Professor Errico Presutti (auth.)3540733043, 9783540733041
Collective behavior in systems with many components, blow-ups with emergence of microstructures are proofs of the double, continuum and atomistic, nature of macroscopic systems, an issue which has always intrigued scientists and philosophers. Modern technologies have made the question more actual and concrete with recent, remarkable progresses also from a mathematical point of view. The book focuses on the links connecting statistical and continuum mechanics and, starting from elementary introductions to both theories, it leads to actual research themes. Mathematical techniques and methods from probability, calculus of variations and PDE are discussed at length.
Table of contents :
Front Matter….Pages i-xxix
Front Matter….Pages 15-15
Thermodynamic limit in the Ising model….Pages 17-73
The phase diagram of Ising systems….Pages 75-98
Mean field, Kac potentials and the Lebowitz–Penrose limit….Pages 99-118
Stochastic Dynamics….Pages 119-154
Front Matter….Pages 155-155
Non-local, free energy functionals….Pages 157-204
Surface tension, Gamma convergence, Wulff shape….Pages 205-228
One dimensional interfaces….Pages 229-270
Front Matter….Pages 271-271
Ising systems with Kac potentials….Pages 273-287
The LMP model and the Pirogov–Sinai strategy….Pages 289-316
Phase transitions in the LMP model….Pages 317-400
DLR measures and the ergodic decomposition….Pages 401-432
Back Matter….Pages 434-469
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