= Finite-system analyticity
{title2=$F_N=-k_BT\log Z_N$}
For finitely many <microstates> with finite <energies> $E_\alpha(h)$ that are <real analytic functions> of a real control $h$, the <canonical partition function> at $T>0$ is a finite positive sum:
$$
Z_N(T,h)=\sum_\alpha e^{-E_\alpha(h)/(k_BT)}>0.
$$
It and the resulting <Helmholtz free energy> are locally <real analytic functions> of $T$ and $h$. Hence such a finite <Ising model> has no literal nonanalytic equilibrium <phase transition>. A limit of these <real analytic functions> can nevertheless fail to be a <real analytic function>; the narrow but smooth crossover can become a sharp <phase transition>.<Statistical Field Theory by David Tong/1 From Spins to Fields>[§1.2.1, discussion following equation (1.14): finite sums, analyticity, and macroscopic sharpness]{ref}
Finite particle number alone is not the hypothesis of a finite sum. Classical <phase space> is continuous, and a system of <quantum states> can have infinitely many <energy eigenvalues>: its <partition function> requires convergence and local bounds that permit differentiation or an analytic extension of the integral or <thermal trace>. A divergent <partition function>, a singularly changed <statistical Hamiltonian>, or the zero-<temperature> limit falls outside the finite-positive-sum argument. Observing a steep change in a finite sample therefore motivates a limiting model; it does not measure an actual infinity.
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