# Course Details for A.Y. 2019/2020

#### Name:

**Mathematical Physics 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

- [Elementary notions of thermodynamics]
Definition of the macroscopic system and macroscopic variables. I and II thermodynamic principles,
thermodynamic entropy and its properties. Irreversibility. Other thermodynamic potentials: free energy and the Gibbs free energy. Variational principles. Van der Waals equation and Maxwell correction. Phase transitions end phase diagram.
- [Lorentz gas] Definition of the microscopic model on the 2-dimensional lattice with 2 types of obstacles. Microscopic dynamic. Macroscopic scale and macroscopic observables. A simplified model with only one type of obstacles: existence of the macroscopic limit. Chaos hypothesis on the initial state: strong and weak convergence of the probability density in the macroscopic limit in this hypothesis. Boltzman linear equation: rigorous and heuristic derivation. Properties. Entropy production. Microscopic reversibility vs macroscopic irreversibility: explanation of the paradox.
- [Boltzmann Gas]
Nonlinear Boltzmann equation and its properties.
Heuristic derivation of the equation. Equilibrium states and Maxwell distribution. Entropy production and irreversibility. Variational principle for the entropy. Relative entropy. Thermodynamic of the
Boltzmann gas. Thermodynamic entropy and Shannon entropy. Boltzmann principle and relation between thermodynamics and statistical mechanics. The relation between thermodynamic pressure and kinetic pressure. Other variational principles for other potentials. State equation of Boltzmann gas
- [Interacting Models: Ising e Lattice gas]
Physic phenomenology and mathematical representation. Extension to these models of the variational principles obtained for the Boltzmann gas. Equilibrium measures. Statistical ensembles: microcanonical, canonical and gran canonical ensembles. Finite volume Gibbs measure and partition function. Pressure. Thermodynamic limit of the pressure. Thermodynamic of the Ising model. Legendre transform and relation between thermodynamic potentials. Mentions to the ensemble equivalence. Introduction to phase transitions.
- [The Curie-Weiss Model] Definition of the model, gran canonical and canonical partition function. Non-equivalence of the ensembles and Maxwell construction. Phase Transitions and e non-analyticity of the free energy and of the pressure.
- [The one-dimensional Ising Model] Transfer matrix and calculation of the pressure. Analyticity of the pressure and absence of phase transition. Mean value of a spin with respect to the Finite volume Gibbs measure and thermodynamic limit
- [Definition of Gibbs states (in the h thermodynamic limit).
Definition of phase transition in terms of Gibbs states.
GKS E FKG inequalities (without prove) and their implications.
Existence of the two extremal states $+$ e $-$ and their translational invariance.
Equivalence between the absence of spontaneous magnetization and absence of phase transition. Low temperatures representation of the 2-dimensional model: Peierls argument and phase transition. Hight temperatures representation of the one dimensional model and absence of phase transition.
Equivalence of the two definitions of phase transition (without prove). Phase Diagram for the Ising model. CennMentions on the DLR measure.

### 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, thermodynamics.

### 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..
* *
- M. Falcioni and A. Vulpiani, Meccanica Statistica Elementare , Springer-Verlag Italia. 2015.
* *
- M. Kardar, Statistical Physics of Particles , Cambridge University Press. 2007.
* *
- S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems:A Concrete Mathematical Introduction
* *

### 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: 22 luglio 2019, 10:17*