Dynamic of the group of renormalization : A study via partial differential equations

Author(s)

Leonardo Fernandes Guidi

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2003-12-10

Examining board members

Domingos Humberto Urbano Marchetti; Nestor Felipe Caticha Alfonso; Joao Carlos Alves Barata; Aldo Procacci; Clodoaldo Grotta Ragazzo

• Advisor: Domingos Humberto Urbano Marchetti

Abstract

We have considered in this thesis two distinct topics related to classic models in equilibrium statistical mechanics. The first one is the analysis of semilinear parabolic partial differential equations given by a suitable limit (size of block L 1) in the renormalization group for the dipole gas in any dimension d>1. The other topic is the construction of a majorant function (, z) for the thermodynamic -1 whose potential admits a scale decomposition in terms of some stable potential. We are capable to demonstrate the well-posedness (existence, uniqueness and continuous dependence of solutions) for Coulomb gas equations and the global asymptotic convergence of the flow to one of its countably many equilibrium solutions. The dipole gas equations are technically more difficult and lack the results weve achieved in Coulomb gas but, despite its difficulties, we can establish the uniqueness of the trivial solution as a equilibrium ane and its stabilish. At least for hierarchical models, the established results give a definite answer to Gallovotti and Niclolòs conjecture of na infinite of phase transitions. The majorant function is constructed as the solution of a first order quase-linear partial differential equation. By means of the characteristics method we are able to relate its solution (the majorant) to Lamberts W-function whose series expansion possess a singularity given by W-function allows better estimates for Mayer series convergence. (AU)