[page 63, §1]

[63.2.1.1] Analysis of finite size effects [1] has become an
indispensible tool in the numerical simulation of critical
phenomena [2, 3, 4, 5].
[63.2.1.2] According to the nonrigorous renormalization group derivations
of finite size scaling [6] the singular part of the
free energy

(1.1) | ||||

(1.2) |

where

[63.2.2.1] More heuristically there are several possibilities to introduce
finite size scaling through a scaling hypothesis.
[63.2.2.2] One such method
[7, 8] assumes that the probability density

(1.3) |

where

(1.4) |

where

[64.1.1.1] Despite their very plausible and seemingly general character finite
size scaling relations are not generally valid [10].
[64.1.1.2] Violations of finite size scaling are closely related to violations of
hyperscaling relations [10, 11].
[64.1.1.3] These violations have been rationalized
via the so called mechanism of “dangerous irrelevant” variables
[12] or by saying that the correlation length

[64.1.2.1] Given the scaling Ansatz (1.3) another well known problem with
present finite size scaling theory concerns integrals of the scaling
function appearing in (1.3).
[64.1.2.2] To see this calculate the finite size scaling form for the absolute
moments of order

(1.5) |

where the new scaling function

(1.6) |

[64.1.2.3] From these moments one finds for the ratio related to the renormalized coupling constant the result

(1.7) |

which implies that

(1.8) |

in the finite size scaling limit for which

[64.2.1.1] Let me summarize the objectives of this work resulting from the above exposition of two problems with current finite size scaling theory. [64.2.1.2] The first objective is to provide general criteria for the validity or violation of finite size scaling. [64.2.1.3] The second objective is to investigate the finite size scaling functions and finite size amplitude ratios in the ensemble limit.

[64.2.2.1] Methodically, the results of this paper follow directly from a
recently introduced classification theory of phase transition
[18, 19, 20, 21, 22, 23].
[64.2.2.2] Let me briefly
outline the basic idea. Within the classification theory it was
shown that each phase transition in thermodynamics as well as in
statistical mechanics is characterized by a set of generalized
Ehrenfest orders plus a set
of slowly varying functions.
[64.2.2.3] This
classification is macroscopic in the sense that it involves only
thermodynamic averages while conformal field theory focusses on
microscopic higher order correlation functions.
[64.2.2.4] The classification
in thermodynamics [18] is based upon the application of
fractional calculus, the one in statistical mechanics [22]
rests upon the theory of limit distributions for sums of independent
random variables.
[64.2.2.5] The latter theory, which cannot be employed in the
traditional way of performing the scaling limit, became applicable
by introducing a fundamentally new scaling limit, which was called
ensemble limit.
[64.2.2.6] In the ensemble limit critical
systems decompose into an infinite ensemble
of infinitely large, yet uncorrelated blocks.
[64.2.2.7] The classification
schemes in thermodynamics and statistical mechanics are mathematically
very different but can be related to each other by studying the
fluctuations in the ensemble of blocks.
[64.2.2.8] The difference between the
classification schemes is found to be related to violations of
hyperscaling.
[64.2.2.9] Moreover, a thermodynamic form of scaling, called
finite ensemble scaling, emerges from the classification.
[64.2.2.10] The basic idea of this paper is to regard finite ensemble scaling
as a macroscopic or thermodynamic form of finite size scaling.
[64.2.2.11] Thus the limit distributions in the classification theory ought
to be related to the probability distributions, such as