Symplectic formulation of statistical physics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:49:06Z http://mathoverflow.net/feeds/question/102656 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102656/symplectic-formulation-of-statistical-physics Symplectic formulation of statistical physics Tobias 2012-07-19T12:23:57Z 2012-07-22T07:09:36Z <p>Does there exists a symplectic formulation of statistical physics?</p> <p>I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated symplectic, but I do not know anything which tries to relate them 'symplectilly'. Partial results are also welcome!</p> http://mathoverflow.net/questions/102656/symplectic-formulation-of-statistical-physics/102674#102674 Answer by Francois Ziegler for Symplectic formulation of statistical physics Francois Ziegler 2012-07-19T14:40:29Z 2012-07-22T07:09:36Z <p>You want to read Chapter IV "Statistical Mechanics" in <em><a href="http://www.ams.org/mathscinet-getitem?mr=1461545" rel="nofollow">Structure of Dynamical Systems</a></em> (1970 French original available <a href="http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm" rel="nofollow">here</a>) by J.-M. Souriau, one of the pioneers of symplectic mechanics.</p> <p>Given a symplectic manifold $X$ on which a Lie group $G$ acts with moment map $\Psi$, Souriau defines the <em>Gibbs states</em> as the probability measures on $X$ that maximize entropy for a given mean value of $\Psi$. He shows (thm 16.219) that they have the form<br> $$\text{const}\times e^{-\langle\Psi(\cdot),\beta\rangle}\lambda, \qquad (\lambda=\text{Liouville measure})$$ for some $\beta\in\mathfrak g$ which generalizes the "inverse temperature" when $G=\mathbf R=$ {time translations}. The rest of the Chapter is devoted to the study of these states; in particular when $G=$ SO(3), $\beta$ can be interpreted as a rotation vector, and the fact that planets revolve in a common plane as the equality of their equilibrium $\beta$s. See also <em>Définition covariante des équilibres thermodynamiques</em>, Suppl. Nuovo Cimento <strong>1</strong> (1966), 203–216.</p> <p>Later Souriau developed a general-relativistic viewpoint on dissipative processes which explains why they preserve the mean value of $\Psi(x)$. Thus, quoting from <a href="http://www.ams.org/mathscinet-getitem?mr=2335767" rel="nofollow">this summary</a> to whet your appetite: "the first principle of thermodynamics [loses] its primitive status and [becomes] a necessary consequence of the invariance of the symplectic structure in gravitational gauge transformations." For more details, see <em><a href="http://dx.doi.org/10.1007/BFb0063682" rel="nofollow">Thermodynamique et géométrie</a></em>, Lecture Notes in Math. <strong>676</strong> (1978), 369–397 or <a href="http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025" rel="nofollow">scanned preprint</a>.</p>