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Controlling General Polynomial Networks

Communications in Mathematical Physics volume 328, pages 1255–1274 (2014)Cite this article

Abstract

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 et al. in (Commun Math Phys 201:657–697, 1999).

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References

  1. Eckmann J.-P., Hairer M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Eckmann, J.-P., Hairer, M., Rey-Bellet L.: Non-equilibrium steady states for networks of springs. In preparation

  3. Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Entropy production in nonlinear, thermally driven Hamiltonian systems. J. Stat. Phys. 95, 305–331 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. 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, 657–697 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Eckmann J.-P., Zabey E.: Strange heat flux in (an)harmonic networks. J. Stat. Phys. 114, 515–523 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Hairer, M.: A probabilistic argument for the controllability of conservative systems. arxiv:math-ph/0506064

  7. Hairer M.: How hot can a heat bath get?. Commun. Math. Phys. 292, 131–177 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Hörmander, L.: The Analysis of Linear Partial Differential Operators I–IV. New York: Springer, 1985

  9. Jurdjevic, V.: Geometric control theory. Cambridge; New York: Cambridge University Press, 1997

  10. Rey-Bellet, L.: Ergodic properties of Markov processes. Open Quantum Syst. II 139 (2006)

  11. Rey-Bellet L., Thomas L.E.: Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215, 1–24 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Villani C.: Hypocoercivity. Mem. Am. Math. Soc. 202, iv+141 (2009)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Physique Théorique, Université de Genève, Geneva, Switzerland

    N. Cuneo & J. -P. Eckmann

  2. Section de Mathématiques, Université de Genève, Geneva, Switzerland

    J. -P. Eckmann

Authors
  1. N. Cuneo
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  2. J. -P. Eckmann
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Correspondence to J. -P. Eckmann.

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Communicated by H. Spohn

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Cuneo, N., Eckmann, J.P. Controlling General Polynomial Networks. Commun. Math. Phys. 328, 1255–1274 (2014). https://doi.org/10.1007/s00220-014-1966-4

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  • DOI: https://doi.org/10.1007/s00220-014-1966-4

Keywords

  • Equivalence Class
  • Heat Bath
  • Controllable Particle
  • Vandermonde Determinant
  • Commun Math Phys