Analogy between mechanics and thermodynamics

A formal resemblance between thermodynamics and mechanics has been found. In this respect, the fundamental ideas of quantum theory, can be obtained from this analogy, We have deduced a correspondence between Caratheodory's theorem and Bohr's stationary states principle. Using the deduced correspondence rules, we obtain the quantification rule of Bohr-Sommerfield, Also. starting from the theory of thermodynamic fluctuations, the Heisenberg uncertainty principle has been deduced without use of any quantum assumption. Our results show then that the Bohr-Sommerfield rule and Heisenberg's principle are the mechanical versions of thermodynamics laws. Exactly as the Schrodinger equation and the quantum mechanics pulse are the mechanical versions of the geometrical optics laws.