OVERVIEW

The course deals with the foundations of Statistical Physics, with applications to the Physics of Matter. An introduction to the mathematical tools required to follow the classes will be provided.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide an introduction to the methods of Statistical Physics for systems of classical and quantum particles (Bose-Einstein and Fermi-Dirac statistics). The mathematical tools, typical of many-variable calculus, necessary for the study and characterization of stochastic processes and the related probability distributions will also be introduced.

AIMS AND LEARNING OUTCOMES

The principal aim and expected outcome of the course is for the student to acquire a critical comprehension of the foundations and of the mathods of Statistical Physics, of the microscopical interpretation of the laws of Thermodynamics and of the physical consequences stemming from assemblies of a large number of idential non-interacting particles (classical or quantum).

PREREQUISITES

The prerequisites stem from the courses "Fisica Generale" and "Istituzioni di Matematiche". At the beginning of this course a very coicise review of some key mathematical aspects will be provided to ease the reader into the topic.

TEACHING METHODS

Lectures will be delivered at the blackboard, in front of a live audience (no streaming unless strictly required e.g. by weather issues). The results of some interactive numerical simulation may be shown by the teacher.

SYLLABUS/CONTENT

1. Review and extension of some aspects of calculus

a. A brief overview about: single-valued real functions of a single variable, their approximation via the Taylor polynomial. Definite integral, both proper and improper.

b. Scalar functions of many real variables: many-variables derivatives. Applications: approximation of a function depending on many variables, some notion about maxima and minima of many variable functions. Elementary notions about many-variable integration: systems of coordinates and jacobian, with applications.

2. Aspects of probabiliy and statistic

a. Causal vs. stochastic variables. Discrete and continuous probability distributions and their general properties.

b. Average (expected) value and variance of a stochastic variable, moments of a distribution, higher moments (brief commentary).

c. Binomial distribution and its properties. Applications: random walk in one dimension, the diffusion equation in 1D.

d. Normal or Gaussian distribution and its properties.

e. Elementary discussion of the central limit theorem.

3. Statistical Physics

a. Extending the laws of classical mechanics to systems with a large number of particles: the need for a statistical approach.

b. The phase space in classical mechanics and its relevance for many-body systems.

c. Internal energy of a thermodynamical system: microscopical interpretation.

d. Microscopical meaning of entropy and its postulates.

e. Microscopical interpretation of the Principle zero of thermodynamics.

f. Microscopical interpretation of the concept of heat and of the first law of thermodynamics.

g. Heat propagation, the Fourier equation.

h. Fluctuations and their relevance/irrelevance (hints).

4. Thermodynamical ensembles and their statistical properties

a. Elementary introduction to the concepts of microcanonical, canonical and grand-canonical ensembles.

b. Canonical ensembles, the Gibbs distribution.

c. Partition function.

d. Average values through the partition function.

e. Connection between the partition function and thermodynamical quantities (hints).

5. Applications of the methods of Statistical Physics

a. The classical ideal gas: Maxwell-Boltzmann distribution. Heat capacity of a classical ideal gas.

b. Ideal quantum gases: the Bose-Einstein and the Fermi-Dirac distributions.

c. Applications of the Bose-Einstein distribution: the black body radiation, heat capacity of a solid within the Einstein model.

d. Applications of the Fermi-Dirac distribution: the specific heat of metals within the Debye model.

RECOMMENDED READING/BIBLIOGRAPHY

The bibliography will be communicated by the teacher during the first lecture. At any time, said bibliography will always be found on the Aulaweb page for the course.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

According to the official Academic time table.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam is composed of two steps: a short written paper and an oral examination.

ASSESSMENT METHODS

The written paper will require the students to solve some simple problem, which will allow to assess the concrete operative skills acquired by the student - also concerning the mathematical concepts introduced at the beginning of the course. During the oral examination the theoretical informations acquired by the student, and their integration within the background of their education in Physics, will be assessed.

FURTHER INFORMATION

Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.