= Surface correction to thermodynamic free energy
{title2=$F=Vf+O(A)$}
For finite-range interactions and bounded wall couplings, only sites within an interaction range of the wall have altered local environments. In a regular container of linear size $L$ and dimension $d$, the <surface area> scales as $A\sim L^{d-1}$ while <volume (thermodynamics)> scales as $V\sim L^d$. A boundary contribution of order $A$ therefore gives an order-$1/L$ correction to <free-energy density>. Away from a <critical phenomenon>, a regular surface contribution often has the form $F=Vf+A\gamma+\cdots$, with $\gamma$ a wall <Helmholtz free energy> per area. Critical <finite-size rounding>, edges and long-range interactions require separate care; vanishing relative boundary size does not guarantee a simple surface expansion in every regime.
An exact check uses the zero-field nearest-neighbour <Ising model> on a chain of $N\ge3$ sites with fixed coupling $J>0$ and $K=J/(k_BT)$. The open chain has $N-1$ bonds, while the periodic ring has $N$. Their finite <partition functions> and dimensionless <Helmholtz free energies> per spin $g_N=F_N/(Nk_BT)$ are
$$
Z_N^{\rm open}=2(2\cosh K)^{N-1},\qquad
Z_N^{\rm ring}=(2\cosh K)^N+(2\sinh K)^N,
$$
$$
g_\infty=-\log(2\cosh K),\qquad
g_N^{\rm open}-g_\infty=\frac{\log\cosh K}{N},\qquad
g_N^{\rm ring}-g_\infty=-\frac{\log[1+(\tanh K)^N]}{N}.
$$
For the open chain, sum the last <Ising spin> successively: each removed bond contributes $2\cosh K$, leaving a factor of two. For the ring, the <transfer matrix for the one-dimensional Ising model> has <eigenvalues> $2\cosh K$ and $2\sinh K$, and its $N$th power's <trace> gives the displayed sum. The two ends are a constant-size boundary in one dimension, so the open-chain total correction is $k_BT\log\cosh K$, and its per-spin correction is exactly proportional to $1/N$.
\Image[/ising-chain-finite-boundary-free-energy.png]
{title=Exact boundary corrections for a finite Ising chain and ring}
Both sets of finite results approach the same $g_\infty$ at the fixed plotted $K=0.8$. Multiplying the correction by $N$ exposes the open-chain boundary term, while the ring correction vanishes exponentially at fixed finite $K$. This model's <thermodynamic limit> remains a <real analytic function> at every positive <temperature>: having a <thermodynamic limit> does not require a <phase transition>.
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