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Theory of nonequilibrium first-order phase transitions for stochastic dynamics
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21.The “fundamental intuition” of phase coexistence has been incorporated in other definitions of phase transition, some of them stemming from the algebra approach (Refs. 25 and 26). In this language a phase transition is expressed by the nonuniqueness of the infinite volume Gibbs measure. As indicated, we do not know how to bring infinite volume (the thermodynamic limit) into so mild a form of nonequilibrium as metastability, nor do we think that such a limit best reflects the physics (cf. Ref. 6).
22.The “remainder” term in Eq. (3.10) must be small in some sense for our phases to be well-defined, but the measure of smallness in that equation can probably be replaced by something weaker.
23.The form for in this equation is adopted for convenience only. In general need not be writable as a sum of one-dimensional projections and a Jordan form would be required. See the remark following Eq. (3.4).
24.By “boundary” of (say) we mean points such that their distance, in terms of from any point in is small.
25.D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969).
26.D. Ruelle, “The C*-algebra approach to statistical mechanics,” in Statistical Mechanics at the Turn of the Decade, edited by E. G. D. Cohen (Marcel Dekker, New York, 1971), p. 67–79.
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