Physics

SERGIO CAPRARA

GENERAL OBJECTIVES:
The first main objective of the course is to introduce the students to the fundamental concepts of Statistical Mechanics (SM). Generally speaking the students must understand how to derive the thermodynamic description of a macroscopic system from its microscopic laws. This knowledge is of obvious relevance for their background in Physics and for the more specialised courses of the next years.
SPECIFIC OBJECTIVES:
A - Knowledge and understanding
OF 1) basic concepts in equilibrium SM and in probability theory
OF 2) computation rules in classical SM (microcanonical, canonical and grand canonical ensembles); equivalence between the different rules and their application to non-interacting systems;
OF 3) Basic concepts and example of interacting systems (Van Der Waals equation).
OF 4) Fundamental concepts and computation rules in Quantum SM; application to the quantum perfect gas, Bose-Einstein and Fermi-Dirac distributions;
OF 5) Thermodynamic properties of quantum bosonic and fermionic systems; high and low temperature limits; Sommerfeld expansion for a Fermionic gas; specific heat in a solid; Debye theory.
OF 6) Brief summary of quantisation of the electromagnetic radiation, black body radiation.
B - Application skills
OF 7) The second objective is to prepare the students to actively solve problems in physics where SM concepts are required. This will happen at first with problems structured with a conceptual scheme similar to the one discussed and applied during the course. However, as their preparation progresses, students are also expected to use SM concepts for solving new problems in different applications.
C - Autonomy of judgment
OF 8) The third and more ambitious objective is to teach the students to think in probabilistic and statistic terms, using concepts and methods from SM as a powerful problem solving tool, both in physics and in different fields (e.g. socio-economic systems, biological systems, medical applications)
D - Communication skills
OF 9) Besides having a clear understanding of the new acquired concepts in SM, the studend should correspondingly acquire the ability to communicate and transmit these concepts in a clear and direct way.
E - Ability to learn
OF 10) The students should become able to read and understand scientific books and articles where SM concepts are involved and should be able to deepen autonomously their knowledge in this field.

a) It is essential to know the basic concepts: i) of classical mechanics and analytical mechanics (in particular the Hamiltonian formalism); ii) of the theory of probability; iii) of thermodynamics as thermodynamic variables, state functions (internal energy, entropy, free energy); iv) of quantum mechanics (in particular, of quantum state, of energy levels); v) of the description of the free particle and of the quantum harmonic oscillator.

b) It is important to be familiar with the main probability distributions (Gaussian, Poissonian, Binomial, etc.) and with examples of physical phenomena that are regulated by these distributions; know the basics of mathematical analysis and distribution theory (e.g., the Dirac delta).

Statistical Mechanics and Thermodynamics

Introduction: Statistical Mechanics and Thermodynamics.
Time-reversal invariance, Poincare’s Theorem and definition of thermodynamic equilibrium.
Extensive and intensive variables.
Phase space, region of motion and thermodynamical observables.
Equilibrium and reversible transformations.
Work, heat and Entropy.
Fundamental postulate of Statistical Mechanics.
Statistical ensembles.
Liouville’s theorem.
Micfocanonical ensemble.
Ergodic hypothesis.
Trajectories and volume in the region of motion.
Lengths and volumes in high-dimensional spaces.
Peak integration and Stirling approximation.



Brief summary of Probability theory

Statistical probability.
General definition and probability space.
Independent events.
Conditional probability.
Law of complete probability.
Discrete and continuous random variables.
Probability and density distribution of probability for one or more variables.
Marginalization, conditional probability, averages.
Transformation of random variables.
Sum of random variables, change of random variables, linear transformation.
Binomial and Poisson probability distributions.
Gaussian probability density for one or more variables.
Fourier transform of the Gaussian probability density and integral representation of the Dirac’s delta.
Generating function of moments, characteristic function, of a probability distribution.
Gaussian distribution and its moments.
Large number law.
Cumulants of a distribution and generating function: Gaussian distribution and its moments.
Central limit theory.

Statistical Mechanics: rules of calculations

Average value and probability distribution of the macroscopic variables.
Summary of thermodynamica potentials.
Subsystems and equilibrium conditions.
Maxwell-Boltzmann distribution.
Equipartition theorem and virial theorem.
Entropy of the perfect gas.
Gibbs paradox and Sackur-Tetrode equation.
Barometric equation of the perfect gas.
Perfect gas of harmonic oscillators.
Perfect gas of relativistic classic particles.
Rule at constant temperature: Canonical ensemble.
Rule at constant pressure: (T,P) ensemble.
Energy fluctuations in the canonical ensemble.
Equipartition theorem in the canonica ensemble.
Rule for open systems at constant temperature: gran canonical ensemble.
Density fluctuations.
Interacting classical gas, virial expansion.

Quantum Stsatistical mechanics

Nernst postulate (Third principle of thermodynamics).
Fundamental postulate of quantum statistical mechanics.
Density matrix operator.
Quantum ensembles.
Quantum ideal gases: Bose and Fermi cases.
Occupation numbers and state functions of quantum ideal gases.
Fermi/Bose gas: low-density and high-temperature limit.
Fermi gas: high-density and low-temperature limit, Fermi energy.
Bose gas: high-density and low-temperature limit, Bose-Einstein condensation.
Gas of quantum harmonic oscillators (phonons): Einstein and Debye models.
The Black body radiation.

1) Statistical Mechanics, K. Huang, Wiley (1987)
2) Statistical Physics - Volume 5, L.D. Landau and E.M. Lifshitz, Elsevier Ltd. (1980)

Lectures and exercises at the blackboard.
Lectures will introduce and explain the various concepts, both fundamental and secondary.
Exercises have a twofold aim: first they clarify and exemplify the concepts treated during the lectures and, secondly, they allow the students to acquire the skills to autonomously solve the problems at different levels of difficulty. Lectures and exercises are strictly interconnected and there will be no specific and precise separation and scheduling between the two

Students will benefit from direct discussion with the teacher and the opportunity to discuss ideas and methodologies with colleagues. The succession of topics developed, the exercises carried out, and the supporting material will be made available on the teacher's website https://www2.phys.uniroma1.it/doc/caprara/SCaprara/sergio-caprara-MS.html

The exam is formed by two complementary parts: a written and an oral part.
The written part (which usually lasts 2.5-3 hours) aims at testing the skills of the student in solving problems of classical and quantum statistical mechanics similar to those presennted durinng the course. The oral part, usually taking 30-40 minutes, is about the most relevant topics of the course. To pass the exam the student should be able to presennt a topic or repeat a calculation presented during the lectures. The student will be asked to apply the learned methods to exercises or examples and situatoins similar to those discussed in the course.
In the evaluation account will be taken of:
- correctness and completeness of the exposed concepts;
- clarity and accuracy of the presentation;
- capability of analytically develop the theoretical topics;
- skills in problem solving (methods and results).