Noé Cuneo's homepage, Université Paris Diderot (Paris 7)

Noé Cuneo
Université Paris Diderot,
UFR de Mathématiques,
Bâtiment Sophie Germain,
8 place Aurélie Nemour,
75205 Paris CEDEX 13

+33 1 57 27 91 25
cuneo [at] lpsm.paris

Information

I am a Maître de conférences (~ tenured assistant professor) in Mathematics at the Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université Paris Diderot (Paris 7), France.

Here is my CV (click here for the French version) and here are my arXiv, ORCID and Google Scholar pages.

Current research

I am mainly interested in stochastic processes, dynamical systems and mathematical physics. More specifically, I have been working on the following topics:

  • Nonequilibrium chains and networks of oscillators and rotors (Hamiltonian systems interacting with stochastic heat baths at different temperatures).
  • Averaging/KAM methods for highly oscillatory dynamics.
  • Large deviations for dynamical systems and stochastic processes.
  • Entropy production and related fluctuation relations/theorems.
  • Statistics of repeated quantum measurement processes.

Please do not hesitate to get in touch if you are also interested in such topics.

Publications and preprints

  • Non-Equilibrium Steady States for Networks of Oscillators, with J.-P. Eckmann, M. Hairer and L. Rey-Bellet. Electronic Journal of Probability 23(55): 1-28, 2018. Abstract, published version, preprint

    Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.

  • Large Deviations and Fluctuation Theorem for Selectively Decoupled Measures on Shift Spaces, with V. Jakšić, C.-A. Pillet and A. Shirikyan. arXiv:1712.09038, 2017. Abstract, preprint

    We establish the Level-1 and Level-3 Large Deviation Principle (LDP) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle-Lanford functions, and the exposition is essentially self-contained.

  • Fluctuation Theorem and Thermodynamic Formalism, with V. Jakšić, C.-A. Pillet and A. Shirikyan. arXiv:1712.05167, 2017. Abstract, preprint

    We study the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.

  • Energy Dissipation in Hamiltonian Chains of Rotators, with J.-P. Eckmann and C. E. Wayne. Nonlinearity 30 (11), R81, 2017. Selected in the Nonlinearity Highlights of 2017 collection. Abstract, published version, preprint

    We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.

  • On the relaxation rate of short chains of rotors interacting with Langevin thermostats, with C. Poquet. Electronic Communications in Probability 22(35): 1-8, 2017. Abstract, published version, preprint

    In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.

  • Ph.D thesis: Non-equilibrium Steady States for Hamiltonian Chains and Networks, 2016. Supervisor: J.-P. Eckmann. Manuscript

  • Non-equilibrium steady states for chains of four rotors, with J.-P. Eckmann. Communications in Mathematical Physics, 345(1): 185–221, 2016. Abstract, published version, preprint

    We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.

  • Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, with J.-P. Eckmann and C. Poquet. Nonlinearity 28 (2015), 2397–2421, 2015. Abstract, published version, preprint

    We consider a chain of three rotors (rotators) whose ends are coupled to stochastic heat baths. The temperatures of the two baths can be different, and we allow some constant torque to be applied at each end of the chain. Under some non-degeneracy condition on the interaction potentials, we show that the process admits a unique invariant probability measure, and that it is ergodic with a stretched exponential rate. The interesting issue is to estimate the rate at which the energy of the middle rotor decreases. As it is not directly connected to the heat baths, its energy can only be dissipated through the two outer rotors. But when the middle rotor spins very rapidly, it fails to interact effectively with its neighbours due to the rapid oscillations of the forces. By averaging techniques, we obtain an effective dynamics for the middle rotor, which then enables us to find a Lyapunov function. This and an irreducibility argument give the desired result. We finally illustrate numerically some properties of the non-equilibrium steady state.

  • Controlling General Polynomial Networks, with J.-P. Eckmann. Communications in Mathematical Physics, 328(3):1255–1274, 2014. Abstract, published version, preprint

    We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for Hörmander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in [Eckmann, Pillet, Rey-Bellet, 1999].

Comments, questions, and mistake reports are more than welcome.

Positions

  • 2018-..., Maître de conférences, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université Paris Diderot, France.
  • 2017-2018, Postdoc, Laboratoire de mathématiques AGM, Université de Cergy-Pontoise, France.
  • 2016-2017, Postdoc, Department of Mathematics and Statistics, McGill University, Montreal, Canada.
  • 2012-2016, Ph.D., Physics, University of Geneva (UNIGE), Switzerland. Advisor: J.-P. Eckmann.

Recent talks

  • 14 Mar. 2019 (upcoming), Institut Denis Poisson, Université d'Orléans, Séminaire Orléans.
  • 14 Feb. 2019 (upcoming), Université de Bordeaux, Dynamique Quantique GDR Conference.
  • 1 Feb. 2019 (upcoming), Université Paris Descartes, Séminaire de probabilités du MAP5.
  • 29 Jan. 2019 (upcoming), LPSM, Sorbonne Université, Paris, Séminaire de Probabilités.
  • 18 Dec. 2018, Université Paris-Dauphine, Séminaire du CEREMADE Analyse-Probabilités.
  • 09 Nov. 2018, Duke University, Durham, Applied Math And Analysis Seminar.
  • 01 Nov. 2018, CRM, Université de Montréal, Entropic Fluctuation Relations in Mathematics and Physics workshop.
  • 24 July 2018, ICMP 2018, McGill University, Montreal, contributed talk.
  • 11 June 2018, Université Paris Diderot, Stochastic Problems in Mathematical Physics and Economics Seminar (link).
  • 28 Mar. 2018, Centre de Physique Théorique, Marseille, Séminaire Systèmes dynamiques quantiques et classiques.
  • 22 Mar. 2018, Département de Mathématiques d'Orsay, Université Paris-Sud, Séminaire de Probabilités et Statistique.
  • 20 Mar. 2018, LPSM, Sorbonne Université, Paris, Séminaire de Théorie Ergodique.
  • 16 Mar. 2018, Institut de mathématiques, Université de Neuchâtel, Rencontres de mathématiques.
  • 20 Feb. 2018, Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Séminaire de Probabilités et Statistique.
  • 08 Feb. 2018, LPSM, Université Paris Diderot, "Modélisation" Workshop.
  • 15 Jan. 2018, Université de Cergy-Pontoise, Séminaire Géométrie, EDP et Physique Mathématique.
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