Abstract: Different types of experimental observations of macroscopic systems have led to different mathematical descriptions of their behavior. The descriptions (called mesoscopic dynamical theories) form a family called a multiscale dynamics. For example, classical equilibrium thermodynamics, fluid mechanics, kinetic theory, and particle mechanics represent multiscale dynamics composed of four different mesoscopic dynamical theories formulated on four different levels of description. The mesoscopic dynamical theories are autonomous, they differ in the amount of details that their take into account (they belong to different levels of description) and in the domain of their validity. My objective is to formulate multiscale dynamics that includes a unified mathematical formulation of individual mesoscopic dynamical theories and a unified formulation of relations among them.

Lecture 1 Mesoscopic dynamics is a combination of Hamiltonian dynamics (inherited from mechanics of ∼ 1023 particles composing the macroscopic systems) and gradient dynamics (arising due to the neglect of certain details). In the first lecture I will present Hamiltonian dynamics as it arises in particle mechanics, continuum (and generalized continuum) mechanics, and kinetic theories.

Lecture 2 How do ignored details influence the time evolution of important features of the behavior of macroscopic systems? What features can we afford to ignore and what we have to keep? It is thermodynamics (i.e. the nonmechanical concept of entropy and the Maximum Entropy Principle) that address these questions. In this second lecture I will formulate (static) multiscale thermodynamics in the unifying setting of contact geometry (i.e. geometry in which Legendre transformations are the natural transformations).

Lecture 3 It is the mesoscopic time evolution (combining the Hamiltonian and the gradient dynamics) that maximizes the entropy. The multiscale dynamics finds its 1 natural formulation in the setting of contact geometry. The unified formulation is in fact a (dynamic, nonequilibrium) multiscale thermodynamics (applicable on all levels of description and also for externally driven macroscopic systems).

Lecture 4 In the fourth lecture I will present some specific applications (in addition to specific illustrations presented in all three previous lectures). The choice of the applications will depend on the particular interests of the participants of the seminar. I suggest: rheology of complex fluids (polymeric fluids, colloids, and immiscible blends), kinetic theories and their relation to hydrodynamic and extended hydrodynamic theories, Cattaneo-type heat conduction.

Miroslav Grmela is a theoretical physicist working in the domain of multiscale thermodynamics (equilibrium, nonequilibrium, and statistical) and in continuum mechanics and kinetic theory of complex fluids. He received his PhD from Czechoslovak Academy of Sciences. He worked in the Nuclear Research Institute of the Czechoslovak Academy of Sciences, at École Polytechnique in Montrèal, and as a visiting professor, at Université Pierre et Marie Curie in Paris, ETH in Zürich, UNAM in Mexico, and Universitat Autonoma de Barcelone in Barcelona. In 1996 he organized the first International Workshop of Non-Equilibrium Thermodynamics (IWNET). Sevenths IWNET workshop took place in Holland in 2015.

In 1983, Miroslav Grmela initiated a systematic investigation of an abstract multiscale thermodynamics that is seen, from the physical point of view, as a theory of relations among different levels of description of macroscopic systems, and from the mathematical point of view, as a combination of symplectic and gradient dynamics that is put into the context of contact geometry. Miroslav Grmela is an author of more than 200 scientific publications. In 2014 he received a prize (Accelerator Research Grant) from Natural Sciences and Engineering Research Council of Canada.