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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> thermodynamics </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6252/#Item_8" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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<p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesSouriau'>In terms of symplectic geometry (Souriau)</a></li> <li><a href='#irreversible_thermodynamics'>Irreversible thermodynamics</a></li> <li><a href='#ReferencesRelativisticThermodynamics'>Relativistic thermodynamics</a></li> <li><a href='#further_generalizations'>Further generalizations</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>It is practically impossible to model a macroscopic <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> in terms of the microscopic kinematical and dynamical variables of all its <a class="existingWikiWord" href="/nlab/show/particles">particles</a>. 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 <a class="existingWikiWord" href="/nlab/show/statistical+mechanics">statistical mechanics</a> or statistical physics.</p> <p>One special case of hierarchical reduction is the limit of large volumes #<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, in which the number of particles (of each species) per volume, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">/</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">N/V</annotation></semantics></math>, stays constant. This is called the <strong><a class="existingWikiWord" href="/nlab/show/thermodynamic+limit">thermodynamic limit</a></strong> 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 <strong><a class="existingWikiWord" href="/nlab/show/equilibrium">equilibrium</a> thermodynamics</strong>, 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 <strong>nonequilibrium thermodynamics</strong>.</p> <p>One of the basic characteristics of a thermodynamical system is its <a class="existingWikiWord" href="/nlab/show/temperature">temperature</a>, which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, enthalpy etc.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/temperature">temperature</a>, <a class="existingWikiWord" href="/nlab/show/inverse+temperature">inverse temperature</a>, <a class="existingWikiWord" href="/nlab/show/absolute+zero">absolute zero</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boltzmann+constant">Boltzmann constant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boltzmann+distribution">Boltzmann distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+thermodynamics">generalized second law</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intensive+and+extensive">intensive and extensive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenstate+thermalization+hypothesis">eigenstate thermalization hypothesis</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Introductions:</p> <ul> <li>John Denker, <em>Modern Thermodynamics</em>, (<a href="http://www.av8n.com/physics/thermo">web</a>, <a href="http://www.av8n.com/physics/thermo-laws.pdf">pdf</a>)</li> </ul> <p>Mathematically rigorous treatments:</p> <ul> <li> <p>Constantin Carathéodory, <em>Untersuchung über die Grundlagen der Thermodynamik</em>, Math. Annalen 67, 355-386</p> </li> <li> <p>Elliott H. Lieb, Jakob Yngvason, <em>The Physics and Mathematics of the Second Law of Thermodynamics</em>, Phys.Rept. 310 (1999) 1-96 (<a href="https://arxiv.org/abs/cond-mat/9708200">arXiv:cond-mat/9708200</a>)</p> </li> <li> <p>Elliott H. Lieb, Jakob Yngvason, <em>A Guide to Entropy and the Second Law of Thermodynamics</em>, Notices Amer. Math. Soc., 45, (1998) 571-581 (<a href="https://arxiv.org/abs/math-ph/9805005">arXiv:math-ph/9805005</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p>Wikipedia: <a href="http://en.wikipedia.org/wiki/Thermodynamics">thermodynamics</a>, <a href="http://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation">fundamental thermodynamic relation</a></p> </li> <li> <p>Azimuth Project, <em><a href="http://www.azimuthproject.org/azimuth/show/Thermodynamics">Thermodynamics</a></em></p> </li> </ul> <h3 id="ReferencesSouriau">In terms of symplectic geometry (Souriau)</h3> <p>A covariant formalization of thermodynamics in terms of <a class="existingWikiWord" href="/nlab/show/moment+maps">moment maps</a> in <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> is due to</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Thermodynamique et géométrie</em>, Lecture Notes in Math. 676 (1978), 369–397 (<a href="http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025">scan</a>)</p> </li> <li id="IglesiasSouriau84"> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a> <em>Heat, cold and Geometry</em>, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (<a href="http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm">web</a>, <a href="http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf">pdf</a>, <a href="https://doi.org/10.1007/978-94-009-7022-9_5">doi:978-94-009-7022-9_5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, chapter IV “Statistical mechanics” of <em>Structure of dynamical systems. A symplectic view of physics</em> . 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</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Essai de «thermodynamique rationnelle» des milieux continus</em>, Annales de l’I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 1-24 (<a href="http://www.numdam.org/item/AIHPA_1981__34_1_1_0">numdam:AIHPA_1981__34_1_1_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, 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)</p> </li> </ul> <p>Review includes</p> <ul> <li id="Marle16">Charles-Michel Marle, <em>From tools in symplectic and Poisson geometry to Souriau’s theories of statistical mechanics and thermodynamics</em>, Entropy 2016, 18(10), 370 (<a href="https://arxiv.org/abs/1608.00103">arXiv:1608.00103</a>)</li> </ul> <p>The Souriau model of thermodynamics has been extented for “geometric science of information” (Koszul information geometry) with a general definition of <a class="existingWikiWord" href="/nlab/show/Fisher+metric">Fisher metric</a>, Euler-Poincaré equation and variational definition of Souriau thermodynamics, as in:</p> <ul> <li> <p>Frederic Barbaresco, <em>Koszul information geometry and Souriau geometric, temperature</em>, Capacity of Lie Group Thermodynamics, MDPI Entropy, n°16, 4521-4565 (2014) <a href="http://www.mdpi.com/1099-4300/16/8/4521/pdf">pdf</a>; <em>Symplectic structure of information geometry: Fisher metric and Euler-Poincaré equation of Souriau Lie group thermodynamics</em>, GSI’15, Springer LCNS <strong>9389</strong>, 529-540 (2015) <a href="https://doi.org/10.1007/978-3-319-25040-3_57">doi</a></p> </li> <li> <p>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</p> </li> <li id="Amari87"> <p>Shun-ichi Amari, Chapter 2: <em>Differential Geometrical Theory of Statistics</em> in <em>Differential geometry in statistical inference</em>, Institute of Mathematical Statistics Lecture Notes - Monograph Series 1987, 19-94 (<a href="https://projecteuclid.org/euclid.lnms/1215467059">euclid:1215467059</a>)</p> </li> <li> <p>A. Bravetti, C. S. Lopez-Monsalvo, F. Nettel, <em>Contact symmetries and Hamiltonian thermodynamics</em>, <a href="http://arxiv.org/abs/1409.7340">arxiv/1409.7340</a></p> </li> </ul> <h3 id="irreversible_thermodynamics">Irreversible thermodynamics</h3> <p>A survey of <a class="existingWikiWord" href="/nlab/show/irreversible+thermodynamics">irreversible thermodynamics</a> is in</p> <ul> <li>Ivan Vavruch, <em>Conceptual problems of modern irreversible thermodynamics</em>, Chem. Listy 96 (2002) (<a href="http://www.chemicke-listy.cz/docs/full/2002_05_01.pdf">pdf</a>)</li> </ul> <p>For more on this see also <em><a class="existingWikiWord" href="/nlab/show/rational+thermodynamics">rational thermodynamics</a></em>.</p> <ul> <li>Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, Christopher Perry, <em>Fluctuating work: from quantum thermodynamical identities to a second law equality</em>, Phys. Rev. X 6, 041017 <a href="http://dx.doi.org/10.1103/PhysRevX.6.041017">doi</a></li> </ul> <h3 id="ReferencesRelativisticThermodynamics">Relativistic thermodynamics</h3> <p>Making sense of thermodynamics when taking into account <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> and ultimately, possibly, <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> (<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>) is notoriously subtle (even ignoring the issue of <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a>).</p> <blockquote> <p>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.</p> <p>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.</p> </blockquote> <p>(quote from Moller, <em><a href="https://archive.org/details/theoryofrelativi029229mbp">The theory of relativity</a>, 1952</em>)</p> <p>A standard textbook has been</p> <ul> <li id="Tolman34">Richard Tolman, <em>Relativity, Thermodynamics and Cosmology</em>, Oxford 1934, reprinted by Dover 1987</li> </ul> <p>but Tolman’s approach has been called into question, see e.g.</p> <ul> <li>Christian Fronsdal, <em>Relativistic thermodynamics</em>, 2014 (<a href="http://fronsdal.physics.ucla.edu/system/files/Gen.Rel_.LMP_.pdf">pdf</a>)</li> </ul> <p>See also</p> <ul> <li> <p>Nils Andersson, <em>General relativistic thermo-dynamics</em>, survey talk 2014 (<a href="http://www.dpg-physik.de/dpg/pbh/aktuelles/pdf/Andersson.pdf">pdf</a>)</p> </li> <li> <p>Sean A. Hayward, <em>Relativistic thermodynamics</em> (<a href="https://arxiv.org/abs/gr-qc/9803007">arXiv:gr-qc/9803007</a>)</p> </li> <li> <p>Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb, <em>What is the entropy of the universe?</em>, Class. Quant. Grav.26:145005, 2009 (<a href="https://arxiv.org/abs/0801.1847">arXiv:0801.1847</a>)</p> </li> </ul> <h3 id="further_generalizations">Further generalizations</h3> <p>Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like <a class="existingWikiWord" href="/nlab/show/Matsubara+formalism">Matsubara formalism</a> in QFT.</p> <p>Mathematical analogies and generalizations include also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>, <em>Algorithmic thermodynamics</em>, <a href="http://math.ucr.edu/home/baez/thermo.pdf">pdf</a>, <a href="http://golem.ph.utexas.edu/category/2010/02/algorithmic_thermodynamics.html">cafe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+Marcolli">M. Marcolli</a>, <a class="existingWikiWord" href="/nlab/show/Ryan+Thorngren">Ryan Thorngren</a>, <em>Thermodynamical semirings</em>, <a href="http://arxiv.org/abs/1108.2874">arXiv/1108.2874</a></p> </li> <li> <p>M. Zinsmeister, <em>Thermodynamic formalism and holomorphic dynamical systems</em>, Amer. Math. Soc. 2000.</p> </li> <li> <p>I. Itenberg, G. Mikhalkin, <em>Geometry in the tropical limit</em>, <a href="http://arxiv.org/abs/1108.3111">arXiv/1108.3111</a></p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> in relation to thermodynamics:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Thanu+Padmanabhan">Thanu Padmanabhan</a>, <em>Gravity and/is Thermodynamics</em>, Current Science, <strong>109</strong> (2015) 2236-2242 [<a href="https://doi.org/10.18520/v109/i12/2236-2242">doi:10.18520/v109/i12/2236-2242</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thanu+Padmanabhan">Thanu Padmanabhan</a>, <em>Exploring the Nature of Gravity</em>, talk notes [<a href="https://arxiv.org/abs/1602.01474">arXiv:1602.01474</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 6, 2024 at 00:27:06. See the <a href="/nlab/history/thermodynamics" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/thermodynamics" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/6252/#Item_8">Discuss</a><span class="backintime"><a href="/nlab/revision/thermodynamics/36" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/thermodynamics" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/thermodynamics" accesskey="S" class="navlink" id="history" rel="nofollow">History (36 revisions)</a> <a href="/nlab/show/thermodynamics/cite" style="color: black">Cite</a> <a href="/nlab/print/thermodynamics" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/thermodynamics" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>