# Course Details for A.Y. 2018/2019

#### 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

- PART I - Zermelo and Loschmidt paradoxes. Reversibility of mechanics. Poincaré recurrence theorem. Scales of time. Example from probability (N coins toss) explains why Zermelo and Loschmidt paradoxes are irrelevant when the Avogadro number N is involved. The continuity equation and incompressibility of the Hamiltonian fluid. The Boltzmann equation and the Stosszahlansatz. The H function and the Boltzmann H theorem. Validity of the molecular chaos assumption (Stosszahlansatz). Equilibrium and Maxwell-Boltzmann distribution. The limit of Grad. The Ehrenfest model: the H theorem and the molecular chaos.
- PART II – The bi-dimensional Lorentz gas on a lattice. The distribution of obstacles and the motion of electrons. Expected value of a test function in the limit of low density of obstacles.
The expected velocity. Expected value and standard deviation of the position. Random initial conditions and the joint probability of velocity and position at a time tau. The linear Boltzmann equation for the joint probability of velocity and position. The entropy and the non negativity of the entropy production. Reversibility and irreversibility revisited. Equilibrium properties of the Ideal gas from the Maxwell-Boltzmann distribution: the Joule's law, the ideal gas law and the entropy as a function of density and temperature.
- PART III - Classical statistical mechanics: the micro-canonical ensemble. Entropy and the zeroth law, the
first law and the second law (Clausiuss statement). The ideal gas with the micro-canonical ensemble. The
internal energy and the equation of state. Mixing entropy and the Gibbs paradox. Maxwell distribution of
velocity and heat capacities. Two-level systems. Classical and quantum harmonic oscillators. The canonical
ensemble: physical definition. Mathematical definition and equivalence with the micro-canonical ensemble in the
thermodynamical limit. Canonical examples: ideal gas, two-level systems. Non-interacting spins in a external
magnetic field: free energy, entropy, energy, magnetization and susceptibility.
- PART IV - The one-dimensional, first neighbors, Ising model in presence of an external magnetic field: transfer
matrices method. Low and high temperature limits: free energy, entropy, energy, magnetization and suscepti-
bility. The mean field (fully connected) Ising model or Curie-Weiss model. The spontaneous magnetization in
absence of external magnetic field and the phase transition. The critical exponents near critical temperature and
their scaling. Behavior of energy and heat capacity close to the critical temperature and to the zero temperature.
Discontinuities of thermodynamical functions and the order of a phase transition.
- PART V -The grand canonical ensemble: the chemical potential and the grand potential. The perfect gas and
a system of non-interacting spins in a external magnetic field. Basic facts of quantum mechanics. Fermi and
Bose statistics: the role of spin. Simple model o non interacting particles with a single state of zero energy
and (thermodynamically) many states of identical positive energy: the fermionic case (Fermi energy) and the
bosonic case (Bose-Einstein condensation). Ideal quantum gases (Fermi and Bose): average occupation number,
pressure, density and energy. High temperature limit and the classical ideal gas. Fermi gas: Fermi energy and
Fermi pressure. Bose gas: transition and Bose-Einstein condensation. Bose-Einstein condensation for a toy
model o non interacting particles with degenerate energy levels equally spaced.

### 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.
* *

### 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: 05 febbraio 2019, 10:59*