[p. 2466l, §1]

A number of recent publications [1, 2, 3, 4, 5]
has reopened the discussion concerning the classification of phase transitions.
In [1] a thermodynamic classification of normal
and anomalous first-order phase transitions was given.
In [3] and [4] it was shown that also continuous phase transitions
can be usefully classified by extending the thermodynamic classification scheme of Ehrenfest.
Based on the generalized classification scheme a class of phase transitions
having order less than unity was conjectured to exist [5]
and it was shown that this transition type is allowed by the laws of classical thermodynamics.
However, the identification of the corresponding
statistical-mechanical classification theory remained incomplete.
Transitions of order less than will be called
anequilibrium transitions in this paper.

[p. 2466l, §2]

My objective in this paper is to discuss in more detail the statistical-mechanical (SM)
and the thermodynamical (TD) classification theories.
To this end the classification scheme introduced in [3] and [4] will first be refined.
Phase transitions of order less than unity [5] will be discussed thermodynamically.
Then the statistical-mechanical classification is shown to be related to finite-size-scaling theory
and the breakdown of hyperscaling

[p. 2466r, §2]

other than through the traditional mechanism [6].
Next it is shown that canonical descriptions of systems which are a subregion
of an infinite sample may require a renormalization of temperature.
Finally some general consequences for the dynamics of critical systems will be discussed.

[p. 2466r, §3]

Discontinuities and divergences of thermodynamic potentials
along curves which cross a critical manifold can
be characterized mathematically through their generalized
orders (with ) as well as their strengths [3, 4].
Besides providing a convenient language the classification scheme
implies multiscaling for thermodynamic phase transitions [3, 4].
Multiscaling has been defined as a scaling form in which the critical exponents
are functions of the scaling variables [7].
The simplest form of multiscaling occurs at a multicritical point
but recently more interesting cases have been discussed [8, 9, 10, 11, 12, 13, 14, 15, 16].
The derivation of multiscaling from analytic continuation
of the Ehrenfest scheme represents a derivation
of thermodynamic scaling results without the use of renormalization-group ideas.

[p. 2466r, §4]

Given these results it was natural to ask whether phase transitions
of order are thermodynamically admissible or not.
A rash answer would be negative because such transitions appear
to violate thermodynamic stability requirements.

[p. 2467l, §0]

A more cautious response, however, is useful.
It was shown in [5] that transitions having
are allowed by the laws of thermodynamics.
Here the consequences of this discovery for statistical mechanics will be studied
in more detail while more general consequences have been discussed elsewhere [17].