[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 -dimensional simple cubic lattice with lattice spacing
in âblock geometryâ, i. e., having finite side length in all directions.
Let be a scalar observable associated with each lattice point.
The lattice represents a discretization of a large but finite
statistical-mechanical system [41].
Let the lengths and the parameters
of the statistical-mechanical system be such that

(3.1) |

Thus the system decomposes into a large number of uncorrelated blocks of linear extension . The ensemble limit is defined as the simultaneous limit in which

(3.2a) |

Two cases can be distinguished: In the critical ensemble limit

(3.2b) |

while for the noncritical ensemble limit

(3.2b) |

[p. 2469r, §5]

If denotes the number of uncorrelated
blocks of size and is the number of sites
in each block then is the total number of lattice sites.
The correlation length diverges in units of in the critical ensemble limit
but stays finite in the noncritical ensemble limit.
Note also that in the critical ensemble limit while remains finite
in the massive scaling limit or the finite-size scaling limit [37].
The critical ensemble limit generates an infinite ensemble of uncorrelated blocks.
This feature allows the application of standard limit theorems
for uncorrelated or weakly dependent variables.

[p. 2469r, §6]

Let denote the scalar observable at lattice site
inside block .
Then

(3.3a) |

are the block sums or block variables for block and

(3.3b) |

is the ensemble sum or ensemble variable for the total system. The are random variables and so are and . Let

(3.4) |

denote the normed and centered ensemble sum and let be the probability distribution function of . Assuming translation invariance the block variables are uncorrelated and identically distributed. Therefore the limiting distribution of is stable in the critical ensemble limit [28, 34, 35]. More precisely, if

(3.5) |

[p. 2470l, §0]

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

(3.6) |

where are constants whose ranges are

(3.7a) | |||

(3.7b) | |||

(3.7c) | |||

(3.7d) |

and

(3.8) |

The constant is called the index of the stable distribution while the parameter characterizes its symmetry or skewness.

[p. 2470l, §1]

If the limit in (3.5) exists and then the norming
constants must have the form [28, 35]

(3.9) |

where the function is slowly varying at infinity. The case corresponds to the critical ensemble limit, while for the limiting distribution is degenerate, i. e., concentrated at a single point, corresponding to the noncritical ensemble limit.

[p. 2470l, §2]

The preceding limit theorem implies that in the limit
the distribution function of the block sums can be approximated as

(3.10) |

where the notation is introduced for stable distributions of index . The objective in the next section will be to establish a large- scaling result for . To obtain it more information on the common distribution of the individual block variables is required.

[p. 2470l, §3]

The limiting distributions of the individual block
variables are independent of because of translation
invariance and they belong to the domain of attraction of a stable law.
The class of possible block variable limits
can thus be characterized as follows [35]:
In order that the characteristic function of
belongs to the domain of attraction of a stable law
whose characteristic function has the logarithm
with , ,
and as in (3.8)-(3.10),
it is necessary and sufficient that in the neighborhood of the origin

(3.11) |

[p. 2470r, §3]

where is a constant and is a slowly varying function for .

[p. 2470r, §4]

Equations (3.9) or (3.11) show that each ensemble limit
is labeled by a set of numbers , , with
ranges as in (3.7) and a slowly varying function .
While differentiates between critical and noncritical limits ,
, and differentiate between different critical ensemble limits.
This characterization is reminiscent of the thermodynamic classification scheme
and suggests a closer correspondence.
To establish such a correspondence it is necessary
to relate the generalized orders in the thermodynamical classification scheme
to the numbers occurring in the characterization of ensemble limits.
This will be done in the next section.

[p. 2470r, §5]

The purpose of the present section is to investigate the dependence
of the probability distribution for ensemble sums in the limit of large .
The scaling relations emerging from this analysis will be called finite-ensemble
scaling because they are closely related to finite-size-scaling relations
by virtue of the similarity between the critical ensemble limit defined above
and the finite-size-scaling limit [37].
The question is how to choose the norming
and centering constants , in (3.10)
given the characterization (3.11) for the individual block variables .

The centering constants in Eq. (3.10) can be eliminated from the problem by setting where

(3.12) |

and with this choice (3.10) becomes

(3.13) |

Although the general form of is known from (3.9) it remains to establish the relationship between the slowly varying functions in (3.9) and (3.11). Once this relation is established Eq. (3.12) represents a finite scaling formula for a system in which the individual block variable limits are characterized by (3.11).

[p. 2470r, §6]

The limiting distribution functions of the individual
block variables have characteristic functions as given by (3.11).
Introduce

(3.14) |

where the slowly varying function is defined through

(3.15) |

and is the slowly varying function appearing in (3.11). For sufficiently large the norming constants

[p. 2471l, §0]

are chosen as

(3.16) |

[p. 2471r, §0]

which is possible because for
and is continuous in a neighborhood of zero.
Then for small [35]

(3.17) |

[p. 2471l, §0]

It follows that for sufficiently large and
this determines in terms of and as

(3.18) |

where is the conjugate slowly varying function to defined in Eq. (3.18). The slowly varying function appearing in (3.9) is thus given as

(3.19) |

in terms of appearing in the limiting distributions (3.11) for the individual blocks.

[p. 2471l, §1]

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

(3.20) |

More interesting than the ensemble variables are the ensemble averages defined as . The probability distribution function for the ensemble averages has the finite-ensemble-scaling form

(3.21) |

If is expressed in terms of the system size they are seen to be closely related to finite-size-scaling theory. Note that Eqs. (3.20) and (3.21) are derived without reference to a particular model or approximate critical Hamiltonian such as the Landau-Ginzburg-Wilson Hamiltonian. They are generally valid for all translationâinvariant critical systems, i. e., systems for which the basic limit distribution (3.5) is not degenerate.

[p. 2471l, §2]

It is now possible to consider the correspondence
between the statistical classification in terms of , , and
and the thermodynamic classification in terms of and .

[p. 2471r, §2]

To do this the index in (3.20) and (3.21)
must be related to the critical exponents.
This is immediately possible from (3.21)
by considering for example the order parameter .
Setting , taking the derivative with respect to in Eq. (3.21)
and using one finds that the th moment
of the order parameter scales with system size as .
Comparing to standard finite-size-scaling theory [36, 37, 38, 39, 40]
relates to the thermodynamic exponents as

(3.22) |

where and are the order parameter and susceptibility exponents, and , was defined in (2.5). Similarly for the energy density one finds

(3.23) |

where is the thermal order of (2.5). This suggests that the correspondence between the statistical and the thermodynamic classification of phase transitions is given generally as . Note that second-order (i. e., self-conjugate) phase transition occupy again a special place in the statistical classification scheme because of the bound in (3.7a). This fact will be related below to violations of hyperscaling relations.

[p. 2471r, §3]

Anequilibrium phase transitions with order correspond
to stable limit distributions with index .
Thus anequilibrium transitions are not only predicted by
equilibrium thermodynamics but also by equilibrium statistical mechanics.
The fact that anequilibrium transitions restrict the range of equilibrium temperatures
as in (2.17) is mirrored by the fact that expectation values of
averages diverge in the critical ensemble limit
for anequilibrium critical points with .
This implies that the traditional formulation of statistical mechanics
becomes inapplicable at anequilibrium critical points just as
traditional thermodynamics becomes inapplicable.

[p. 2471r, §4]

While the general correspondence between and
is reassuring it is not sufficient to establish the existence
of anequilibrium phase transitions in statistical mechanics.
To demonstrate their existence requires a possibly exact calculation
of the partition sum for a concrete statistical-mechanical model.
It is possible to demonstrate the existence of anequilibrium transitions in

[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
where the sum runs over all nearest-neighbour pairs
of lattice sites , and the continuous spin variables
have a Gaussian single spin measure proportional to .
The limiting free-energy density for the one-dimensional Gaussian model
is well known and it reads

(3.24) |

where and denotes Boltzmannâs constant. The exact free-energy density (3.24) for the one-dimensional Gaussian model exhibits an anequilibrium transition of order at the critical temperature .

[p. 2472l, §1]

The identification cannot hold for all values of
because is required by (3.7).
The new restriction is now seen to be related
to the violation of hyperscaling and the breakdown of finite-size scaling for
thermal fluctuations in dimensions .
Consider the class of statistical-mechanical models
obeying the Lebowitz inequality for the four-point functions
and infrared bounds for the two-point functions [33].
For such models the susceptibility exponent obeys
and the correlation-function exponent obeys .
Then using , the Fisher inequality ,
the hyperscaling relation , and relation (3.23)
the following chain of inequalities is obtained:

(3.25) |

For general models hyperscaling may fail at because there are distributions with nonalgebraic tails within the domain of attraction of the normal law. Note that in this way the inequality provides a general mechanism for the breakdown of hyperscaling independent of identifying dangerous irrelevant variables in a particular model. Analogous breakdown phenomena are expected to occur for critical fluctuations in observables other than the energy density.