= Thermodynamic limit
{wiki}
The <thermodynamic limit> describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and <volume (thermodynamics)> $V$ tend to infinity with <number density> $n=N/V$ fixed. Hold the chosen <intensive and extensive thermodynamic quantities> such as <temperature> $T$ and an applied <magnetic field> fixed along that sequence. In the <canonical ensemble> the controls are $T,N,V$; in the <grand canonical ensemble> one instead fixes $T$ and <chemical potential>, and the limiting <number density> is a result, possibly with a jump at <phase coexistence>. These are different specifications of a limiting experiment.
The useful limit is usually an <intensive and extensive thermodynamic quantities> normalization of the <Helmholtz free energy>, not a finite limit of the total <extensive quantity> itself:
$$
F_{N,V}=-k_BT\log Z_{N,V},\qquad
f(T,n)=\lim_{\substack{N,V\to\infty\\N/V=n}}\frac{F_{N,V}}{V},\qquad
a(T,n)=\lim_{\substack{N,V\to\infty\\N/V=n}}\frac{F_{N,V}}{N}=\frac{f(T,n)}n.
$$
Here $Z_{N,V}$ is the <canonical partition function>, $k_B$ is the <Boltzmann constant>, $f$ is <free-energy density>, and $a$ is <Helmholtz free energy> per particle. Thus $F_{N,V}=Vf+o(V)$ when the limit exists; finite systems need not be exactly <extensive quantities> because of the <surface correction to thermodynamic free energy>.
Specify the interactions and the growing containers too. For a conventional fluid, use a fixed <stable pair potential> with a <tempered pair potential> tail, and containers whose boundary layer of any fixed thickness occupies a vanishing fraction of the <volume (thermodynamics)>; uniformly enlarged cubes work, a tube of fixed cross-section does not give the same bulk limit. For a finite-range <Ising model>, keep the <lattice> spacing and bond coupling fixed while enlarging all intended spatial directions. Ordinary bounded wall interactions or <periodic boundary conditions> can then give the same bulk <free-energy density>, although <boundary conditions> can select different <spontaneous symmetry breaking> states. This is a set of hypotheses, not a theorem that every interaction has a boundary-independent limit.<The large deviation approach to statistical mechanics by Hugo Touchette>[§V.F, printed pp. 55–57: stable and tempered interactions and existence of the thermodynamic limit]{ref}
The <Curie–Weiss model> below deliberately uses a different limiting prescription: its all-to-all coupling is divided by $N$. Without that factor its interaction <energy> would grow as $N^2$, rather than as an <extensive quantity>. The model has no spatial wall or <correlation length> to compare with a container; it is an exactly summable illustration of <finite-size rounding>, not a quantitative model of the fluid experiment below.
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