What is the definition of the thermodynamic limit of a thermodynamic quantity?

Question

Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem:

Theorem: In the thermodynamic limit, the pressure: $$\psi(\beta,h) := \lim_{\Lambda \uparrow \mathbb{Z}^{d}}\psi_{\Lambda}^{\#}(\beta, h) $$ is well-defined and independent of the sequence $\Lambda \uparrow \mathbb{Z}^{d}$ and of the type of the boundary condition $\#$.

Here, I'm using the same notation and conventions from chapter 3 of Velenik and Friedli's book. The notation $\Lambda \uparrow \mathbb{Z}^{d}$ stands for the convergence in the sense of Van Hove.

Definition [Convergence in the sense of Van Hove] A sequence $\{\Lambda_{n}\}_{n\in \mathbb{N}}$ of (finite) subsets of $\mathbb{Z}^{d}$ is said to converge to $\mathbb{Z}^{d}$ in the sense of Van Hove if all three properties listed below are satisfied:

(1) $\{\Lambda_{n}\}_{n\in \mathbb{N}}$ is an increasing sequence of subsets.

(2) $\bigcup_{n\in \mathbb{N}}\Lambda_{n} = \mathbb{Z}^{d}$

(3) $\lim_{n\to \infty}\frac{|\partial^{in}\Lambda_{n}|}{|\Lambda_{n}|} = 0$, where $|X|$ denotes the cardinality of the set $X$ and $\partial^{in}\Lambda:=\{i\in \Lambda: \hspace{0.1cm} \exists j \in\Lambda^{c} \hspace{0.1cm} \mbox{with} \hspace{0.1cm} |i-j|=1 \}$

My point here is the following. Convergence in the sense of Van Hove is a notion of convergence of sets, not functions of sets. But what does $\lim_{\Lambda\uparrow \mathbb{Z}^{d}}\psi^{\#}_{\Lambda}(\beta, h)$ mean?

Answer 1

It means that if you consider any sequence of sets $(\Lambda_n)_{n\in\mathbb{N}}$ converging to $\mathbb{Z}^d$ in the sense of van Hove, then the sequence of numbers $(\psi_{\Lambda_n}^\#(\beta,h))_{n\in\mathbb{N}}$ is convergent. (Moreover, the theorem claims that the limit is independent of the sequence chosen and that the limiting function has nice properties.)

(It may be more natural to consider nets rather than sequences, but we refrained from that generality in the book since this was not necessary in the cases we discuss.)