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Variational discretization of the nonequilibrium thermodynamics of simple systems. (English) Zbl 1394.37113

Based on a Lagrangian variational formulation of nonequilibrium thermodynamics previously developed by the authors [F. Gay-Balmaz and H. Yoshimura, J. Geom. Phys. 111, 169–193 (2017; Zbl 1404.37033); F. Gay-Balmaz and H. Yoshimura, J. Geom. Phys. 111, 194–212 (2017; Zbl 1357.37051)], in the present paper they obtain a class of variational integrators by discretization. For simplicity, the class of problems considered are the so-called simple and closed systems: only one scalar thermal variable (the entropy) and a finite set of mechanical variables are sufficient to describe the state of the system, and in addition no power due to matter transfer between the system and the exterior is assumed to take place.
The Lagrangian variational formulation for nonequilibrium thermodynamics extends Hamilton’s principle of classical mechanics by allowing the inclusion of irreversible phenomena. Once established, a discrete version is proposed and the corresponding properties concerning structure preservation are obtained. In particular, it is shown that the variational integrator constructed by applying the discretization process preserves a discrete version of a property generalizing symplecticity when the effects of friction and temperature are taken into account. Regularity conditions to be satisfied by the system are also established which guarantee the existence of a discrete flow, solution of the discrete version of the variational principle.
Three different variational discretization schemes are deduced: they correspond, respectively, to extensions of the Verlet scheme, the variational midpoint rule and the symmetrized Lagrangian variational integrator. These methods are subsequently tested on a mass-spring-friction system moving in a closed room filled with an ideal gas. Since the exact solution is known for this simple example, the above integration schemes are tested and the corresponding numerical solution is compared with the exact values of the entropy, temperature and internal energy over time, showing in all case a correct behavior.

MSC:

37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
49S05 Variational principles of physics
80M30 Variational methods applied to problems in thermodynamics and heat transfer
70H30 Other variational principles in mechanics
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
49M15 Newton-type methods
49M25 Discrete approximations in optimal control
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References:

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