# STATISTICAL PHYSICS

## Learning outcomes of the course unit

Understanding the bases of quantum statistical mechanics and learning some important applications.

Learning to use the second quantization formalism for many body systems.

Making judgements: students have do demonstrate that they improved their critical abilities in statistical physics, that they can study autonomously and that they can develop numerical simulation codes and analyze the results with critical attitude.

Communication skills: students have to demonstrate that they can expose topics of statistical physics effectively. In particular, they must be able to introduce these topics in a clear and accessible way, not only for a specialist in the field, but also for a physicist with a different background.

Learning skills: students have do demonstrate that their knowledge of statistical physics is robust enough that they can comprehend the main topics in the field, including specialized ones not treated during the course. They must be able to develop numerical simulations on these topics autonomously.

## Prerequisites

Basic knowledge of quantum mechanics and classical statistical mechanics

## Course contents summary

Main results in classical statistical mechanics.

Mixed states in quantum statistical mechanics, density operator, statistical entropy.

Fundamental principle of statistical mechanics, quantum ensembles (microcanonical, canonical, T-P, gran-canonical). Paramagnets, molecular vibro-rotations, specific heat of solids.

Identical particles, Fock spoace, second quantization, photons, ideal quantum gases, Hubbard model, spin-waves.

Fluctuations. Phase transitions. Mean-field approximation. MonteCarlo method.

Computer simulations (Matlab): Heisenberg model in canonical ensemble. MonteCarlo method for the 2d Ising model. Exact-diagonalization study of the Hubbard model.

## Course contents

Introduction.

Ergodic principle.

Basic notions of classical statistical mechanics : microcanonical and canonical ensembles, energy probability distribution, ideal gas, equipartition theorem.

Quantum statistical mechanics: density operator, statistical entropy, Liouville-Von Neumann equation.

Fundamental postulate of statistical mechanics.

Microcanonical ensemble: probability distribution of an internal observable, harmonic oscillators (Einstein model for the heat capacity of solids), spontaneous evolution after removal of a constraint, thermal contact, exchange of volume and matter, Gibbs paradox, Maxwell-Boltzmann approximation for identical particles.

Introduction to transport theories (linear nonequilibrium thermodynamics), diffusion equation.

Canonical ensemble: free energy, probability distribution of an internal observable, energy probability distribution, spontaneous evolution after removal of a constraint, equilibrium between subsystems, canonical pressure vs mechanical pressure, heat and work, pressure and virial, canonical distribution for identical and independent particles, monoatomic ideal gas in Maxwell-Boltzmann approximation, paramagnetism, diamagnetism, , polyatomic ideal gas in Maxwell-Boltzmann approximation (molecular rotations and vibrations), heat capacity of solids, blackbody radiation, susceptibility and fluctuations (static linear response).

TP ensemble (introduction): application to point defects in a crystal.

Grand-canonical ensemble: chemisorption, barometric formula, quantum ideal gases (Fermi-Dirac and Bose-Einstein distributions), high-temperature expansion, application to white dwarf stars.

Phase transitions: examples, order parameter, long-range order and divergence of fluctuations, Landau models, critical exponents, Ising model, order-disorder transition in solid solutions, one-dimensional Ising model, spin-1 Ising, Potts, Heisenberg, XY models. Montecarlo simulations.

Fock space, observables on Fock space, creation and annihilation operators, representation of 1- and 2-body observables, Hubbard model, harmonic oscillator, boson representation of spins, Holstein-Primakoff for ferromagnets, electron gas in Hartree-Fock approximation.

Numerical simulations (MATLAB): canonical description of the quantum Heisenberg model on clusters, Montecarlo simulation for the two-dimensional ising model, Mean-field model, transfer-matrix method, Hubbard model with a finite number of orbitals.

## Recommended readings

Lecture notes.

Huang - Statistical Mechanics

Yeomans - Statistical Mechanics of Phase Transitions

Bruus and Flensberg - Introduction to Quantum field theory in condensed matter physics

## Teaching methods

Lectures and computer simulations.

## Assessment methods and criteria

Oral examination, usually consisting in five questions covering the main topics of the course. The final grade is calculated as the average score obtained on the five questions. The student can also optionally expose the results of a numerical simulation on a topic related with the course. This replaces one of the five questions.

## Other informations

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