Finite-size scaling

This page summarizes the idea behind a scaling collapse and how it maps onto the package.

The scaling hypothesis

At a continuous phase transition, the correlation length diverges as   , where is the control parameter (temperature, coupling, probability, ...) and its critical value. In a finite system of linear size , is cut off by , and close to an observable depends on and only through the ratio . This leads to the scaling form

where is the critical exponent of (e.g. for the order parameter, for the susceptibility) and is a universal function. Rescaling    and   maps the data of all system sizes onto , a scaling collapse. The collapse only works for the right parameters, so finding the best collapse determines , and .

Choosing the scaling function

The presets of ScalingFunction cover the common observables. Each preset has two or three parameters p1, p2, p3, which you can name with p_names:

observable	behaviour at	preset	parameters
dimensionless: Binder cumulant, , spanning probability	independent of	:x	,
order parameter (magnetization, cluster density)		:xy	, ,
susceptibility, correlation time		:xny	, ,

All presets rescale x as  . :xy multiplies y by , :xny by , so that the third parameter is a positive exponent in both cases. Other scaling forms, e.g. with logarithms or corrections to scaling, need a custom scaling function.

Measuring the collapse

To compare the rescaled curves, a quality function interpolates them and measures their spread; lower values mean a better collapse. The default, MultipleSplines, interpolates every system size separately and computes the variance between the curves on a grid of points (weighted by the error bars, if given). The curves are only compared where all system sizes overlap. The keyword dx restricts this further to a window of rescaled x values: the scaling form holds best close to , i.e. for small  .

Things that limit the accuracy

• Corrections to scaling. The scaling form is the leading term for large and small  . Small system sizes and wide windows dx bias the parameters. Compare fits with the smallest sizes removed and with a narrower window; see the percolation example.

• Correlated parameters. and (and and the y exponent) are usually correlated. The default profile error analysis takes this into account, see Error analysis.

• Correlated data. Curves computed from the same Monte Carlo samples (e.g. histogram reweighting) are correlated. None of the error methods takes this into account.