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](https://www.unige.ch/math/folks/velenik/smbook/Ising_Model.pdf) 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?



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


