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Hydrodynamic Limit for a Hamiltonian System with Boundary Conditions and Conservative Noise

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Abstract

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy converge to the solution of the Euler system of equations in the smooth regime.

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References

  1. Bernardin, C., Olla, S.: Thermodynamics and non-equilibrium macroscopic dynamics of chains anharmonic oscillators (in progress). Available at https://www.ceremade.dauphine.fr/~olla/

  2. Fritz J., Funaki T., Lebowitz J.L.: Stationary states of random Hamiltonian systems. Probab. Theory Relat. Fields 99(2), 211–236 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fritz J., Liverani C., Olla S.: Reversibility in infinite Hamiltonian systems with conservative noise. Commun. Math. Phys. 189(2), 481–496 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Fritz J.: Microscopic theory of isothermal elastodynamics. Arch. Ration. Mech. Anal. 201(1), 209–249 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  6. Li, T.-T., Yu, W.-C.: Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke University Mathematics Series, vol. V, (1985)

  7. Liverani C., Olla S.: Ergodicity in infinite Hamiltonian systems with conservative noise. Probab. Theory Relat. Fields 106(3), 401–445 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Olla S., Varadhan S., Yau H.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Olla, S.: Microscopic derivation of an isothermal thermodynamic transformation (2013). arXiv:1310:079v1

  10. Varadhan, S.R.S.: Large Deviation and Application. SIAM, Philadelphia, (1984)

  11. Yau H.T.: Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22(1), 63–80 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Correspondence to Nadine Braxmeier-Even.

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Communicated by J. Fritz

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Braxmeier-Even, N., Olla, S. Hydrodynamic Limit for a Hamiltonian System with Boundary Conditions and Conservative Noise. Arch Rational Mech Anal 213, 561–585 (2014). https://doi.org/10.1007/s00205-014-0741-1

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  • DOI: https://doi.org/10.1007/s00205-014-0741-1

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