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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.

Continuum mechanics. --- Scaling laws (Statistical physics). --- Calculus --- Atomic Physics --- Mathematics --- Physics --- Physical Sciences & Mathematics --- Scaling laws (Statistical physics) --- Statistical mechanics. --- Micromechanics. --- Mechanics of continua --- Ratio and proportion (Statistical physics) --- Scale invariance (Statistical physics) --- Scaling hypothesis (Statistical physics) --- Scaling phenomena (Statistical physics) --- Mathematics. --- Dynamics. --- Ergodic theory. --- Physics. --- Dynamical Systems and Ergodic Theory. --- Mathematical Methods in Physics. --- Composite materials --- Solid state physics --- Microstructure --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Mechanics --- Quantum statistics --- Statistical physics --- Thermodynamics --- Physical laws --- Ranking and selection (Statistics) --- Differentiable dynamical systems. --- Mathematical physics. --- Physical mathematics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Force and energy --- Statics

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Partial differential equations --- Fluid mechanics --- Infiltratie (Statistische fysica) --- Infiltration (Physique statistique) --- Percolation (Statistical physics) --- 51 --- Lattice theory --- Statistical physics --- Mathematics --- 51 Mathematics

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In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDE's. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analysed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases. All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms. In general researchers interested in the relations between PDE’s and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.

Partial differential equations --- Operational research. Game theory --- Probability theory --- Mathematics --- Mathematical physics --- Classical mechanics. Field theory --- Statistical physics --- Thermodynamics --- Mechanical properties of solids --- Physics --- Materials sciences --- Heat engines. Steam engines --- Fuels --- differentiaalvergelijkingen --- thermodynamica --- waarschijnlijkheidstheorie --- theoretische fysica --- stochastische analyse --- statistiek --- wiskunde --- ingenieurswetenschappen --- fysica --- kansrekening --- dynamica --- warmteoverdracht

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Ergodic theory. Information theory --- Mathematical physics --- wiskunde --- fysica --- informatietheorie