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\section*{thermodynamics}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesSouriau}{In terms of symplectic geometry (Souriau)}\dotfill \pageref*{ReferencesSouriau} \linebreak
\noindent\hyperlink{irreversible_thermodynamics}{Irreversible thermodynamics}\dotfill \pageref*{irreversible_thermodynamics} \linebreak
\noindent\hyperlink{ReferencesRelativisticThermodynamics}{Relativistic thermodynamics}\dotfill \pageref*{ReferencesRelativisticThermodynamics} \linebreak
\noindent\hyperlink{further_generalizations}{Further generalizations}\dotfill \pageref*{further_generalizations} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \textbf{[[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 \textbf{[[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 \textbf{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.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[entropy]], [[relative entropy]], [[second law of thermodynamics]], [[generalized second law of thermodynamics|generalized second law]], [[KMS state]]


\item [[intensive and extensive]]


\item [[thermal quantum field theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Introductions:

\begin{itemize}%
\item John Denker, \emph{Modern Thermodynamics}, (\href{http://www.av8n.com/physics/thermo}{web}, \href{http://www.av8n.com/physics/thermo-laws.pdf}{pdf})

\end{itemize}
Mathematically rigorous treatments:

\begin{itemize}%
\item Constantin Carathéodory, \emph{Untersuchung über die Grundlagen der Thermodynamik}, Math. Annalen 67, 355-386


\item Elliott H. Lieb, Jakob Yngvason, \emph{The Physics and Mathematics of the Second Law of Thermodynamics}, Phys.Rept. 310 (1999) 1-96 (\href{https://arxiv.org/abs/cond-mat/9708200}{arXiv:cond-mat/9708200})


\item Elliott H. Lieb, Jakob Yngvason, \emph{A Guide to Entropy and the Second Law of Thermodynamics}, Notices Amer. Math. Soc., 45, (1998) 571-581 (\href{https://arxiv.org/abs/math-ph/9805005}{arXiv:math-ph/9805005})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia: \href{http://en.wikipedia.org/wiki/Thermodynamics}{thermodynamics}, \href{http://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation}{fundamental thermodynamic relation}


\item Azimuth Project, \emph{\href{http://www.azimuthproject.org/azimuth/show/Thermodynamics}{Thermodynamics}}



\end{itemize}
\hypertarget{ReferencesSouriau}{}\subsubsection*{{In terms of symplectic geometry (Souriau)}}\label{ReferencesSouriau}

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

\begin{itemize}%
\item [[Jean-Marie Souriau]], \emph{Thermodynamique et g\'e{}om\'e{}trie}, Lecture Notes in Math. 676 (1978), 369--397 (\href{http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025}{scan})


\item [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]] \emph{Heat, cold and Geometry}, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (\href{http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm}{web}, \href{http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf}{pdf}, \href{https://doi.org/10.1007/978-94-009-7022-9_5}{doi:978-94-009-7022-9\_5})


\item [[Jean-Marie Souriau]], chapter IV ``Statistical mechanics'' of \emph{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. Birkh\"a{}user Boston, Inc., Boston, MA, 1997


\item [[Patrick Iglesias-Zemmour]], \emph{Essai de «thermodynamique rationnelle» des milieux continus}, Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 1-24 (\href{http://www.numdam.org/item/AIHPA_1981__34_1_1_0}{numdam:AIHPA\_1981\_\_34\_1\_1\_0})



\end{itemize}
Review includes

\begin{itemize}%
\item Charles-Michel Marle, \emph{From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics}, Entropy 2016, 18(10), 370 (\href{https://arxiv.org/abs/1608.00103}{arXiv:1608.00103})

\end{itemize}
The Souriau model of thermodynamics has been extented for ``geometric science of information'' (Koszul information geometry) with a general definition of [[Fisher metric]], Euler-Poincar\'e{} equation and variational definition of Souriau thermodynamics, as in:

\begin{itemize}%
\item Frederic Barbaresco, \emph{Koszul information geometry and Souriau geometric, temperature}, Capacity of Lie Group Thermodynamics, MDPI Entropy, n\textdegree{}16, 4521-4565 (2014) \href{http://www.mdpi.com/1099-4300/16/8/4521/pdf}{pdf}; \emph{Symplectic structure of information geometry: Fisher metric and Euler-Poincar\'e{} equation of Souriau Lie group thermodynamics}, GSI'15, Springer LCNS \textbf{9389}, 529-540 (2015) \href{https://doi.org/10.1007/978-3-319-25040-3_57}{doi}


\item Shun-ichi Amari, Chapter 2: \emph{Differential Geometrical Theory of Statistics} in \emph{Differential geometry in statistical inference}, Institute of Mathematical Statistics Lecture Notes - Monograph Series 1987, 19-94 (\href{https://projecteuclid.org/euclid.lnms/1215467059}{euclid:1215467059})


\item A. Bravetti, C. S. Lopez-Monsalvo, F. Nettel, \emph{Contact symmetries and Hamiltonian thermodynamics}, \href{http://arxiv.org/abs/1409.7340}{arxiv/1409.7340}



\end{itemize}
\hypertarget{irreversible_thermodynamics}{}\subsubsection*{{Irreversible thermodynamics}}\label{irreversible_thermodynamics}

A survey of [[irreversible thermodynamics]] is in

\begin{itemize}%
\item Ivan Vavruch, \emph{Conceptual problems of modern irreversible thermodynamics}, Chem. Listy 96 (2002) (\href{http://www.chemicke-listy.cz/docs/full/2002_05_01.pdf}{pdf})

\end{itemize}
For more on this see also \emph{[[rational thermodynamics]]}.

\begin{itemize}%
\item \'A{}lvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, Christopher Perry, \emph{Fluctuating work: from quantum thermodynamical identities to a second law equality}, Phys. Rev. X 6, 041017 \href{http://dx.doi.org/10.1103/PhysRevX.6.041017}{doi}

\end{itemize}
\hypertarget{ReferencesRelativisticThermodynamics}{}\subsubsection*{{Relativistic thermodynamics}}\label{ReferencesRelativisticThermodynamics}

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]]).

\begin{quote}%
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.


\end{quote}
(quote from Moller, \emph{\href{https://archive.org/details/theoryofrelativi029229mbp}{The theory of relativity}, 1952})

A standard textbook has been

\begin{itemize}%
\item Richard Tolman, \emph{Relativity, Thermodynamics and Cosmology}, Oxford 1934, reprinted by Dover 1987

\end{itemize}
but Tolman's approach has been called into question, see e.g.

\begin{itemize}%
\item Christian Fronsdal, \emph{Relativistic thermodynamics}, 2014 (\href{http://fronsdal.physics.ucla.edu/system/files/Gen.Rel_.LMP_.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Nils Andersson, \emph{General relativistic thermo-dynamics}, survey talk 2014 (\href{http://www.dpg-physik.de/dpg/pbh/aktuelles/pdf/Andersson.pdf}{pdf})


\item Sean A. Hayward, \emph{Relativistic thermodynamics} (\href{https://arxiv.org/abs/gr-qc/9803007}{arXiv:gr-qc/9803007})


\item Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb, \emph{What is the entropy of the universe?}, Class. Quant. Grav.26:145005, 2009 (\href{https://arxiv.org/abs/0801.1847}{arXiv:0801.1847})



\end{itemize}
\hypertarget{further_generalizations}{}\subsubsection*{{Further generalizations}}\label{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

\begin{itemize}%
\item [[John Baez]], Mike Stay, \emph{Algorithmic thermodynamics}, \href{http://math.ucr.edu/home/baez/thermo.pdf}{pdf}, \href{http://golem.ph.utexas.edu/category/2010/02/algorithmic_thermodynamics.html}{cafe}
\item [[M. Marcolli]], Ryan Thorngren, \emph{Thermodynamical semirings}, \href{http://arxiv.org/abs/1108.2874}{arXiv/1108.2874}
\item M. Zinsmeister, \emph{Thermodynamic formalism and holomorphic dynamical systems}, Amer. Math. Soc. 2000.
\item I. Itenberg, G. Mikhalkin, \emph{Geometry in the tropical limit}, \href{http://arxiv.org/abs/1108.3111}{arXiv/1108.3111}

\end{itemize}
[[!redirects irreversible thermodynamics]]



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