# nLab thermodynamics

## Surveys, textbooks and lecture notes

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

It is practically impossible to model a macroscopic physical system in terms of the microscopic kinematical and dynamical variables of all its particles. Thus one makes a hierarchical reduction in which this complexity is reduced to a small number of collective variables. The theoretical framework for such reductions for systems is statistical mechanics or statistical physics.

One special case of hierarchical reduction is the limit of large volumes #$V$, in which the number of particles (of each species) per volume, $N/V$, stays constant. This is called the thermodynamic limit in statistical physics. Under some standard assumptions like homogeneity (spacial and possibly directional) and stability (no transitory effects), there is a small number of collective variables characterizing the system. Such a description can be (and historically was) postulated as an independent self-consistent phenomenological theory even without going into the details of statistical mechanics; such a description is called equilibrium thermodynamics, which is believed to be deducible from statistical mechanics, as has been partially proved for some classes of systems. Sometimes transitional finite-time phenomena are described either statistically by studying stochastic processes or by a more elaborate hierarchical form of thermodynamics, so-called nonequilibrium thermodynamics.

One of the basic characteristics of a thermodynamical system is its temperature, which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, entropy, enthalpy etc.

## References

### General

Introductions:

• John Denker, Modern Thermodynamics, (web, pdf)

Mathematically rigorous treatments:

• Constantin Carathéodory, Untersuchung über die Grundlagen der Thermodynamik, Math. Annalen 67, 355-386

• Elliott H. Lieb, Jakob Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, Phys.Rept. 310 (1999) 1-96 (arXiv:cond-mat/9708200)

• Elliott H. Lieb, Jakob Yngvason, A Guide to Entropy and the Second Law of Thermodynamics, Notices Amer. Math. Soc., 45, (1998) 571-581 (arXiv:math-ph/9805005)

### In terms of symplectic geometry (Souriau)

A covariant formalization of thermodynamics in terms of moment maps in symplectic geometry is due to

• Jean-Marie Souriau, Thermodynamique et géométrie, Lecture Notes in Math. 676 (1978), 369–397 (scan)

• Patrick Iglesias-Zemmour, Jean-Marie Souriau Heat, cold and Geometry, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (web, pdf, doi:978-94-009-7022-9_5)

• Jean-Marie Souriau, chapter IV “Statistical mechanics” of Structure of dynamical systems. A symplectic view of physics . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

• Patrick Iglesias-Zemmour, Essai de «thermodynamique rationnelle» des milieux continus, Annales de l’I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 1-24 (numdam:AIHPA_1981__34_1_1_0)

• Jean-Marie Souriau, J.-M. Mécanique statistique, groupes de Lie et cosmologie. In Colloques int. du CNRS; numéro 237; Aix-en-Provence, France, 1974; pp. 24–28, 59–113. English translation by F. Barbaresco, April, 2020. Available online: https://www.academia.edu/42630654/Statistical_Mechanics_Lie_Group_and_Cosmology_1_st_part_Symplectic_Model_of_Statistical_Mechanics(access on 20 April 2020)

Review includes

• Charles-Michel Marle, From tools in symplectic and Poisson geometry to Souriau’s theories of statistical mechanics and thermodynamics, Entropy 2016, 18(10), 370 (arXiv:1608.00103)

The Souriau model of thermodynamics has been extented for “geometric science of information” (Koszul information geometry) with a general definition of Fisher metric, Euler-Poincaré equation and variational definition of Souriau thermodynamics, as in:

• Frederic Barbaresco, Koszul information geometry and Souriau geometric, temperature, Capacity of Lie Group Thermodynamics, MDPI Entropy, n°16, 4521-4565 (2014) pdf; Symplectic structure of information geometry: Fisher metric and Euler-Poincaré equation of Souriau Lie group thermodynamics, GSI’15, Springer LCNS 9389, 529-540 (2015) doi

• Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498. https://www.mdpi.com/1099-4300/22/5/498

• Shun-ichi Amari, Chapter 2: Differential Geometrical Theory of Statistics in Differential geometry in statistical inference, Institute of Mathematical Statistics Lecture Notes - Monograph Series 1987, 19-94 (euclid:1215467059)

• A. Bravetti, C. S. Lopez-Monsalvo, F. Nettel, Contact symmetries and Hamiltonian thermodynamics, arxiv/1409.7340

### Irreversible thermodynamics

A survey of irreversible thermodynamics is in

• Ivan Vavruch, Conceptual problems of modern irreversible thermodynamics, Chem. Listy 96 (2002) (pdf)

• Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, Christopher Perry, Fluctuating work: from quantum thermodynamical identities to a second law equality, Phys. Rev. X 6, 041017 doi

### Relativistic thermodynamics

Making sense of thermodynamics when taking into account special relativity and ultimately, possibly, general relativity (gravity) is notoriously subtle (even ignoring the issue of Bekenstein-Hawking entropy).

Shortly after the advent of the relativity theory, Planck, Hassenoerl, Einstein and others advanced separately a formulation of the thermodynamical laws in accordance with the special principle of relativity. This treatment was adopted unchanged including the first edition of this monograph. However it was shown by Ott and indepently by Arzelies, that the old formulation was not quite satisfactory, in particular because generalized forces were used instead of the true mechanical forces in the description of thermodynamical processes.

The papers of Ott and Arzelies gave rise to many controversial discussions in the literature and at the present there is no generally accepted description of relativistic thermodynamics.

(quote from Moller, The theory of relativity, 1952)

A standard textbook has been

• Richard Tolman, Relativity, Thermodynamics and Cosmology, Oxford 1934, reprinted by Dover 1987

but Tolman’s approach has been called into question, see e.g.

• Christian Fronsdal, Relativistic thermodynamics, 2014 (pdf)

• Nils Andersson, General relativistic thermo-dynamics, survey talk 2014 (pdf)

• Sean A. Hayward, Relativistic thermodynamics (arXiv:gr-qc/9803007)

• Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb, What is the entropy of the universe?, Class. Quant. Grav.26:145005, 2009 (arXiv:0801.1847)

### Further generalizations

Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like Matsubara formalism in QFT.

Mathematical analogies and generalizations include also

• John Baez, Mike Stay, Algorithmic thermodynamics, pdf, cafe
• M. Marcolli, Ryan Thorngren, Thermodynamical semirings, arXiv/1108.2874
• M. Zinsmeister, Thermodynamic formalism and holomorphic dynamical systems, Amer. Math. Soc. 2000.
• I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv/1108.3111