# Finite-size scaling analysis¶

## The finite-size scaling ansatz¶

Consider a system with some parameter $$\varrho$$, which undergoes a phase transition at a critical value $$\varrho_c$$. Divergences in the correlation length $$\xi$$ and of fluctuations (e.g. the susceptibility $$\chi$$) characterize the critical point in infinite systems. Typically, in the critical region a diverging quantity $$A_\infty(\varrho)$$ scales as $$|\varrho - \varrho_c|^{-\zeta}$$ with some critical exponent $$\zeta$$. In fact, this critical behavior should also hold in systems of finite size $$L$$ at scales much larger than the characteristic length scale $$\xi$$. Here and in the following, the characteristic length scale $$\xi$$ is the correlation length in the infinite system ($$L \to \infty$$). As the correlation length diverges as $$\xi \sim |\varrho - \varrho_c|^{-\nu}$$ for $$\varrho \to \varrho_c$$, we have

$A_L(\varrho) \sim |\varrho - \varrho_c|^{-\zeta} \sim \xi^{\zeta / \nu}, \qquad (L \gg \xi, \varrho \to \varrho_c).$

For $$L \ll \xi$$, the system size $$L$$ takes over the role as the cutoff, such that we expect

$A_L(\varrho) \sim L^{\zeta/\nu}, \qquad (L \ll \xi, \varrho \to \varrho_c).$

These considerations constitute the finite-size scaling ansatz [NB99][BH10][FB72]

$A_L(\varrho) = \xi^{\zeta/\nu} f(L / \xi), \qquad (L \to \infty, \varrho \to \varrho_c),$

with

$\begin{split}f(x) \begin{cases} = \text{const.} & \text{for } |x| \gg 1, \\ \sim x^{\zeta/\nu} & \text{for } x \to 0. \end{cases}\end{split}$

The scaling function $$f(x)$$ is a dimensionless function of the dimensionless ratio $$L/\xi$$ of the finite system size and the infinite-system correlation length. This ratio controls the finite-size effects. The conventional scaling function is $$\tilde{f}(x) = x^{-\zeta} f(x^\nu)$$ [NB99][BH10] such that

$A_L(\varrho) = L^{\zeta/\nu} \tilde{f}\left(L^{1/\nu} (\varrho - \varrho_c)\right), \qquad (L \to \infty, \varrho \to \varrho_c),$

with

$\begin{split}\tilde{f}(x) \begin{cases} = \text{const.} & \text{for } x \to 0 \quad (L \ll \xi), \\ \sim L^{-\zeta/\nu} (\varrho - \varrho_c)^{-\zeta} & \text{for } |x| \gg 1 \quad (L \gg \xi). \end{cases}\end{split}$

## The data collapse method¶

A simulation experiment yields the quantity $$a_{L, \varrho}$$ at system size $$L$$ and parameter $$\varrho$$, with standard error $$da_{L, \varrho}$$. The scaling function is

$\tilde{f}\left(L^{1/\nu} (\varrho - \varrho_c) \right) = L^{-\zeta/\nu} A_L(\varrho).$

Thus, plotting $$L^{-\zeta/\nu} a_{L, \varrho}$$ against $$L^{1/\nu}(\varrho-\varrho_c)$$ should let the experimental data collapse onto the single curve $$\tilde{f}(x)$$. For this to happen, the critical parameter value $$\varrho_c$$ and the critical exponents $$\zeta, \nu$$ need to be correct. These assumptions hold for $$L \to \infty$$, with systematic errors at finite sizes [NB99][BH10].

## The intersection method¶

At a first-order transition where the order parameter $$P$$ remains constant on both sides of the transition, its critical exponent is $$\beta = 0$$.

So we have

$P_L(\varrho) = \tilde{P}\left(L^{1/\nu}(\varrho-\varrho_c)\right),$

and hence, $$P_L(\varrho_c) = \tilde{P}(0)$$ independent of the system size $$L$$. Thus, the common intersection point of the measured curves $$p_{L, \varrho}$$ yields an estimate of the threshold $$\varrho_c$$. This estimate is unbiased with regards to the critical exponents, and “should be free” from systematic errors due to finite system size [BH10].

## Implementation in the fssa package¶

### Routines¶

 fssa.fssa.scaledata Scale experimental data according to critical exponents

### Classes¶

 fssa.fssa.ScaledData A namedtuple for scaledata()