Non-equilibrium steady states for networks of oscillators. (English) Zbl 1397.82033

Summary: 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.


82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
60B10 Convergence of probability measures
37A60 Dynamical aspects of statistical mechanics
Full Text: DOI arXiv Euclid


[1] P. Carmona, Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths, Stochastic Process. Appl. 117 (2007), no. 8, 1076–1092. · Zbl 1132.60069
[2] F. Conrad and M. Grothaus, Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials, J. Evol. Equ. 10 (2010), no. 3, 623–662. · Zbl 1239.82015
[3] B. Cooke, D. P. Herzog, J. C. Mattingly, S. A. McKinley, and S. C. Schmidler, Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard-Jones-like repulsive potential, Commun. Math. Sci. 15 (2017), no. 7, 1987–2025. · Zbl 1386.37053
[4] N. Cuneo and J.-P. Eckmann, Controlling general polynomial networks, Commun. Math. Phys. 328 (2014), no. 3, 1255–1274. · Zbl 1294.82025
[5] N. Cuneo and J.-P. Eckmann, Non-equilibrium steady states for chains of four rotors, Commun. Math. Phys. 345 (2016), no. 1, 185–221. · Zbl 1343.60092
[6] N. Cuneo, J.-P. Eckmann, and C. Poquet, Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, Nonlinearity 28 (2015), no. 7, 2397–2421. · Zbl 1328.82029
[7] N. Cuneo and C. Poquet, On the relaxation rate of short chains of rotors interacting with Langevin thermostats, Electron. Commun. Probab. 22 (2017), 8 pp. · Zbl 1380.82028
[8] A. Dymov, Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy, ArXiv e-prints (2017).
[9] J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys. 212 (2000), no. 1, 105–164. · Zbl 1044.82008
[10] J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Stat. Phys. 95 (1999), no. 1-2, 305–331. · Zbl 0964.82051
[11] J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys. 201 (1999), no. 3, 657–697. · Zbl 0932.60103
[12] J.-P. Eckmann and E. Zabey, Strange heat flux in (an)harmonic networks, J. Stat. Phys. 114 (2004), no. 1-2, 515–523. · Zbl 1060.82038
[13] M. Grothaus and P. Stilgenbauer, A hypocoercivity related ergodicity method for singularly distorted non-symmetric diffusions, Integr. Equ. Oper. Theory 83 (2015), no. 3, 331–379. · Zbl 1361.37007
[14] M. Hairer, A probabilistic argument for the controllability of conservative systems, ArXiv Mathematical Physics e-prints (2005).
[15] M. Hairer, How hot can a heat bath get?, Commun. Math. Phys. 292 (2009), no. 1, 131–177. · Zbl 1190.82022
[16] M. Hairer, On Malliavin’s proof of Hörmander’s theorem, Bull. Sci. Math. 135 (2011), no. 6-7, 650–666. · Zbl 1242.60085
[17] M. Hairer and J. C. Mattingly, Slow energy dissipation in anharmonic oscillator chains, Comm. Pure Appl. Math. 62 (2009), no. 8, 999–1032. · Zbl 1169.82012
[18] M. Hairer and J. C. Mattingly, Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, 2011, pp. 109–117. · Zbl 1248.60082
[19] M. Hairer, A. M. Stuart, and J. Voss, Analysis of SPDEs arising in path sampling. II. The nonlinear case, Ann. Appl. Probab. 17 (2007), no. 5-6, 1657–1706. · Zbl 1140.60329
[20] R. Z. Has’minskii, Stochastic stability of differential equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, Translated from the Russian by D. Louvish.
[21] D. P. Herzog and J. C. Mattingly, Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials, ArXiv e-prints (2017).
[22] R. Höpfner, E. Löcherbach, and M. Thieullen, Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin-Huxley model, ESAIM Probab. Stat. 20 (2016), 527–554. · Zbl 1355.60104
[23] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. · Zbl 0156.10701
[24] B. Leimkuhler, E. Noorizadeh, and F. Theil, A gentle stochastic thermostat for molecular dynamics, J. Stat. Phys. 135 (2009), no. 2, 261–277. · Zbl 1179.82065
[25] J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185–232. · Zbl 1075.60072
[26] S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W. Glynn. · Zbl 0925.60001
[27] R. Raquépas, A note on Harris’ ergodic theorem, controllability and perturbations of harmonic networks, ArXiv e-prints (2018).
[28] D. Revuz and M. Yor, Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften, vol. 293, Springer-Verlag, Berlin, 1999. · Zbl 0917.60006
[29] L. Rey-Bellet, Ergodic properties of Markov processes, Open quantum systems. II, Lecture Notes in Math., vol. 1881, Springer, Berlin, 2006, pp. 1–39. · Zbl 1126.60057
[30] L. Rey-Bellet and L. E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Commun. Math. Phys. 215 (2000), no. 1, 1–24. · Zbl 1017.82028
[31] L. Rey-Bellet and L. E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Commun. Math. Phys. 225 (2002), no. 2, 305–329. · Zbl 0989.82023
[32] D. W. Stroock, Partial differential equations for probabilists, Cambridge Studies in Advanced Mathematics, vol. 112, Cambridge University Press, Cambridge, 2008. · Zbl 1145.35002
[33] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, pp. 333–359. · Zbl 0255.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.