Physics

IRENE ROSANA GIARDINA

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 mandatory to know the basic concepts: i) of classical mechanics and anaytical mechanics (in particular the Hamiltonian formalism); ii) of probability theory; iii) of thermodynamics, like thermodynamic variables, state functions (internal energy, entropy, free energy); iv) quantum mechanics (in particular quantum state, energy levels); v) quantum free particle and harmonic oscillator.
b) it is important to be familiar with the main probability distributions (Gaussian, Poissonian, Binomial, etc.) and of physicsl examples that are ruled by these distributions; basic concepts of calculus and of distribution theory (e.g. 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.
Fundamentals on the ISING model (mean-field)

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

- K. Huang (H), "Statistical Mechanics", (Wiley, 1987)
- L.D. Landau e E.M. Lifsits (LL), "Fisica Statistica" (parte I), (Editori Riuniti, 1978)
- J. Sethna: "Statistical Mechanics: entropy, order parameters and complexity" (Oxford Un. Press, 2006)
- S-K Ma, Statistical Mechanics, World Scientific, Singapore (1985)

The detailed program and specific references to books and book chapters can be found in the teacher's website
https://sites.google.com/uniroma1.it/irene-giardina/teaching/meccanica-statistica

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.

Attendance to lectures is on voluntary basis even though strongly recommended to not only master the theoretical part, but also to learn in class how to solve problems.

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 during 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).