= Finite-size rounding
<Finite-size rounding> replaces a limiting discontinuity or divergence by a finite crossover or peak. For a spatial system near a <continuous phase transition>, the bulk description ceases to apply unchanged when the <correlation length> becomes comparable with the sample size. How the crossover width and location depend on size also depends on the model, geometry and <boundary conditions>; there is no universal $1/N$ rule for every <phase transition>.
An exact finite example is the <Curie–Weiss model> with $s_i=\pm1$, $J>0$, and a conjugate field $h$ in <energy> units. Set $K=J/(k_BT)$ and $u=h/(k_BT)$. Its <statistical Hamiltonian> and finite <partition function> are
$$
H_N=-\frac{J}{2N}\left(\sum_i s_i\right)^2-h\sum_i s_i,\qquad
Z_N(K,u)=\sum_{k=0}^N\binom Nk\exp\left[\frac{K}{2N}(2k-N)^2+u(2k-N)\right].
$$
The <binomial coefficient> counts the <microstates> with $k$ positive <Ising spins>. The diagonal terms in the square add a configuration-independent constant to the <energy>; retaining them fixes the normalization used here. Divide each summand by $Z_N$ to obtain the <probability mass function> of $m_k=(2k-N)/N$. In particular,
$$
\langle m\rangle=\frac1N\partial_u\log Z_N,\qquad
\partial_u\langle m\rangle=N\operatorname{Var}(m).
$$
The <variance> expression makes the finite response nonnegative, while <spin inversion symmetry> gives $\langle m\rangle=0$ exactly at $u=0$ for every plotted $N$.
\Image[/curie-weiss-finite-size-rounding.png]
{title=Exact finite Curie–Weiss rounding in field and temperature}
Every curve is computed from the finite <partition function>, without selecting one minimum of the <Bragg-Williams free energy of the Ising model>. The field panel holds $K=1.5$ fixed and shows increasingly rapid switching between opposite <spin magnetizations>. The <temperature> panel uses $u=0$: its nonzero $\langle|m|\rangle$ measures the typical magnitude, while the mean <spin magnetization> stays zero. The reference $k_BT/J=1$ marks the limiting <critical temperature>, not a singular point of any finite curve.
A short limiting argument explains the field panel. Combining the <binomial coefficient> with its leading <entropy> gives a <Helmholtz free energy> per spin whose stationarity equation is $m=\tanh(Km+u)$. For $K>1$ and $u=0$ the two ordered minima have $m=\pm m_*$, where $m_*=\tanh(Km_*)$. For large $N$ well below the <critical temperature>, their relative <Boltzmann weight> is approximately $e^{2Num_*}$, giving $\langle m\rangle\approx m_*\tanh(Num_*)$. Thus this two-peak approximation has a field width of order $1/(Nm_*)$ in $u$. It is not valid at the <critical temperature>, where the peaks merge; the figure uses the exact finite sum throughout.
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