# Course Details for A.Y. 2014/2015

#### Name:

**Fisica Matematica 1 / Mathematical Physics 1**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 1^{st} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

This course aims to enable students to make a connection between mechanics
and thermodynamics

### Course Content

- PART I - (Thermodynamics: The three + one laws.)
Introduction to thermodynamics. The Zeroth law and the temperature. The first law: work, heat and internal energy. Differential forms: specific heat at constant pressure, specific heat at constant volume and other thermodynamical quantities. The second law and the entropy. Properties of the entropy. Reversible and irreversible processes; adiabatic, isochoric, isobaric and isothermal processes. The Carnot cycle and the Otto cycle; thermal engines, efficiency and the Carnot efficiency. The third law: equivalence of definitions and the behavior of entropy and specific heat at temperatures close to the absolute zero.
- PART II - (Thermodynamics: Thermodynamic potentials.)
Convex regions and convex functions. Convexity, concavity and monotony of the entropy and of the internal energy. The entropy and the fundamental equation. Continuity of convex functions. The Legendre transform. Convexity, epigraphs, closures, closed convex hulls and the inverse transform. Existence of directional derivatives of convex functions. Legendre transforms: the free energy and the grand canonical potential. The grand canonical potential: pressure and chemical potential. Properties of the thermodynamic potentials: convexity, concavity and monotony. Phase transitions (definition) and magnetism (basic).
- PART III - (Statistical Mechanics: Micro-canonical ensemble.)
Shannon entropy and the principle of maximum entropy. Entropy and the micro-canonical ensemble. Entropy of the ideal gas: Gibbs paradox and indistinguishability. Elementary classical model for a crystal and the law of Dulong and Petit. Elements of Quantum Mechanics. Eigenvalues of the Hamiltonian: harmonic oscillator, Coulomb potential and free particles in a box. The spin and the statistics of Fermi and Bose. Entropy in quantum statistical mechanics. Laplace's principle. Entropy, internal energy, magnetization, susceptibility and specific heat of a system of non-interacting spins in an external magnetic field (paramagnetic salts).
- PART IV - (Statistical Mechanics: Canonical ensemble.)
Free energy and magnetization of a system of non-interacting spins in an external magnetic field. Proof of the equivalence of canonical and micro-canonical ensembles. The Gibbs measure, the Shannon entropy and the thermodynamic entropy. The general Ising model. Method of transfer matrices for the one-dimensional Ising model with interactions between nearest-neighbors and external magnetic field. Mean field model (Weiss-Ising) with and without external magnetic field. Magnetization and susceptibility: phase transition. Existence of the thermodynamic limit for systems with short-range interactions.
- PART V - (Statistical Mechanics: Grand-canonical ensemble.)
Equivalence between canonical and grand-canonical ensembles. Grand canonical potential of a system of non-interacting indistinguishable particles (Fermi and Bose statistics). Statistical mechanics of a system of non-interacting fermions with a single vanishing energy state and a macroscopic number of states with the same positive energy. Same model for bosons: Bose-
Einstein condensation. Ideal gas of fermions, energy, pressure and entropy. High and low temperature limits: the Fermi sea. Ideal gas of bosons, energy, pressure and entropy. High and low temperature limits Bose-Einstein condensation.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- The student should acquire the basic notions of Thermodynamics and Statistical Mechanics.
- The student should be able to solve problems of Thermodynamics and Statistical Mechanics. Moreover the student should be able to recognize when the acquired notions are useful for the comprehension of other topics.
- The student should be able to understand problems of Thermodynamics and Statistical Mechanics and understand similiarities and differences among them.
- The student should be able to explain the connection between Thermodynamics and Statistical Mechanics.
- The student should have acquired the ability of reading and understanding more advanced topics in Statistical Mechanics.

### Prerequisites and Learning Activities

Probability theory, classical mechanics

### Assessment Methods and Criteria

Written and oral examination

### Textbooks

- R. J. Baxter, Exactly Solved models in Statistical Mechanics , Academic Press. 1989.
* *
- E. Presutti, Lezioni di Meccanica Statistica , Aracne . 1995..
* *
- P. Fornasini, Lezioni di Termodinamica avanzata , Università di Trento. 2013.
* *
- T. C. Dorlas, Statistical Mechanics. Fundamentals and Model Solutions , IOP Publishing Ltd . 1999.
* *
- M. Kardar, Statistical Physics of Particles , Cambridge University Press. 2007.
* *
- E. Fermi, Thermodynamics , Dover. 1956.
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 18 febbraio 2015, 16:01*