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Cohomological geometry of differential equations
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On the management of thermodynamic processes V. V. Lychagin |
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Abstract: At the beginning of the talk, three geometric approaches to thermodynamics will be discussed. The first approach is the Gibbs energy approach, which will be reformulated in terms of contact geometry and where the description of substances (the so-called equations of state) is given in terms of Legendre submanifolds in thermodynamic contact phase spaces, and thermodynamic processes, as well as controls, will be given by contact vector fields. The second approach is based on information geometry and follows the principle of maximum entropy, also known as the principle of minimum information gain or Occam's razor. Both of these approaches lead us to the same model of thermodynamics, but they also introduce important new concepts, such as the Gibbs-Duhem principle and Riemannian structures on Legendre submanifolds. The third approach is based on the geometry of jet spaces (or the geometry of differential equations), and it provides a more convenient apparatus for the practical description and calculation of both equations of state and thermodynamic processes, taking into account phase transitions. Thermodynamic process control will be understood as a thermodynamic process that does not destroy the process in question, but allows it to be accelerated or slowed down. The set of controls forms a Lie algebra, in which the Lie algebra of symmetries is a Lie subalgebra. We will present equations that depend on the equations of state of the medium and allow us to find control processes, as well as illustrate their application in the case of adiabatic processes. If time permits, phase transitions of the first, second and higher orders will be considered, both in thermodynamic processes and in controls, as well as their connection with Arnold's theory on the singularities of projections of Lagrangian manifolds. Please download the formula file td.pdf and keep it handy during the talk so that the speaker can refer to it. Language: English |
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