Geometric “quantization” of thermodynamics and statistical corrections at critical points

Gibbs thermodynamics is treated as a two-dimensional manifold, called Lagrangian by the author, in a 4-dimensional phase space in which the temperature and pressure play the role of coordinates and the entropy and volume the role of momenta. The tunneling canonical operator determines the asymptotic behavior of the partition function. This treatment is generalized to the case of thermodynamics with intensive and extensive coordinates. A new axiomatics of thermodynamics and quasithermodynamics is given.