F.\text{val}(X, a, i) = \sigma.\omega(X, a)$$

Let $N$ be a net of observables and $\sigma$ be a state of the infinite system (a net state). Suppose $F$ is a thermodynamic family of states $(\omega_i)_{i \in I}$ defined on regions $(\Lambda_i)_{i \in I}$. If for every index $i$, the local state $\omega_i$ is equal to the restriction of the global state $\sigma$ to the region $\Lambda_i$ (i.e., $\omega_i = \sigma|_{\Lambda_i}$), then the thermodynamic limit state of the family $F$ is equal to $\sigma$.

Let $F$ be a thermodynamic family of finite-volume states $(\omega_i)_{i \in I}$ on a local net of observables $N$, and let $\sigma$ be the thermodynamic limit state associated with the ultrafilter $\mathcal{U}$ on $I$. Let $\alpha$ be a symmetry action of a group $G$ on the net, where $\tau_{g,X}$ denotes the transformation mapping an observable $a$ in region $X$ to its translate in region $g \cdot X$. If for every group element $g \in G$, every local region $X$, and every local observable $a \in A(X)$, the expectation values satisfy the asymptotic invariance condition $$\lim_{i \to \mathcal{U}} \left( \text{val}_F(g \cdot X, \tau_{g,X}(a), i) - \text{val}_F(X, a, i) \right) = 0,$$ where $\text{val}_F(X, a, i)$ is the expectation value of $a$ in the $i$-th finite-volume state (defined as $\omega_i(a)$ if $X$ is contained in the $i$-th volume and $0$ otherwise), then the thermodynamic limit state $\sigma$ is invariant under the action $\alpha$.