VI Conclusion

[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 “nonequilibrium temperatures” 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 ω~t<1. The latter systems exhibit algebraically decaying stationary states and nonexponential relaxation given by H-functions. Given these results it seems not too far fetched to suggest that anequilibrium transitions are promising candidates for the elusive glass transition although much more experimental and theoretical work is required to establish this point.