2022/2023

2022/2023

DM270

MAT/07 (MATHEMATICAL PHYSICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (29/09/2022 - 13/01/2023)

6

60 lesson hours

Italian

ORAL TEST

VIRGA EPIFANIO GUIDO (titolare) - 6 ECTS

The title of this course could be condensed in a single word: Entropy. It is a course on the second law of thermodynamics, properly phrased in mathematical terms, but with an eye on the historical development of the notion of entropy, so central in modern science. Thus, the work of giants such as Clausius, Boltzmann, and Onsager on this topic will be expounded and revised. The only prerequisites are basic courses on Analytical Mechanics and Thermodynamics, which are usually present in the syllabi of most BSc's in Mathematics, Physics, or Engineering.

The aim of this course is to provide students with the ability of mastering the elusive concept of entropy. As a result, they should be able to derive the free energy functional of simple model systems. More importantly, they should learn to tell whether a mathematical model for a phase of matter is compatible with first principles.

1. Entropy in the work of Clausius: Historical background; thermodynamic background; the second law of thermodynamics.
2. Entropy in the work of Carathéodory: Empirical temperature; Carathéodory's axiom; absolute temperature; growth of entropy.
3. Entropy in the work of Boltzmann: Distribution function; entropy flux; reformulation and extrapolation of Boltzmann's entropy; entropy constant and quantisation of phase space.
4. Enthalpy: Equation of state (EOS); special cases; van der Waals equation.
5. Gases and rubber: Energetic and entropic parts of elasticity; entropy-induced elasticity; entropic forces; entropy and disorder; entropy of a rubber rod.
6. Statistical thermodynamics: Canonical ensemble; ergodic hypothesis.
7. Entropy and energy: a modicum of thermodynamics; potential energies; available free energies; minimisation of free energy; competition between energy and entropy.
8. Phase transitions: Molecular interactions; phase equilibrium; phase transition in a van der Waals gas; Onsager's theory for liquid crystals.
9. Shape memory alloys: Phenomena and applications; statistical model for shape memory alloys; entropic stabilisation; pseudo-elasticity.
10. The third law of thermodynamics: Entropy constant and Planck constant; capitulation of entropy.
11. Gibbs paradox: entropy of mixing; psuedo-Gibbs-paradox.
12. Radiation thermodynamics: Photon transport equation; Planck distribution; dissipative and radiative entropy sources.

This course partakes the nature of a Reading Course (as in the English tradition). So, traditional lectures read by the instructor will be accompanied by seminars where students will illustrate their own readings.

1. I. Mueller, W. Weiss, Entropy and Energy, Springer-Verlag, Berlin, 2005, ISBN: 978-3-540-24281-9.
2. I. Mueller, A history of thermodynamics, Springer-Verlag, Berlin, 2007, ISBN: 978-3-540-46226-2.
3. R.H. Swendsen, An introduction to statistical mechanics and thermodynamics, Oxford University Press, Oxford, 2012, ISBN: 978-0-19-964694-4.
4. P. Podio-Guidugli, Continuum thermodynamics, Springer Nature, 2019, ISBN: 978-3-030-11156-4.

Students' seminars, possibly supplemented by questions raised in the public discussion, will form the basis for their final evaluation. Their ability to stand up to criticism will be particularly appreciated.