Section outline

  • General

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    Testi/Bibliography

    Michael Plischke, Birger Bergersen, Equilibrium Statistical Mechanics (3rd edition) World Scientific (PB)

    Luca Peliti, Simone Pigolotti, Stochastic Thermodynamics: An Introduction, Princeton University Press (PP)

    Esame:

    esame orale basato su domande di verifica della comprensione e della capacità di esposizione scientificamente corretta degli argomenti del programma, e sulla discussione degli esercizi svolti a casa

    Exam:

    oral exam with questions aimed to verify the understanding of the program and the ability to convey the topics in a scientifically sound way.  A discussion of the exercises done at home is also part of the exam.

  • Lezione 1

    Introduzione al corso

    Richiami di termodinamica, prima e seconda legge, potenziali termodinamici, energia libera di Helmholtz

    PB Capitolo 1

    Vedere la descrizione generale per la legenda coi libri di testo

    Introduction to the course. Review of thermodynamics. First and second law. Thermodynamic potentials: Helmholtz free energy.
    PB chapter 1

  • Lezione 2

    Energia libera di Gibbs, funzioni di risposta: calore specifico e compressibilità, criteri di equilibrio e stabilità PB capitolo 1

    Ensemble statistici: Microcanonico PB capitolo 2, paragrafo 2.1

    Gibbs free energy. Response functions: specific heats and compressibilities. Conditions for equilibrium and stability. PB chapter 1
    Microcanonical ensemble PB chapter 2, section 2.1

  • Lezione 3

    Ensemble statistici: Microcanonico (cont.), Canonico, Gran Canonico PB capitolo 2, paragrafi 2.1-2.2

    Microcanonical ensemble, canonical ensemble, grand canonical ensemble, PB chapter 2, sections 2.1 - 2.2 

  • Lezione 4

    Generalized ensembles, teacher's notes

    Maximum Entropy Principle, PB 2.5

    Thermodynamics of phase transition, PB section 1.8 (not the discussion on the latent heat
    and the Gibbs phase rule, but you are welcome to read it on your own).

    Introduction to the Ising model, PB 3.1

  • Lezione5

    Mean field theory of the Ising model (cont.), PB section 3.1 

    Peierls' argument PB 3.3

    Bragg-Williams Theory, PB section 3.2.

  • Lezione6

    Critical behaviour of MF theories, PB paragrafo 3.5

    Ising chain, exact solution with h=0, introduction to the transfer matrix formalism PB 3.6

  • Lezione7

    Ising chain, exact solution, correlation length, PB section 3.6
    Landau theory, symmetry considerations, PB section 3.7, and first part of sec. 3.8

    • Exercise set 1 Assignment
      Opened: Sunday, 12 October 2025, 12:00 AM
      Due: Tuesday, 14 October 2025, 5:00 PM

  • Lezione8

    Discussion on exercises, first set.

    Spin-spin correlation in mean field

    For this topic I follow the derivation of L. Peliti, "Statistical mechanics in a nutshell", 5.14

    For a more formal approach see also PB 3.10

  • Lezione9

    Spin-spin correlation in mean field (cont.), correlation length, Ginzburg criterion

    For this topic I follow the derivation of L. Peliti, "Statistical mechanics in a nutshell", pages 150-153, 160-161.

    For a more formal approach see also PB 3.10

  • Lezione10

    Non-Equilibrium Systems in linear regime, Onsager's Regression Hypothesis and Time Correlation
    Functions, fluctuation-dissipation theorem, DC 8.1, 8.2, 8.5

    DC= David Chandler, Introduction to Modern Statistical Mechanic, Oxford University Press.

    Stochastic dynamics, PP 2.5

    Master equations,  PP 2.6

  • Lezione11

    Master equations,  Perron Frobenius theorem, steady states and equilibrium states, detailed balance:  PP 2.6

  • Lezione12

    More on detailed balance

    Example of master equations: Fermi Golden rule, birth and death process

    Clock Model

    Stochastic trajectories : PP 2.7

  • Lezione13

    More on stochastic trajectories

    Stochastic thermodynamics: transition rates, out-of-equilibrium systems ,  generalized detailed balance condition. Stochastic work and stochastic heat, first law of stochastic thermodynamics. PP 3.1, 3.2

  • Lezione14

    Detailed Balance, general case PP 3.5

    Stochastic Entropy PP 3.6

    Entropy production rate and Schnakenberg formula PP 3.8

  • Lezione15

    Irreversible processes and forward/backward protocols PP 4.1

    Integral fluctuation relation for the total entropy, Jensen's inequality and second law of stochastic thermodynamics  PP4.2

  • Lezione 16

    Detailed fluctuation relation for the total entropy PP 4.5

    Entropy production for a particle on a loop PP 4.3

    Fluctuation relation for the work PP 4.6

  • Lezione17

    More on Jarzynski relation

    Derivation of the diffusion equation, I used a different approach than PP 2.8, but the final equations are the same

    Trajectory probability and fluctuation relation for Langevin equation PP 4.11, 4.12 

    • Exercise set 2 Assignment
      Opened: Monday, 17 November 2025, 12:00 AM
      Due: Tuesday, 18 November 2025, 4:00 PM

  • Lezione 18

    Fluctuation Relations and experiments, slides

    Discussion on exercises, set 2

  • Lezione 19

    More on the exercises

    Discussion on the out-of-equilibrium current in the clock model and its thermodynamic properties

  • Lezione 20

    Uncertainty relation and its application to molecular motors, PP 8.2, 8.3

    Eq. 8.22 in PP is not correct.

    The correct inequality to use is eq. 9 in

    https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.123.110602

  • Lezione 21

    Thermodynamics of Information Maxwell demon and experiments, slides Lezione 18.

    Szilard thought experiment, information theory and stochastic thermodynamics PP 5.1 and 5.3

    Mutual information and stochastic mutual information  PP 5.3

  • Lezione 22

    More on mutual information PP 5.3

    Sagawa - Ueda relation PP 5.4

  • Lezione 23

    Fluctuation Relation for a general feedback protocol, teacher's notes

    Quantum Jarzynski equality: this is not an exam topic

  • Lezione 24

    Discussion on exercises, set 3

    Bennett-Feynman information-fueled engine and information reservoirs PP 5.8