[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

[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

[p. 2466r, §4]

Given these results it was natural to ask whether phase transitions
of order

[p. 2467l, §0]

A more cautious response, however, is useful.
It was shown in [5] that transitions having