Physics

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.

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).

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).

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

Richiami di termodinamica e meccanica classica Teorema di Liouville, ipotesi ergodica e ensemble microcanonico, postulato fondamentale della meccanica statistica Elementi di calcolo della probabilita' (assiomi, proprieta', esempi di distribuzioni notevoli) Ensemble microcanonico: additivita' entropia, teorema equipartizione, gas ideale classico. Riflessioni su concetto di entropia (paradosso di Gibbs, entropia come ignoranza, irreversibilita') Ensemble canonico: funzione di partizione, energia libera di Helmholtz, densita' di probabilita' dell'energia. Oscillatore armonico classico - relazione barometrica - gas di particelle relativistiche Relazioni di fluttuazione-dissipazione e equivalenza tra insieme canonico e microcanonico Gas perfetto. Equipartizione dell'energia. Statistica di Maxwell Boltzmann; condizioni di validita` della meccanica statistica classica. Distribuzione di Maxwell; Insieme gran canonico. Il gran potenziale. Funzioni termodinamiche nell' insieme gran canonico. Equivalenza insieme canonico e gran canonico Gas quantistici. Distribuzioni di Fermi-Dirac e di Bose-Einstein. Limite classico: alte temperature e/o basse densità. Gas di Fermi allo zero assoluto: energia di Fermi, energia media, pressione. Calore specifico di un gas di Fermi alle basse temperature. Condensazione di un gas di Bose-Einstein. Vibrazioni nei solidi, modello di Debye Spettro del corpo nero. Formula di Planck.

Thermodynamics

Statistical Mechanics di Kerson Huang, Wiley Statistical Physics by Roberto Piazza Springer S-K Ma, Statistical Mechanics, World Scientific, Singapore (1985)

in the classroom

Ability to solve statistical mechanics problems Ability to orally explain the topics listed in the program

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

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).

- 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)

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).

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.