[p. 2474l, §2]

The present paper has discussed the existence and
properties of phase transitions of order less than unity

[p. 2474r, §2]

which were termed anequilibrium transitions.
The following results of direct physical relevance were obtained:
(i) Anequilibrium transitions are allowed by the laws of
thermodynamics and they occur in models of statistical mechanics.
(ii) The existence of anequilibrium transitions implies
the existence of ânonequilibrium temperaturesâ at which the system
cannot be described by equilibrium statistical physics.
This result points towards a possible incompleteness
in the foundations of statistical physics.
(iii) The paper has presented a general derivation of
finite-size scaling without use of renormalization-group theory.
(iv) A mechanism for the breakdown of hyperscaling was found
which does not invoke dangerous irrelevant variables.
(V) Anequilibrium transitions exhibit an entropy catastrophy and are asymmetric.
They require a renormalization of temperature if the reservoir in
the canonical ensemble is made of the same substance as the system itself.
The general classification theory predicts modified generators
for the time evolution of macroscopic observables in systems with