## Electronic Communications in Probability

### On the relaxation rate of short chains of rotors interacting with Langevin thermostats

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 35, 8 pp.

Dates
Accepted: 23 May 2017
First available in Project Euclid: 21 June 2017

https://projecteuclid.org/euclid.ecp/1498010648

Digital Object Identifier
doi:10.1214/17-ECP62

Mathematical Reviews number (MathSciNet)
MR3666856

Zentralblatt MATH identifier
1380.82028

Subjects
Primary: NA

#### Citation

Cuneo, Noé; Poquet, Christophe. On the relaxation rate of short chains of rotors interacting with Langevin thermostats. Electron. Commun. Probab. 22 (2017), paper no. 35, 8 pp. doi:10.1214/17-ECP62. https://projecteuclid.org/euclid.ecp/1498010648

#### References

• [1] Philippe 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.
• [2] Ben Cooke, David P Herzog, Jonathan C Mattingly, Scott A McKinley, and Scott C Schmidler, Geometric ergodicity of two–dimensional hamiltonian systems with a lennard–jones–like repulsive potential, arXiv preprint (2011), arXiv:1104.3842.
• [3] Noé Cuneo, Jean-Pierre Eckmann, and C Eugene Wayne, Energy dissipation in Hamiltonian chains of rotators, arXiv preprint (2017), arXiv:1702.06464.
• [4] Noé Cuneo and Jean-Pierre Eckmann, Non-equilibrium steady states for chains of four rotors, Commun. Math. Phys. 345 (2016), no. 1, 185–221.
• [5] Noé Cuneo, Jean-Pierre Eckmann, and Christophe Poquet, Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, Nonlinearity 28 (2015), no. 7, 2397–2421.
• [6] Jean-Pierre Eckmann and Martin Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys. 212 (2000), no. 1, 105–164.
• [7] Jean-Pierre Eckmann, Claude-Alain Pillet, and Luc 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.
• [8] Martin Hairer, How hot can a heat bath get?, Commun. Math. Phys. 292 (2009), no. 1, 131–177.
• [9] Martin Hairer and Jonathan C. Mattingly, Slow energy dissipation in anharmonic oscillator chains, Comm. Pure Appl. Math. 62 (2009), no. 8, 999–1032.
• [10] Yao Li, On the polynomial convergence rate to nonequilibrium steady-states, arXiv preprint (2016), arXiv:1607.08492.
• [11] Yao Li and Lai-Sang Young, Polynomial convergence to equilibrium for a system of interacting particles, arXiv preprint (2016), arXiv:1601.00717.
• [12] Luc Rey-Bellet and Lawrence E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Commun. Math. Phys. 225 (2002), no. 2, 305–329.
• [13] Tatiana Yarmola, Sub-exponential mixing of random billiards driven by thermostats, Nonlinearity 26 (2013), no. 7, 1825–1837.
• [14] Tatiana Yarmola, Sub-exponential mixing of open systems with particle-disk interactions, J. Stat. Phys. 156 (2014), no. 3, 473–492.