Finite-size scaling and data collapse

Suppose a conventional short-range continuous transition has bulk correlation length $\xi\sim|t|^{-\nu}$. A sample of linear size L cannot behave as an infinite collection of independent correlated regions once xi is comparable with L. The crossover condition gives a rounding window of order $L^{-1/\nu}$. The relevant size is linear extent, not total site count; for a d-dimensional cubic sample, N grows as $L^d$.

The usual finite-size ansatz replaces a bulk divergence by a power of L times a scaling function of $tL^{1/\nu}$. Boundary conditions, shape, and normalization conventions must be kept consistent across the size sequence. A susceptibility peak is a finite-size pseudocritical feature, and its location can shift with L. It should not be identified automatically with the infinite-system transition temperature or assigned a universal shift amplitude.

For a scalar reversal-symmetric order parameter, define a Binder quantity from its second and fourth moments. The leading magnetization scale cancels in the ratio. With the convention used here, a narrow symmetric pair of peaks gives two thirds and a centred Gaussian gives zero. Some references multiply this definition by three halves to make the ordered limit one; numerical crossing values cannot be compared without translating that convention.

At an ordinary critical point, a dimensionless Binder ratio or xi divided by L can approach a size-independent value for fixed geometry. Crossings for successive size pairs therefore help estimate the critical temperature. Irrelevant operators generate corrections such as $L^{-\omega}$, causing crossing drift and imperfect overlap. The limiting ratio may be universal within a specified shape and boundary class, but it is not one number common to every geometry or every order-parameter dimension.

Choose the critical temperature and exponents from a physically justified model, then plot the scaled ordinate against the scaled thermal variable for all sizes. Test whether a single smooth scaling function describes statistically compatible data. Correlated Monte Carlo measurements need appropriate uncertainty estimates, and uncertainty in the critical temperature propagates into the horizontal coordinates. Agreement for one or two sizes is weak evidence because correction terms and flexible fitting functions can absorb systematic errors.

Repeat the analysis while removing the smallest sizes, narrowing the thermal window, and including justified correction terms. Compare other observables and dimensionless crossings rather than optimizing visual overlap alone. Above the upper critical dimension, dangerous irrelevant variables can modify the naive size powers; at a BKT transition logarithmic corrections and an essential correlation length require a different scaling treatment. First-order coexistence also has its own volume-dependent rounding, so it should not be forced into a conventional continuous-transition collapse.