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I Introduction

[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 1 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 \lambda\geq 1 (with \lambda\in\mathbb{R}) 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 \lambda<1 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 \lambda<1 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].