1,720,996 research outputs found
Contact geometry and thermodynamics
No full textThese 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
Contact Hamiltonian Dynamics: The Concept and Its Use
No full textWe 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
Thermostat algorithm for generating target ensembles
No full textWe present a deterministic algorithm called contact density dynamics that generates any prescribed target distribution in the physical phase space. Akin to the famous model of Nosé and Hoover, our algorithm is based on a non-Hamiltonian system in an extended phase space. However, the equations of motion in our case follow from contact geometry and we show that in general they have a similar form to those of the so-called density dynamics algorithm. As a prototypical example, we apply our algorithm to produce a Gibbs canonical distribution for a one-dimensional harmonic oscillator
Liouville's theorem and the canonical measure for nonconservative systems from contact geometry
No full textStandard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton's equations, Liouville's theorem and to find the canonical invariant measure. In this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry. In particular, we find out the generalized Hamilton's equations, Liouville's theorem and the microcanonical and canonical measures invariant under the contact flow. Remarkably, the latter measure has a power law density distribution with respect to the standard contact volume form. Finally, we argue on the several possible applications of our results
Thermodynamic curvature and ensemble nonequivalence
No full textIn this work we consider thermodynamic geometries defined as Hessians of different potentials and derive some useful formulas that show their complementary role in the description of thermodynamic systems with 2 degrees of freedom that show ensemble nonequivalence. From the expressions derived for the metrics, we can obtain the curvature scalars in a very simple and compact form. We explain here the reason why each curvature scalar diverges over the line of divergence of one of the specific heats. This application is of special interest in the study of changes of stability in black holes as defined by Davies. From these results we are able to prove on a general footing a conjecture first formulated by Liu, Lü, Luo, and Shao stating that different Hessian metrics can correspond to different behaviors in the various ensembles. We study the case of two thermodynamic dimensions. Moreover, comparing our result with the more standard turning point method developed by Poincaré, we obtain that the divergence of the scalar curvature of the Hessian metric of one potential exactly matches the change of stability in the corresponding ensemble
Scaling symmetries and canonoid transformations in Hamiltonian systems
No full textIn this paper, we investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case, we pay special attention to non-standard (non-canonical) symmetries, in particular scaling symmetries and canonoid transformations, as they provide new interesting tools for the qualitative study of these systems. Our main results are the characterizations of these non-standard symmetries and the analysis of their relation with conserved (or dissipated) quantities
Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries
Full text linkIn this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so-called Kirillov Hamiltonian system. Moreover, we show that if we reduce first by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the configuration space of the symplectic Hamiltonian system is a Lie group , which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie–Poisson structure on the dual space of the Lie algebra of . We also discuss a reconstructionprocess for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.Fil: Bravetti, A.. Universidad Nacional Autónoma de México; MéxicoFil: Grillo, Sergio Daniel. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; ArgentinaFil: Marrero, J. C.. Universidad de La Laguna; EspañaFil: Padrón, E.. Universidad de La Laguna; Españ
Dark energy from geometrothermodynamics
No full textWe investigate a general class of equations of state reproducing the dark energy effects in terms of geometric considerations on thermodynamic interaction. We infer cosmological solutions by combining thermodynamics with contact manifold and Riemannian geometry, showing that the standard ΛCDM model can be treated as a limiting case of a more general approach, providing early time departures as the universe expands. Thus, we interpret the microscopic nature of dark energy through the mathematical formalism of geometrothermodynamics (GTD). In particular, we investigate the thermodynamic nature of a class of cosmological models which reproduce how the universe is currently speeding up. To do so, we aim to describe thermodynamic equilibrium states of the universe through a particular equilibrium space, where the Riemannian metric g becomes a thermodynamical ruler between different states. The particular assumption we made is to consider the metric structure to be invariant under precise Legendre transformations. It turns out that any thermodynamic interaction is determined once the scalar curvature of the equilibrium manifold is known. The consequence of our recipe leads to a class of dark energy equations of state which relates standard pressure to the volume occupied by the fluid itself. In our picture, dark energy is thus determined from constant thermodynamic interaction on the manifold of GTD
Para-Sasakian geometry in thermodynamic fluctuation theory
No full textIn this work we tie concepts derived from statistical mechanics, information theory and contact Riemannian geometry within a single consistent formalism for thermodynamic fluctuation theory. We derive the concrete relations characterizing the geometry of the thermodynamic phase space stemming from the relative entropy and the Fisher–Rao information matrix. In particular, we show that the thermodynamic phase space is endowed with a natural para-contact pseudo-Riemannian structure derived from a statistical moment expansion which is para-Sasaki and η-Einstein. Moreover, we prove that such manifold is locally isomorphic to the hyperbolic Heisenberg group. In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics. Finally, we argue on the possible implications of our results
A geometric approach to the generalized Noether theorem
No full textWe provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented by Zhang P-M et al (2020 Eur. Phys. J. Plus 135 223). Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether’s theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments
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