Dynamical Properties of Gaussian Thermostats

2019

Cosymplectic geometry can be viewed as an odd dimensional counterpart of symplectic geometry. Likely in the symplectic case, a related property which is preservation of closed forms $\omega$ and $\eta$, refers to the theoretical possibility of further understanding a cosymplectic manifold $(M, \omega, \eta)$ from its group of diffeomorphisms. In this paper we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold $(M, \omega, \eta)$ in threefold:first of all, we study cosymplectics counterpart of the Moser isotopy method, a proof of a cosymplectic version of Darboux theorem follows, and we present the features of the space of almost cosymplectic vector fields, this set forms a Lie group whose Lie algebra is the group of all almost cosymplectic diffeomorphisms; $(II)$ we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is $C^0-$closed in the ...

Physical Review E, 2000

Deformations of submanifolds of thermodynamic equilibrium states introduced by continuous contact maps on a phase-space manifold are considered in terms of the geometrical formulation of thermodynamics. The notion of a contact Hamiltonian is recalled in order to give some possible physical interpretations of such a function in terms of statistical quantities describing initial and deformed systems. Using contact flows we propose a very efficient method for constructing continuous families of thermodynamic systems. A few examples show the possible advantages of using contact Hamiltonians.

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2006

We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.

2016

Abstract. In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski. In this paper, we consider a class of invertible maps with discontinuities and unbounded derivatives, which model billiards and other physical systems. Let F be one of these maps, and assume that F preserves a symplectic form and is non-uniformly hyperbolic. The last condition means that the Lyapunov exponents of F are non-zero with respect to the symplectic volume, which is invariant. Our main result (Theorem 3.1) establishes sufficient conditions for a point in the domain of F to have a neighborhood contained up to a set of zero measure in one ergodic component of F. In fact, we prove a stronger result, namely, that the mentioned neighborhood is contained up to a set of zero measure in one Bernoulli component of F. Results of this type are often called ‘local ergodic theorems’. Loc...

arXiv (Cornell University), 2015

The main result of this work is the following: for volume preserving flows on compact manifolds with the C r topology, 1 r ∞ , the closure of every invariant manifold of periodic orbits and singularities is a chain transitive set. We also develop to new local constructions, which surprise by the simplicity of the arguments. One, a local perturbation to change an orbit to a nearby without altering its past. The other is a flow box theorem in the context of volume preserving flows, a result that is well known for Hamiltonians or general flows.

Complex Manifolds

This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C 0−closed in the group Diff∞ (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C 0−limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogu...

Journal of Geometry and Physics

CARNOT LILLE 2024 Conference, 2024

Since the work of Claude Shannon, Entropy has been introduced axiomatically. The recent extension of information geometry to Lie groups via symplectic models of statistical mechanics, initiated by Jean-Marie Souriau (Lie groups thermodynamics), makes it possible to give a purely geometric "constructive" definition of the Entropy. Starting from the symmetry group which acts on the data or the system, the Entropy is constructed as an invariant Casimir function on the symplectic foliation generated by the co-adjoint orbits of the group (co-adjoint orbits constructed by the action of the group on the moment map, where the moment map is a geometrization of Noether's theorem). The level sets of Entropy thus appear as the leaves of this symplectic foliation generated by the co-adjoint action. In the case of the symmetry group of Euclidean space, we find the classical Shannon Entropy, which is therefore a special case of this model. We therefore move from Shannon's axiomatic definition to a geometric “constructive” definition of Entropy. Symmetry generates Entropy via moment map. In the theory of representations of Lie groups introduced by Alexandre Kirillov, we associate with these co-adjoint orbits a symplectic manifold via a 2 KKS form (Kirillov- Kostant -Souriau) and an associated Riemannian metric. As part of the extension of Information geometry to Lie groups, this metric appears exactly like the Fisher-Koszul metric . Thanks to the cohomology of Lie algebra and the Souriau cocycle , we can deal with cases where the co-adjoint operator is not equivariant (case of non-null cohomology) . This model makes it possible to extend the notion of Gaussian probability density for an informational measurement on Lie groups or for homogeneous spaces on which a Lie group acts, by considering the Gibbs density at Maximum Entropy, which has the property of being covariant under the action of the group. We will illustrate this fact by giving the definition of a Gaussian with maximum entropy for the half-plane and the Poincaré disk via the group SL(2,R) and SU(1,1), and for the covariance matrices symmetric positive definite SPD via the group SL( n,R ) and SU( n,n ) (the SDP matrices being considered as the pure imaginary axis of the Siegel half-space). This symplectic foliation structure, level sets of Entropy, also makes it possible to interpret the 2nd principle of thermodynamics by studying its transverse structures. Initiated by Charles Ehresmann and Georges Reeb , the theory of foliations makes it possible to give the geometry structure of thermodynamics, made up of a symplectic foliation (describing non-dissipative dynamics) and a transverse structure given by a Riemannian foliation (describing dissipative dynamics with creation of entropy). The metric which appears on the transverse Riemannian foliation is the dual metric of the Fisher metric, that is to say the Hessian of Entropy. We thus have 2 transverse foliations associated with the 2 dual metrics of information geometry. The dynamics on these 2 transverse foliations is given by the equation of the “metriplectic” flow with a Poisson bracket to move on the symplectic leaves (non-dissipative flow because we move on the Entropy level sets; l entropy being an invariant function of Casimir in coadjoint representation on symplectic leaves) and a metric bracket to move on the transverse Riemannian leaves (production of Entropy at constant Energy; Energy is a function of Casimir for the metric bracket on these leaves which are therefore the level sets of Energy). Recently, it was shown that the metriplectic flow was compatible with Onsager relations in non-equilibrium thermodynamics. The metriplectic flow is used in machine learning for TINNs ( Thermodynamics-Informed Neural Network) and to stabilize bosonic CAT and GKP Qubits of quantum computers (the dissipative Linblad equation is a linear approximation of the metriplectic equation). This model is studied within the framework of 2 European actions, COST CaLISTA ( https://site.unibo.it/calista/en ) and MSCA CaLIGOLA ( https://site.unibo.it/caligola/en ), and also in the “North Paris Basin” seminar on “geometric structures of dissipation”. References : [1] Barbaresco, F. (2022) Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier, https://doi.org/10.1016/bs.host.2022.02 .003 ; https://www.sciencedirect.com/science/article/abs/pii/S0169716122000062 [2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. ( eds ) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17 ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17 [3] Barbaresco F. (2022). Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Non-Linear Lindblad Quantum Master Equation, MDPI special Issue "Geometric Structure of Thermodynamics: Theory and Applications", 2022, Entropy 2022, 24(11), 626; https://doi.org/10.3390/e24111626 [4] Barbaresco F. (2024). Symplectic Foliation Model of Sadi Carnot's Thermodynamics: from Carathéodory's seminal idea to Souriau's Lie Groups Thermodynamics, Celebration of 200 years since Sadi Carnot's Reflections on the motive power of Fire, 1824–2024 & the Carnots , https://carnotlille2024.sciencesconf.org/ resource/page/id/2 [5] Barbaresco, F. (2023). Souriau's Geometric Principles for Quantum Mechanics. In: Nielsen, F., Barbaresco, F. ( eds ) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_39 [6] Barbaresco, F. (2021). Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau's Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map. In: Nielsen, F., Barbaresco, F. ( eds ) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_28 [7] Souriau, JM (1969). Structure of dynamic systems. Dunod , available on line : http://www.jmsouriau.com/structure_des_systemes_dynamics.htm [8] Barbaresco, F. (2024) Symplectic Foliation Model of Sadi Carnot's Thermodynamics: from Carathéodory's seminal idea to Souriau's Lie Groups Thermodynamics, Carnot Lille 2024 Colloquium, to be published published in SPRINGER Book Series History of Physics.

arXiv: Mathematical Physics, 2020

In this work we show that a Legendre transformation is nothing but a mere change of symplectic polarization from the point of view of contact geometry. Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries. We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent i.e. the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds. This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system. We find that to define a Riemannian structure with such properties it is necessary to abandon ...

ma.utexas.edu

We study the behavior of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity.