Statistic Mechanics

Statistical Mechanics provides students with the tools necessary to analyze the properties of macroscopic systems by means of an adequate combination of the laws of mechanics, which describe the behavior of the "particles" of which they are constituted, and statistical methods. Statistical Mechanics also allows an interpretation of fundamental laws of Thermodynamics, such as the Principle of Increase in Entropy. The program of the subject consists of two parts: the first one is dedicated to the Classical Statistical Mechanics, where the "Gibbs method" is described, several ensembles are introduced that allow to describe the behavior of systems in different thermodynamic conditions and study different applications of the theory; The second one describes the methodology of Quantum Statistical Mechanics, analyzing the properties of quantum ideal gases (statistics of Bose-Einstein and Fermi-Dirac) and analyzing the behavior of various systems of interest (electron gas, boson gas, photon gas, etc.).

Learning outcomes:
Regarding the subject of Statistical Mechanics, the student will demonstrate:
·To know the conceptual bases of Statistical Mechanics, its general methodological aspects and some of the most relevant implications (in particular, the irreversibility of macroscopic systems and the relation of matter with Thermodynamics), as well as mastering the use of approximations of classic or quantum ideal gas in different situations and for different systems.

1. INTRODUCTION. Brief review of mathematical statistics. Elementary theory of probabilities. Probability distribution functions. Statistical entropy. Jaynes principle of maximum entropy. Markovian processes: master equation.

2. FUNDAMENTAL CONCEPTS OF STATISTICAL MECHANICS. Systems and ensembles. Microstates and macrostates of a physical system. Mechanical description of the microstates of a physical system. Limit of validity of of the classic description.
Phase space, phase volume and density of states: particles in a box and in a harmonic potential. Liouville's theorem.

3. STATISTICAL ENSEMBLES. Fundamental postulates of Statistical Mechanics. Equilibrium ensembles through the principle of maximum entropy: microcanonical, canonical and grand-canonical ensembles. Generalized ensemble. Evolution towards equilibrium and irreversibility: master equation. Einstein's fluctuation theory.

4. QUANTUM STATISTICAL MECHANICS Two-level systems. Statistical theory of paramagnetism. Einstein model of a solid.

Ideal quantum gases. Systems of identical particles. Partition function of a quantum ideal gas. Bose-Einstein and Fermi-Dirac statistics. Bose gas: Bose-Einstein condensation. Electron gas. Statistical theory of thermal radiation: photon gas. Debye model of the solid: phonon gas.

5. CLASSICAL LIMIT OF QUANTUM STATISTICS: MAXWELL-BOLTZMANN STATISTICS. Dilute limit of quantum statistics: Maxwell-Boltzmann statistics. Statistical mechanics in the classical limit. Ideal gas in the classical limit. Applications.

Basic:

L.M. Varela, H. Montes y T. Méndez, Mecánica Estadística, USC Editora, 2024
Teacher's notes of the subject and collections of solved exercises, which will be available to students in the Virtual Campus of the USC.

Complementary:

B. DIU, C. GUTHMANN, D. LEDERER, B. ROULET. Introduction à la Physique Statistique, Hermann (París, 1989).
T. L. HILL, An Introduction to Statistical Thermodynamics, Dover (New York, 1960).
K. HUANG, Statistical Mechanics, Wiley (New York, 1963).
R. KUBO, Statistical Mechanics. North-Holland (Amsterdam, 1974).
D. A. McQUARRIE, Statistical Mechanics, Harper Collins (Nueva York, 1976).
W. T. GRANDY, Foundations of Statistical Mechanics Reidel Publishing (Dordrecht, 1993).
L. E. REICHL, A Modern Course in Statistical Physics, University of Texas Press (Austin, 1980).
D. CHANDLER, Introduction to Modern Statistical Mechanics, Oxford University Press (New York, 1987).
J. DE LA RUBIA, J. BREY, Mecánica Estadística. Cuadernos UNED (Madrid, 2001).
L. D. LANDAU, E. M. LIFSHITZ Física Estadística. Vol.5 Curso de física teórica. Reverté (Barcelona, 1988).
J. L. CASTILLO y P. L. GARCIA YBARRA. Introducción a la Estadística Mediante Problemas. Sanz y Torres (Madrid, 1994).
C. FERNÁNDEZ TEJERO, J. M. RODRÍGUEZ PARRONDO, 100 problemas de Física Estadística, Alianza Editorial (Madrid, 1996).

BASIC AND GENERAL
CB1 - That students have proven to possess and understand knowledge in a study area that is part of the general secondary education base, and is often found at a level that, while supported by advanced textbooks, also includes some aspects which involve knowledge from the forefront of your field of study.
CB2 - Let students know how to apply their knowledge to their work or vocation in a professional way and possess the skills that are usually demonstrated through the elaboration and defense of arguments and the resolution of problems within their area of study.
CB3 - That students have the ability to gather and interpret relevant data (usually within their area of study) to issue judgements that include a reflection on relevant social, scientific or ethical topics.
CG1 - Own and understand the most important concepts, methods and results of the different branches of Physics, with a historical perspective of its development.
CG2 - Have the ability to gather and interpret relevant data, information and results, obtain conclusions and issue reasoned reports on scientific, technological or other issues that require the use of knowledge of Physics.
CG3 - Apply the acquired theoretical and practical knowledge as well as the capacity for analysis and abstraction in the definition and positioning of problems and in the search for their solutions in both academic and professional contexts.

TRANSVERSAL
CT1 - Acquire capacity for analysis and synthesis.
CT2 - Have organizational and planning capacity.
CT5 - Develop critical reasoning.

SPECIFIC
CE1 - Have a good understanding of the most important physical theories, locating in its logical and mathematical structure, its experimental support and the physical phenomenon that can be described through them.
CE2 - Be able to clearly handle the orders of magnitude and make appropriate estimates in order to develop a clear perception of situations that, although physically different, show some analogy, allowing the use of known solutions to new problems.
CE5 - Be able to perform the essence of a process or situation and establish a model of work of the same, as well as to carry out the necessary approaches in order to reduce the problem to a manageable level. He will demonstrate critical thinking to build physical models.
CE6 - Understand and master the use of mathematical and numerical methods most commonly used in Physics
CE8 - Be able to handle, search and use bibliography, as well as any relevant source of information and apply it to research projects and technical development of projects.

A course will be activated on the Moodle platform of the Virtual Campus, to which information of interest to students will be uploaded, as well as various teaching materials.

The general methodological guidelines established in the Memory of the Degree in Physics of the USC will be followed. The classes will be face-to-face and the distribution of expository and interactive hours will follow that specified in the Grade Report.

The tutorials can be face-to-face or online. If they are online, they will require an appointment, which is also recommended in person.

In the first opportunity, the evaluation of each student will be made through continuous evaluation made up of the following components:
a) Two classroom activities that will take place during class hours.
b) Final control.
The student's grade will be the maximum of the weighted average of a) + b) and the grade obtained in control b). To this end, the weight of the final control will be 65% and that of the classroom face-to-face activities 35%.

The student's grade in the second opportunity will correspond to the grade obtained in the corresponding official exam.

The continuous assessment grade is not retained for repeating students.


The qualification of "not presented" will be granted in accordance with the provisions of the regulations on the permanence of Bachelor's and Master's degrees in force at the University of Santiago.

In cases of fraudulent completion of exercises or tests, the following will apply to the provisions of the "Regulations for evaluating students' academic performance and reviewing grades":

"Article 16. Fraudulent performance of exercises or tests.

The fraudulent performance of any exercise or test required in the evaluation of a subject will imply the qualification of failed in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. It is considered fraudulent, among other things, the realisation of plagiarised works or obtained from sources accessible to the public without re-elaboration or reinterpretation and without citations to the authors and the sources ”.

Lectures in large group 24 hours
Lectures in small group 18 hours
Tutoring in very small groups or individualized 3 hours
Autonomous or group study 57 hours
Writing exercises, conclusions or other work 10.5 hours

Attend lectures.
Try to solve the practical exercises that are proposed throughout the course.
Consult the recommended bibliography.

Recommended prerequisites: Classical Mechanics I-II. Thermodynamics and Kinetic Theory. Mathematical Methods I-VI.