In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we review in detail the major features of standard symplectic Hamiltonian dynamics and show that all of them can be generalized to the contact case. (Alessandro Bravetti), hans@ciencias.unam.mx (Hans Cruz), diego.tapias@nucleares.unam.mx (Diego Tapias) 4 Conclusions and perspectives 29 Appendix A Invariants for the damped parametric oscillator 32 Appendix B Equivalence between the contact Hamilton-Jacobi equation and the contact Hamiltonian equations 34
These are the lecture notes for the course given at the “XXVII International Fall Workshop on Geometry and Physics” held in Seville (Spain) in September 2018. We review the geometric formulation of equilibrium thermodynamics by means of contact geometry, together with the associated metric structures arising from thermodynamic fluctuation theory and the use of Hamiltonian flows to describe thermodynamic processes. Finally, we discuss the state of the art of the subject, its connection with other areas of physics, and suggest several possible further directions.
Abstract:We give a short survey on the concept of contact Hamiltonian dynamics and its use in several areas of physics, namely reversible and irreversible thermodynamics, statistical physics and classical mechanics. Some relevant examples are provided along the way. We conclude by giving insights into possible future directions.
It has been shown that contact geometry is the proper framework underlying classical thermodynamics and that thermodynamic fluctuations are captured by an additional metric structure related to Fisher's Information Matrix. In this work we analyze several unaddressed aspects about the application of contact and metric geometry to thermodynamics. We consider here the Thermodynamic Phase Space and start by investigating the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics. Then we present a novel mathematical characterization of first order phase transitions as equilibrium processes on the Thermodynamic Phase Space for which the Legendre symmetry is broken. Moreover, we use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics and we show that the relevant Hamiltonian coincides with the irreversible entropy production along thermodynamic processes. Therefore, we use such property to give a geometric definition of thermodynamically admissible fluctuations according to the Second Law of thermodynamics. Finally, we show that the length of a curve describing a thermodynamic process measures its entropy production.
We propose a novel approach for parameterizing the luminosity distance, based on the use of rational "Padé" approximations. This new technique extends standard Taylor treatments, overcoming possible convergence issues at high redshifts plaguing standard cosmography. Indeed, we show that Padé expansions enable us to confidently use data over a larger interval with respect to the usual Taylor series. To show this property in detail, we propose several Padé expansions and we compare these approximations with cosmic data, thus obtaining cosmographic bounds from the observable universe for all cases. In particular, we fit Padé luminosity distances with observational data from different uncorrelated surveys. We employ union 2.1 supernova data, baryonic acoustic oscillation, Hubble space telescope measurements and differential age data. In so doing, we also demonstrate that the use of Padé approximants can improve the analyses carried out by introducing cosmographic auxiliary variables, i.e. a standard technique usually employed in cosmography in order to overcome the divergence problem. Moreover, for any drawback related to standard cosmography, we emphasize possible resolutions in the framework of Padé approximants. In particular, we investigate how to reduce systematics, how to overcome the degeneracy between cosmological coefficients, how to treat divergences and so forth. As a result, we show that cosmic bounds are actually refined through the use of Padé treatments and the thus derived best values of the cosmographic parameters show slight departures from the standard cosmological paradigm. Although all our results are perfectly consistent with the ΛCDM model, evolving dark energy components different from a pure cosmological constant are not definitively ruled out. Finally, we use our outcomes to reconstruct the effective universe equation of state, constraining the dark energy term in a model independent way.PACS numbers: 98.80.Jk, 98.80.Es
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