III Finite ensemble scaling

[7.2.1.1] The quantity of main interest for finite ensemble scaling [21, 22, 23] is the macroscopic ensemble sum XM⁢N⁢φ→ given by (2.19). [7.2.1.2] The idea is to neglect completely its microscopic definition (2.18) in terms of cell variables, and to consider the mesoscopic block variables XM⁢N⁢φ→⁢y→ as a starting point. [7.2.1.3] The univariate probability distribution of the ensemble variable is defined as

[7.2.1.4] Because the ensemble limit automatically generates independent and identically distributed block variables XM⁢N⁢φ→⁢y→j the standard theory of stable laws [27, 28] can be applied. [7.2.1.5] It yields the existence and uniqueness of limiting distributions for the linearly renormalized ensemble sums

where DN>0 and CN are real numbers. [7.2.1.6] Remember that this holds for sums of arbitrary block variables independent of their microscopic definition. [7.2.1.7] The index M serves only as a reminder for the fact that the ensemble limit is used.

[page 8, §0]   [8.1.1.1] The distribution function PZM⁢N⁢x for ZM⁢N⁢φ→ is given in terms of PXM⁢N⁢x as PXM⁢N⁢DN⁢x+CN and it is thus sufficient to consider PXM⁢N⁢x. [8.1.1.2] The (weak) ensemble limit of these probability distribution functions

exists if and only if H⁢x;ϖX⁢c,ζX⁢c,C⁢c,D⁢c is a stable distribution function whose characteristic function

has the form

for ϖX≠1 and

for ϖX=1. [8.1.1.3] The c-dependence of the parameters ϖX⁢c,ζX⁢c,C⁢c,D⁢c has been suppressed to shorten the notation. [8.1.1.4] The parameters ϖX,ζX,C,D obey

[8.1.1.5] If the limit exists, and D≠0, the constants DN must have the form

where Λ⁢N is a slowly varying function [28], i.e.

for all b>0.

[8.1.1.6] The forms (3.5) and (3.6) of the limiting characteristic functions imply the following scaling relations for the stable probability densities h⁢x;ϖX,ζX,C,D. [8.1.1.7] If ϖX≠1 then

holds, while for ϖX=1 one has

[8.2.0.1] The parameters C and D correspond to the centering and the width of the distribution.

[8.2.1.1] Strictly stable probability densities (i.e. those with ϖX≠1) are conveniently written in terms of Mellin transforms [29, 30]. [8.2.1.2] This representation is useful for computations and involves the general class of H-functions [31, 32]. [8.2.1.3] For 1<ϖX<2 corresponding to equilibrium phase transitions two cases are distinguished. [8.2.1.4] If ζX≠1 then [30, 22]

where ϱ=12-ζXϖX+ζX2 and the definition of the general H-function HP⁢Qm⁢n is given in the appendix. [8.2.1.5] If ζX=1 then for 1<ϖX<2

[8.2.1.6] Similar expressions hold for 0<ϖX<1[30, 22]. [8.2.1.7] The special case ϖX=2 of the general limit theorem (3.3) is the central limit theorem [28] and in this case the stable probability density

is the Gaussian distribution with mean C and variance σ2=4⁢D. [8.2.1.8] Note that the right hand side is independent of ζX in this case. [8.2.1.9] Another special case expressible in terms of elementary functions is ϖX=1,ζX=0 where

is the Cauchy distribution centered at C and having width D.

[8.2.2.1] For sufficiently large but finite N=L/ξd equation (3.3) implies that the distribution function of ensemble variables may be approximately written as

[page 9, §0]   involving a nonuniversal cutoff functionR⁢x,M,N,c such that R⁢0,M,N,c=1 andlimx→±∞⁡R⁢x,M,N,c=0 for all M,N<∞. In the ensemble limit the cutoff function must obey

for all x and c as a result of equation (3.3). [9.1.0.1] Note that equation (3.17) does not hold for the finite size scaling limit. [9.1.0.2] Instead Table I implies that for the finite size scaling limit

if the limit exists, and where now c=L/ξ. [9.1.0.3] The function R⁢x;∞,L/ξd,0 may in general differ from unity, and thus the finite size scaling limit may involve a nonuniversal cutoff function which is absent in the finite ensemble limit.

[9.1.1.1] Wherever possible equation (3.16) will be abbreviated as

to shorten the equations. [9.1.1.2] If the centering constants are now chosen as

then using equations (3.10),(3.11) and (3.8) the basic finite ensemble scaling result [21, 22]

is obtained for the probability density function pXM⁢N⁢x of suitably centered and renormalized ensemble sums. [9.1.1.3] The approximate result (3.21) has formed the basis for the statistical mechanical classification of phase transitions [21, 22].

[9.1.2.1] From the basic result (3.21) the scaling form for the probability density of ensemble averaged block variables X¯M⁢N⁢φ→=XM⁢N⁢φ→/M⁢N is readily obtained using eq. (3.10) as

[9.1.2.2] Setting X=Ψ this result is found to be distinctly different from equation (1.3). This shows that finite ensemble scaling (3.22) and finite size scaling (1.3) are not equivalent.