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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 $s_i=\pm 1$, $J>0$, and a conjugate field $h$ in energy units. Set $K=J/(k_{B}T)$ and $u=h/(k_{B}T)$. Its statistical Hamiltonian and finite partition function areThe 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,The variance expression makes the finite response nonnegative, while spin inversion symmetry gives $⟨m⟩=0$ exactly at $u=0$ for every plotted $N$.

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 $⟨|m|⟩$ measures the typical magnitude, while the mean spin magnetization stays zero. The reference $k_{B}T/J=1$ 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 $m=\pm m_{\ast }$, where $m_{\ast }=\tanh (K{m}_{\ast })$. For large $N$ well below the critical temperature, their relative Boltzmann weight is approximately $e^{2Nu{m}_{\ast }}$, giving $⟨m⟩\approx m_{\ast }\tanh (Nu{m}_{\ast })$. Thus this two-peak approximation has a field width of order $1/(N{m}_{\ast })$ in $u$. It is not valid at the critical temperature, where the peaks merge; the figure uses the exact finite sum throughout.

Consider a stipulated dilute collection of independent two-level systems, each with longitudinal magnetic moment $s\mu$, $s=\pm 1$, and energy $E_s=-s\mu B_0$. The prescribed vacuum/applied magnetic flux density is $B_0={\mu }_0{H}_{\mathrm{app}}$. Ignore interactions, orbital magnetic response, and the small sample-field correction. This is the magnetic version of the existing independent spin-one-half two-level system, with alignment probabilities rather than an interacting Curie–Weiss model.

For temperature $T>0$, the canonical ensemble gives the one-moment partition function $Z_1=2\cosh x$, with $x=\mu B_0/(k_{B}T)$. Taking the expected value over its two Boltzmann weights givesHere $n$ is the number density, $k_B$ the Boltzmann constant, and $\tanh$ the hyperbolic tangent. For $|x|\ll 1$, its small-argument expansion of the hyperbolic tangent gives $M\simeq {\mu }_{0}n{\mu }^2{H}_{\mathrm{app}}/(k_{B}T)$, a Curie's law response in the applied-$H$ convention. For large $|x|$, $M$ approaches the saturation magnetization $n\mu$ with sign set by the field, and its differential magnetic susceptibility falls towards zero.

Both plots use stipulated $n=10^{26}{\mathrm{m}}^{-3}$, $\mu =9.2740101\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, and ${\mu }_0=1.25663706\times 10^{-6}\mathrm{T}\mathrm{m}{\mathrm{A}}^{-1}$. The moment is approximately one Bohr magneton; the chosen vacuum permeability is a rounded model input. At $T=2\mathrm{K}$, $M_s=927.401\mathrm{A}{\mathrm{m}}^{-1}$ and ${\chi }_{\mathrm{model}}(0)=3.91410\times 10^{-4}$. The linear approximation overestimates $M$ by less than $3\%$ for $|x|\le 0.3$, corresponding to $|{\mu }_0{H}_{\mathrm{app}}|\le 0.89324\mathrm{T}$ or $|H_{\mathrm{app}}|\le 0.71082\mathrm{MA}{\mathrm{m}}^{-1}$. These parameters keep the modeled response weak; neglected magnetic feedback is of relative order ${\chi }_{\mathrm{model}}(0)$, rather than an explanation of a material's ordered phase.

At $B_0=0$, cooling increases the zero-field slope as $1/T$. At fixed $B_0\ne 0$, cooling instead gives ${\chi }_{\mathrm{model}}\sim 4{\mu }_{0}n{\mu }^2{e}^{-2\mu |B_0|/(k_{B}T)}/(k_{B}T)$: a saturated moment can hardly change further. The order of the zero-field and zero-temperature limits matters. This independent model has no finite-temperature collective phase transition; the diverging zero-field slope alone does not establish one.

For fractal geometry, let $C_n$ be the $n$th middle-third construction of the Cantor set. It contains $2^n$ intervals of length $3^{-n}$, with total length $(2/3{)}^n$. Each finite $C_n$ contains intervals, so its Hausdorff dimension is one. The limiting set $C={\bigcap }_n{C}_n$ has zero length andIndeed, $2^n$ intervals of scale $3^{-n}$ cover it, giving the upper dimension bound from $2^n{3}^{-ns}\to 0$ whenever $s>\log 2/\log 3$. For the lower bound, assign mass $2^{-n}$ to each level-$n$ interval. Any sufficiently small interval of length $r$ meets only a bounded number of construction intervals at the comparable scale, giving a bound $\mu (I)\le K{r}^{\log 2/\log 3}$. Covering $C$ by such intervals then forces their dimension-weighted sizes to sum to at least $1/K$, yielding the matching lower bound. The noninteger dimension is deducible from a simple iterative rule even though it is absent from every finite-stage set at arbitrarily fine resolution. A finite construction nevertheless exhibits the same scaling over intermediate resolutions. Thus the order of idealization and probing ever smaller scales matters: an actual finite-resolution pattern can have useful fractal behaviour without possessing the exact limiting dimension at all scales.

For a phase transition, finite-spin partition functions are finite sums of positive exponentials at real finite temperature and field. Their free energies are therefore analytic there. Genuine thermodynamic nonanalyticity can appear only after a suitable infinite-system limit. The Curie–Weiss model gives an explicit demonstration. With $s_i=\pm 1$, takeGrouping configurations by $m=N^{-1}{\sum }_i{s}_i$ and using their binomial multiplicities gives the limiting free energy as the minimum ofThe number of possible magnetizations grows only linearly with $N$, so their largest exponential contribution determines the limiting free energy. Differentiating the variational expression givesAt $h=0$ and $\beta J>1$, its two stable minima have magnetizations $\pm m_{\ast }\ne 0$. Every finite zero-field system instead has exactly zero mean magnetization by spin-inversion symmetry. ConsequentlyThis noncommutation of limits explains how spontaneous symmetry breaking emerges. Below the critical temperature, the limiting free energy has a cusp in $h$ at zero, while finite systems have a sharp but smooth crossover. Away from the critical point, the relative weights of the two phases behave approximately as $e^{2\beta Nh{m}_{\ast }}$; the rounded magnetization is correspondingly about $m_{\ast }\tanh (\beta Nh{m}_{\ast })$. Exact singularity belongs to the idealized limit, whereas the narrow crossover and long-lived phases can be physically useful at finite size. The theoretical reduction is the derivation and its approximation control, not a claim that a finite analytic partition function is already nonanalytic.

For superselection, consider an infinite spin lattice with quasi-local observable algebra $\mathcal{A}$. These are zero-temperature equilibrium phases of a ferromagnetic Ising spin Hamiltonian with nearest-neighbour interactions $-J{\sigma }_z^{(i)}{\sigma }_z^{(j)}$. At finite $N$, the all-up and all-down vectors belong to one Hilbert space, and the coherent stateis distinguishable from the corresponding mixture by a suitably chosen global observable. But if $A_R$ acts on only $R<N$ spins, its off-diagonal matrix element between these vectors vanishes: the untouched spins contribute $⟨\uparrow |\downarrow ⟩=0$. Hence its expectations already equal those of the equal-weight mixture, independent of $\theta$. Increasingly global observables can recover the phase at finite $N$, but the infinite product of spin flips is not a norm limit of finite-support observables and is absent from $\mathcal{A}$.

In the infinite-volume limit the up and down phases have disjoint state representations. The averaged magnetization $M_N=N^{-1}{\sum }_i{\sigma }_z^{(i)}$ commutes asymptotically with any fixed local observable: $\Vert [M_N,A_R]\Vert \le 2R\Vert A_R\Vert /N$. In the phase representations its limiting values are $+1$ and $-1$, distinguishing sectors. Finite spin-flip vectors form a dense set on which these averages tend to their respective scalar values, so boundedness extends the strong limits to the phase Hilbert spaces. An intertwiner between the representations would obey $(-I)T=T(+I)$ and must therefore vanish, explaining their disjointness. In the combined representation the observable algebra is block-diagonal between them, so no quasilocal measurement detects their relative phase. This yields an emergent superselection rule relative to the chosen observable algebra. It is not a derivation of physical wave-function collapse. Fractal dimension, a thermodynamic singularity and sector separation each exhibit new limiting structure while retaining an explicit microscopic construction.

Use the positive exponent in the PDF. The TeX introduces a minus sign in the numerator while retaining the positive-sign normalizing constant, which would not define the stated probability distribution. The Curie–Weiss model assigns equal weights to $s$ and $-s$, since $({\sum }_j{s}_j{)}^2$ is unchanged by simultaneous reversal. This spin inversion symmetry pairs every configuration with one having the opposite value of $S_i$, so

Let $C_{12}=\mathbb{E}[S_1{S}_2]\ge 0$, using the supplied correlation property of the Curie–Weiss model. Since $1_{\{S_i=1\}}=(1+S_i)/2$ and each spin has zero expected value,Divide by $\mathbb{P}(S_1=1)=1/2$ to obtainThe expression assumes $n\ge 2$, as required for $S_2$ to exist. It makes the connection between the spin correlation and the requested conditional probability explicit.

If exactly $k$ spins equal $+1$, then $M=(2k-n)/n$. Therefore its possible values are $E_n$, with $k=n(1+m)/2$ for $m\in E_n$. Choosing the $k$ positive spins gives the binomial coefficientEvery such configuration has the same Curie–Weiss model weight because ${\sum }_{i,j}{s}_i{s}_j=({\sum }_i{s}_i{)}^2=n^2{m}^2$. The probability mass function of this spin magnetization is consequentlyand zero elsewhere. Grouping the original partition function by the same count also giveswhich verifies normalization without counting any configuration twice.

A standard sufficient tempering condition is this tail bound beyond a fixed radius, with $ϵ>0$ in spatial dimension $d$. It makes the distant interaction tail integrable; finite-range pair potentials satisfy it. Together with a stable pair potential and suitable growing containers, such assumptions support the conventional thermodynamic limit. A Curie–Weiss model with its size-dependent coupling uses a different prescription.

The thermodynamic limit describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and volume $V$ 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: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; 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.^[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.