[p. 2469l, §2]

Given the thermodynamic classification of phase transitions
it is natural to ask whether anequilibrium phase
transitions and a statistical-mechanical classification corresponding
to the thermodynamic scheme exist for critical behavior in statistical mechanics.
These questions are discussed in the following sections.
Statistical mechanics for noncritical systems
is based on the law of large numbers [30].
This suggests that the theory of critical phenomena
may be founded in the theory of stable laws.
Although natural this idea is usually rejected because the
divergence of correlation lengths and susceptibilities appears
to imply that the microscopic random variables are strongly dependent [31, 32, 33]
while the standard theory of stable laws applies only
to weakly dependent or independent variables [28, 34, 35].

[p. 2469l, §3]

The problem of strong dependence arises from the particular choice
of performing the infinite-volume limit and the continuum limit.
One usually starts from an infinite-volume lattice theory
and then asks for possible continuum (or scaling) limits
of the rescaled infinite-volume correlation functions [33].
Depending on whether the rescaled correlation lengths remain finite
or not one distinguishes the “massive” and the “massless” scaling limit
but in either case the infinite-volume limit has been performed
before taking the scaling limit.

[p. 2469l, §4]

The idea of the present paper for basing a statistical
classification of critical behavior on the theory of stable
laws is related to that of finite-size scaling [36, 37, 38, 39, 40]
and

[p. 2469r, §4]

uses a different method of taking infinite-volume and continuum limits.
Consider a

(3.1) |

Thus the system decomposes into a large number of uncorrelated
blocks of linear extension

(3.2) |

Two cases can be distinguished: In the critical ensemble limit

(3.2) |

while for the noncritical ensemble limit

(3.2) |

[p. 2469r, §5]

If

[p. 2469r, §6]

Let

(3.3) |

are the block sums or block variables for block

(3.3) |

is the ensemble sum or ensemble variable for the total system.
The

(3.4) |

denote the normed and centered ensemble sum and let

(3.5) |

[p. 2470l, §0]

denotes the limiting distribution function of the normed
ensemble sums (3.4) then the characteristic function

(3.6) |

where

(3.7a) | |||

(3.7b) | |||

(3.7c) | |||

(3.7d) |

and

(3.8) |

The constant

[p. 2470l, §1]

If the limit in (3.5) exists and

(3.9) |

where the function

[p. 2470l, §2]

The preceding limit theorem implies that in the limit

(3.10) |

where the notation

[p. 2470l, §3]

The limiting distributions

(3.11) |

[p. 2470r, §3]

where

[p. 2470r, §4]

Equations (3.9) or (3.11) show that each ensemble limit

[p. 2470r, §5]

The purpose of the present section is to investigate the

The centering constants

(3.12) |

and with this choice (3.10) becomes

(3.13) |

Although the general form of

[p. 2470r, §6]

The limiting distribution functions of the individual
block variables

(3.14) |

where the slowly varying function

(3.15) |

and

[p. 2471l, §0]

are chosen as

(3.16) |

[p. 2471r, §0]

which is possible because

(3.17) |

[p. 2471l, §0]

It follows that

(3.18) |

where

(3.19) |

in terms of

[p. 2471l, §1]

Finally Eq. (3.12) gives the finite-ensemble scaling for
the distribution of ensemble variables

(3.20) |

More interesting than the ensemble variables

(3.21) |

If

[p. 2471l, §2]

It is now possible to consider the correspondence
between the statistical classification in terms of

[p. 2471r, §2]

To do this the index

(3.22) |

where

(3.23) |

where

[p. 2471r, §3]

Anequilibrium phase transitions with order

[p. 2471r, §4]

While the general correspondence between

[p. 2472l, §0]

this way.
A concrete example occurs in what is perhaps the simplest model
in the theory of critical phenomena,
namely, the one-dimensional Gaussian model [42].
This finding is important because the Gaussian model is of
central importance in the modern theory of critical phenomena
as the starting point for systematic perturbative calculations [6].
The model Hamiltonian is

(3.24) |

where

[p. 2472l, §1]

The identification

(3.25) |

For general models hyperscaling may fail at