Non-local, free energy functionals

Abstract

AbstractThe basic postulates of the mesoscopic theory are recalled in Chap. 6 and then exemplified in the context of the L–P free energy functional (of Chap. 4). While motivated by a microscopic theory such functionals play in Chap. 6 the role of a primitive notion and Part I of the book is not essential in reading the chapter.In Sect. 6.1 thermodynamics is derived in the thermodynamic limit by solving variational problems. The analysis parallels the one in Sect. 2.3 and applies to more general non-local free energy functionals than the L–P functional. The non-local dynamics (for the L–P functional) is studied in Sect. 6.2, where several properties of the Allen–Cahn equation (which is the gradient flow of the Ginzburg–Landau functional, i.e. the local analogue of the L–P functional) are proved to hold like the Barrier Lemma, super- and subsolutions and the maximum principle. Variational problems with constraints are studied in Sects. 6.3 and 6.4. In the former the constraint fixes the profile outside a bounded region close to equilibrium, and exponential decay from the boundaries is proved. In Sect. 6.4 the free energy cost of deviations from equilibrium is estimated proving “extensive” lower bounds, namely bounds which are proportional to the volume of the region where the deviations occur (the Peierls bounds). Basic notions like local equilibrium, phase indicators and contours are here introduced, extending by a coarse graining procedure the corresponding notions in Sect. 3.1 for the Ising model. Sects. 6.3 and 6.4 have a fundamental role to play in the sequel of the book.The bibliography for Chap. 6 is mainly contained in Sect. 8.7.Mathematics Subject Classification (2000)82B2649Q2074A15