Communications in Mathematical Physics

, Volume 345, Issue 1, pp 185–221 | Cite as

Non-Equilibrium Steady States for Chains of Four Rotors

  • N. Cuneo
  • J.-P. Eckmann
Open Access


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.


Invariant Measure Lyapunov Function External Rotor Heat Bath Central Rotor 
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Département de Physique ThéoriqueUniversité de GenèveGenevaSwitzerland
  2. 2.Section de MathématiquesUniversité de GenèveGenevaSwitzerland

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