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

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?

• Just an aside: for certain stat-mech models thermodynamic quantities are known to depend on the choice of boundary conditions, e.g the free energy of the six-vertex model (which has d=2) with periodic vs domain-wall boundaries. I believe that the conditions of the definition are met; the point is that the local (divergencelessness) constraint of the model causes the choice of boundary conditions to influence a macroscopic portion of the lattice. – Jules Lamers Jul 22 '20 at 9:54

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.)