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Ultimo aggiornamento 7 mag 2018
Autore
Mario Angelelli
Sesso M
Esperimento MMNLP
Tipo Dottorato
Destinazione dopo il cons. del titolo Post-dottorato (Italia)
Università Universita' Del Salento
Strutt.INFN/Ente
Lecce
Titolo Geometric structures in statistical physics: surfaces, amoebas and tropical limit
Abstract Partition function and free energy are fundamental concepts in statistical physics. Their knowledge provides a complete information about macroscopic systems. Hence, the calculation of partition functions and the study of their properties are principal tasks in theoretical physics. This research is devoted to the investigation of certain geometric and algebraic aspects of these objects. First, a basic statistical mapping is introduced as a generalization of the standard formula for free energy. This mapping defines a hypersurface with particular geometric characteristics. In this way, one can associate geometric structures to statistical and physical quantities. In particular, the Gauss-Kronecker curvature is identified as an adequate parameter for the description of physical interactions and symmetries. This approach is further extended to include metastability and instability with the introduction of the concept of statistical amoebas. A fine structure for statistical amoebas is associated with the stratification of the space of parameters coming from the zero locus of the partition functions. The issues of composition and dominance are captured by the tropical skeleton of a statistical model. Tropical algebra is introduced in statistical physics by means of a suitable limit for Boltzmann's constant. Concrete examples are studied, with particular focus on highly (exponentially) degenerate models whose physical manifestations (residual entropy, limiting temperatures) are underlined. This scheme is formalized in a microscopic-macroscopic correspondence, where microsystems (i.e. elements of a set) are related to a macroscopic description (subsets of a set) within the composition rules of tropical algebras. This correspondence is explored through physical consequences, e.g. relations with ultrametrics, non-equilibrium and tropical equilibrium, dependence on the choice of ground energy.
Anno iscrizione 2014
Data conseguimento 18 lug 2017
Luogo conseguimento Lecce
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