The purpose is to illustrate the theoretical approach to the analysis of critical phenomena through the renormalization group theory in real space. We will also introduce numerical algorithms for the analysis of critical properties. We expect an in- depth understanding of this topics. Ising model. Absence of phase transition in one dimension: free energy argument and exact solution with transfer matrices. Kramers-Wannier duality.
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The purpose is to illustrate the theoretical approach to the analysis of critical phenomena through the renormalization group theory in real space. We will also introduce numerical algorithms for the analysis of critical properties. We expect an in- depth understanding of this topics. Ising model. Absence of phase transition in one dimension: free energy argument and exact solution with transfer matrices. Kramers-Wannier duality. Exact solution in two dimensions. Fluctuation dissipation theorem.
Widom Scaling and scale relations. Correlation functions and extension of mean field theory. General formulation of the renormalization group in real space. Scaling of free energy and relevant eigenvalues. Ising model on a triangular bidimensional lattice: renormalization at lowest cumulant order.
Renormalization of a Potts model on a hierarchical lattice. Monte Carlo methods: Markovian stochastic processes, detailed balance, ergodicity. Spin flipping: Metropolis algorithm.
Cluster flipping: Fortuin Kasteleyn transformation, the cluster count Hoshen-Kopelman alogorithm, and Swendsen-Wang algorithm. Lectures, including the illustration of the theory, and applications. Huang, Statistical Mechanics; L. Landau and E. Lifshitz, Statistical Physics, Part 1; L. Kadanoff, Statistical Physics. Statics, Dynamics and Renormalization, D.
Landau and K. Students have access to transcripts of lectures on the homepage of the professor, and it is possible to stream on demand recording of lessons.
Final oral examination with discussion of topics related to course program and related exercises. Skip to main content. Corso di Second cycle degree in Physics. Academyc year when starting the degree:. Academyc year when helding the course:. Course type:. Compulsory subjects, characteristic of the class. Standard lectures hours:. Basic principles of classical and quantum statistical mechanics.
MECCANICA STATISTICA II
Information on the course unit. ECTS: details. Course unit organization. Prerequisites: Statistical Mechanics course given at the third year of the laurea triennale Thermodynamics Target skills and knowledge: After completing the course the student should be able to understand and explain the basic concepts and the use of advanced techniques in statistical mechanics.
We'd like to understand how you use our websites in order to improve them. Register your interest. An effective Microcanonical Thermodynamics of self gravitating systems SGS is proposed, analyzing the well known obstacles thought to prevent the formulation of a rigorous Statistical Mechanics SM , as those due to the formal unboundedness of available phase space and to the unscreened, long range, nature of the interaction. The latter feature entails the well known in equivalence of statistical ensembles, puts clearly into question the meaning, for these systems, of the Thermodynamic Limit , and rules out the use of canonical and grand-canonical ensembles. As to the first obstacle , we argue nevertheless that a hierarchy of timescales exist such that, at any finite time , the volume of the effectively available region of phase space is indeed finite, and that the dynamics satisfies a strong chaos criterion, leading to a fast , increasingly uniform, spreading of orbits over an effectively invariant subset of the constant N,V,E surface; thus leading to the definition of a secularly evolving, generalized microcanonical ensemble , which allows to define an almost extensive effective entropy and to derive self-consistent definitions for other thermodynamic variables, giving thus an orthode for SGS. Moreover, a Second Law-like criterion allows to single out the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of SGS.