Abstract
We show that elements of control theory, together with an application of Harris’ ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form
in \(\mathbf {R}^d\), where A encodes the harmonic part of the force and \(-F\) corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: A is dissipative, the pair (A, B) satisfies the Kalman condition, F grows sufficiently slowly at infinity (depending on the dimension d), and the vector fields in the equation of motion satisfy the weak Hörmander condition in at least one point of the phase space.
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Agrachev, A.A., Kuksin, S., Sarychev, A.V., Shirikyan, A.: On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations. Annales de l’Institut Henri Poincaré (B) Probability and Statistics 43(4), 399–415 (2007)
Agrachev, A.A., Sarychev, A.V.: Navier–Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech. 7(1), 108–152 (2005)
Bogachev, V.I.: Gaussian Measures, Number 62. American Mathematical Society, Providence (1998)
Carmona, P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stoch. Process. Appl. 117(8), 1076–1092 (2007)
Coron, J.-M.: Control and Nonlinearity, Volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2007)
Cuneo, N., Eckmann, J.-P., Hairer, M., Rey-Bellet, L.: Non-equilibrium steady states for networks of oscillators. Electron. J. Probab. 23, 1–28 (2018)
Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)
Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in nonlinear, thermally driven hamiltonian systems. J. Stat. Phys. 95(1), 305–331 (1999)
Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201(3), 657–697 (1999)
Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6(4), 504–515 (1965)
Hairer, M., Mattingly, J.C.: Yet another look at Harris’ ergodic theorem for Markov chains. In: Seminar on Stochastic Analysis, Random Fields and Applications VI, volume 63, pp. 109–117. Springer, Berlin (2011)
Harris, T.E.: The existence of stationary measures for certain Markov processes. Proc. Third Berkeley Symp. Math. Stat. Probab. 2, 113–124 (1956)
Jakšić, V., Pillet, C.-A., Shirikyan, A.: Entropic fluctuations in thermally driven harmonic networks. J. Stat. Phys. 166(3), 926–1015 (2017)
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer, Berlin (2012)
Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225(2), 305–329 (2002)
Shirikyan, A.: Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations. J. Funct. Anal. 249, 284–306 (2007)
Shirikyan, A.: Controllability implies mixing I. Convergence in the total variation metric. Uspekhi Matematicheskikh Nauk 72(5), 165–180 (2017)
Tropper, M.M.: Ergodic and quasideterministic properties of finite-dimensional stochastic systems. J. Stat. Phys. 17(6), 491–509 (1977)
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Communicated by Christian Maes.
The author would like to thank Armen Shirikyan for introduction to these questions and crucial suggestions for this particular application, Noé Cuneo and Vojkan Jakšić for informative discussions, as well as the Département de mathématiques at Université Cergy–Pontoise, where part of this research was conducted, for its hospitality. The research of the author was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec–Nature et technologies (FRQNT) and the NonStops project of the Agence nationale de la recherche (ANR-17-CE40-0006-02).
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Raquépas, R. A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks. Ann. Henri Poincaré 20, 605–629 (2019). https://doi.org/10.1007/s00023-018-0740-0
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DOI: https://doi.org/10.1007/s00023-018-0740-0