Statistical Physics

General characterization

Prerequisites

Mechanics and Thermodynamics

Bibliography

A Thermodynamics and Statistical Mechanics, W. Greiner, L. Neise L., H. Stocker 1997 Springer
B Introdução à Física Estatística, J.P. Casquilho, P.I. Cortez Teixeira, 2015 IST Press
C Statistical and Thermal Physics, F. Reif 1985 McGraw-Hill
D Statistical Physics, F. Mandl, 1988, Wiley

Teaching method

Exposition of the "theory" for 45 min. 

Application exercises during the rest of the class. The first exercises are very "basic" in order to clarify and stabilize the concepts introduced during the first part of the lesson.

  During these two parts, questions are made to probe and correct (mis-) understanding that students have about some phenomena.

Evaluation method

Article 1 - Classes
1. Theoretical-practical classes are mandatory, with attendance being recorded, for all students without previous attendance.
2. Students with prior attendance are exempt from theoretical-practical classes.
3. No justification for occasional absences is accepted. Each student must manage the possibility of not attending 1/3 of the classes for potential commitments or unforeseen situations, including occasional illness.
4. At the beginning of the semester, a provisional schedule of classes and test dates will be provided.

Article 2 - Assessment

1. The assessment for this course unit (CU) will be differentiated for students with and without attendance.
2. The assessment for students with attendance is carried out through two tests (each test score rounded to one decimal place) or an exam. The final grade (CF) is the average, rounded to the nearest whole number, of the test grades (CT) or the exam grade.
3. The assessment for students without attendance consists of coursework (30%) and tests or an exam (70%). Each coursework grade is rounded to one decimal place, and the final coursework grade (CB) is the average of the coursework grades, rounded to the nearest whole number.
4. There will be at least five coursework assignments throughout the semester. From the date of publication on the CLIP platform (after announcement in class), students will have one week to complete each assignment. The coursework can be completed in groups of up to two students and must be submitted on paper during class. Under no circumstances will submissions via email or after the deadline be accepted. Failure to submit coursework within the defined period will result in a score of zero. Students may be required to justify their coursework in person.

Article 3 - Attendance
1. A student who attends at least 2/3 of the theoretical-practical classes will obtain attendance.
2. The list of students with prior attendance will be available on CLIP under "Support Documentation > Others" by the end of the first week of classes.

Article 4 - Final Grade
1. The final grade (CF) is calculated using the following formula, rounded to the nearest whole number:
o CF = CT × 0.7 + CB × 0.3 (attendance in the current academic year)
o CF = CT (attendance in a previous academic year)
2. Any student with attendance and a final grade equal to or greater than ten will pass this Course Unit.
3. If the final grade is greater than 16, the student is required to take an additional assessment (e.g., oral exam).
4. Failure to attend the additional assessment implies that the student accepts the final grade of 16.

Article 5 - Grade Improvement
1. A student wishing to improve their grade must follow the legal procedures for enrolment.
2.If the final grade after improvement exceeds 16, the conditions outlined in points 3 and 4 of Article 4 apply.

Article 6 - Classroom Conduct
To ensure that everyone benefits from the learning experience, students must adhere to the following rules in class:
a. Punctuality: You should be present in the classroom at the start of the class. The lecturer may deny entry to students who are more than five minutes late. The classroom is not a café where you can enter and leave at will.
b. Unless requested by the lecturer, mobile phones must not be used during the class. Taking photos with any device is not permitted.

Article 7 - Assessment Moments
1. Each test will cover the material taught up to the last class that has not yet been assessed.
2. Although the tests are not cumulative, due to the nature of the topics covered in this course unit (CU), it is possible that an assessment may refer to knowledge from previously assessed material.
3. The schedule and rooms for the tests or exam will be published on the CLIP platform on the day of the assessment.
4. Registration for tests is mandatory and must be done exclusively through the CLIP platform. The registration period opens two weeks before the assessment and closes one week before. If a student does not arrive within the first fifteen minutes of the test, they will be considered unregistered.
5. Unregistered students may only take the test if, fifteen minutes after the test begins, there are available seats and test booklets for all interested students.
6. The test ends at the same time for all students (registered and unregistered).
7. During the test, students are not allowed to have a mobile phone, watch, or any electronic communication device. Failure to comply with this rule will always be considered fraud.
8. Upon entering the room, students must have the following items in hand, which they may take to their seat: ID card, scientific calculator (non-programmable and non-graphical), writing materials (only answers written in pen without traces of pencil will be corrected). All other belongings must be placed where the invigilating lecturer directs.
9. During the test, consultation of any personal or external materials is prohibited. Failure to comply with this rule will always be treated as fraud.
10. Test booklets are not required for assessments. Each student will be provided with a test booklet, which cannot be unbound, and a formula sheet.
11. Any instance of fraud during an assessment will be handled in accordance with the faculty''s assessment regulations.

Subject matter

Chapter I – Classical Thermodynamics
a) First Law – Heat, Work, Internal Energy, Energy Conservation.
b) Second Law – Irreversibility, Entropy, Carnot Cycle. Entropy of Ideal Gas – Gibbs Paradox.
c) Thermodynamic Potentials – Helmholtz Free Energy, Enthalpy, Gibbs Free Energy. Chemical Potential. Applications.
d) System Stability – Combination of the 1st and 2nd Law. Equilibrium State.

Chapter II – Probability Concepts
a) Basic Statistical Concepts – Sample Space, Event, Probability, and Frequency. Continuous and Discrete Variables.
b) Probability Distribution – Binomial Distribution, Poisson Distribution, Gaussian Distribution. Central Limit Theorem. Approximations. Mean Value, Variance, and Relative Fluctuations.
c) Random Walk – General Case 1D and 3D. Applications.

Chapter III – Microcanonical System
a) Isolated System – Macrostates, Microstates. Statistical Weight Ω(E,V,N).
b) Fundamental Postulates of Statistical Physics – Microcanonical Distribution. Ergodic Theorem, Equiprobability Postulate. System in and out of Equilibrium.
c) System Definitions – Statistical Entropy, Temperature, Pressure, Chemical Potential.
d) Schottky Defect Concentration – Calculation of Concentration, Arrhenius Law. Application.

Chapter IV – Canonical System
a) Boltzmann Distribution – Canonical System, Partition Function.
b) Application of Thermodynamic Potentials – Calculation of the System''s Average Energy, Fluctuations. Applications.
c) Third Law – Heat Capacity, Dulong-Petit Law, Einstein Model, Debye Law, Nernst Theorem.
d) Application – Two-Level System, Partition Function of N Equivalent Systems.

Chapter V – Classical Ideal Gas
a) Definition of Classical Ideal Gas – Identical Particles, Degrees of Freedom, Monoatomic Gas.
b) Partition Function – Calculation, Energy Equipartition Theorem, Specific Heat.
c) Pressure and Internal Energy – Calculation. Maxwell-Boltzmann Distribution.
d) Classical-Quantum Limit – Classical Limit of State Occupation, Quantum Concentration, Significance of the Limit.

Chapter VI – Macrocanonical System
a) Gibbs Distribution – Thermal and Diffusion Equilibrium, Partition Function.
b) Fermi-Dirac Distribution – Definition, Factorisation of the Grand Partition Function, High Temperature.
c) Bose-Einstein Distribution – Definition, Low Temperature, Classical and Quantum Regimes. Bose-Einstein Condensate.
d) Classical-Quantum Limit – Classical Limit of State Occupation, Quantum Concentration, Significance of the Limit.

Chapter VII – Quantum Gas
a) Free Electrons – Electronic Density in Metals, Sommerfeld Model.
b) Fermi Sphere at T = 0 – Electron Distribution in Reciprocal Space (multi-dimensions), Fermi Energy, Fermi Temperature, Fermi Wavelength, Fermi Velocity.
c) Specific Heat of Electron Gas – Evolution of Fermi Distribution with Temperature, Calculation.
d) Magnetisation of an Electron Gas – Calculation, Comparison with Ideal Gas, Pauli Susceptibility. Applications.

Programs

Programs where the course is taught: