# ANALYTICAL AND STATISTICAL MECHANICS

## Learning outcomes of the course unit

The student will acquire the basic theoretical concepts in Lagrangian and Hamiltonian mechanics. She/he will understand the principles leading to the study of macroscopic systems and the basic concepts in statistical mechanics. The student will be able to apply these methods to calculate the thermodynamical properties of macroscopic systems at equilibrium, starting from the statistical distribution of microscopic variables in phase space, in simple physical systems. She/he will develop learning skills and she/he will be able to identify the relevant points in a physical problem, the validity of relations and their applicability.

## Prerequisites

A solid understanding of calculus and basic familiarity with linear algebra; two semesters of introductory calculus-based physics (mechanics and termodynamics)

## Course contents summary

Introduction to Analytical Mechanics.

Statistical Mechanics of Microcanonical, Canonical and Gran Canonical Ensembles.

Applications of the classical ensembles.

## Course contents

- Classical Mechanics in an arbitrary reference frame. Constraints, virtual displacements, generalized lagrangian coordinates. The Lagrangian of a physical systems and the Lagrange equations. Symmetries and conservation laws. Noether's theorem. Small oscillations, normal modes. The Legendre transform and the Hamiltonian. Hamilton's equations. Configuration space and phase space. Poisson brackets.

- Variational principles and Lagrange and Hamilton equations. Elements of calculus of variations. Canonical transformations. Elements of perturbation theory. Examples of relevant Lagrangians and Hamiltonians of physical systems: central forces, changed particles in an electromagnetic field. Infinite degrees of freedom: the vibrating string.

- The statistical description of a macroscopic system. Systems with many degrees of freedom and classical mechanics. Average values without dynamics: statistical ensembles and probability measures. Liouville theorem. The problems of the microscopic approach. Temporal averages and the ergodic hypothesis. Recurrence times and macroscopic variables. How and if equilibrium is reached.

Brief review of thermodynamics: extensive and intensive variables, thermodynamic potentials, Legendre transformations, response functions.

-Microcanonical distribution. Boltzmann entropy and its properties. Additivity. Microcanonical classical ideal gas. Gibbs paradox.

- Canonical distribution. The partition function and the Helmotz free energy. Energy fluctuation in the canonical ensemble. Fluctuation and response. Maxwell distribution. Equipartition. Equivalence between microcanonical e canonical ensembles. Canonical Ideal Gas.

- Gran canonical distribution. Gran canonical partition function and pressure. Chemical potential. Gran canonical Ideal Gas.

## Recommended readings

H. Goldstein- C. Poole - J. Safko, Classical Mechanics

L.D. Laundau - E.M. Lifsits, Mechanics

L.D. Laundau - E.M. Lifsits, Statistical Physics

Lecture notes.

K. Huang - Statistical Mechanics

## Teaching methods

Lectures and exercices

## Assessment methods and criteria

midterm exam; final written and oral examination.

## Other informations

Support activity: tutor activity during the course, material from web sites on advanced subjects