Journal of Statistical Physics

, Volume 174, Issue 1, pp 219–257 | Cite as

Memory Effects in the Fermi–Pasta–Ulam Model

  • Graziano AmatiEmail author
  • Hugues Meyer
  • Tanja Schilling


We study the intermediate scattering function (ISF) of the strongly-nonlinear Fermi–Pasta–Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics simulations we distinguish two limit regimes, as the system behaves as an ideal gas at high temperature and as a harmonic chain for low excitations. At intermediate temperatures the ISF relaxes to equilibrium in a nontrivial fashion. We then calculate analytically the Taylor coefficients of the ISF to arbitrarily high orders (the specific, simple shape of the two-body interaction allows us to derive an iterative scheme for these). The results of the recursion are in good agreement with the numerical ones. Via an estimate of the complete series expansion of the scattering function, we can reconstruct within a certain temperature range its coarse-grained dynamics. This is governed by a memory-dependent Generalized Langevin Equation (GLE), which can be derived via projection operator techniques. Moreover, by analyzing the first series coefficients of the ISF, we can extract a parameter associated to the strength of the memory effects in the dynamics.


FPU model FPU system Fermi–Pasta–Ulam model Fermi–Pasta–Ulam system Coarse-graining Generalized Langevin equation Intermediate scattering function Memory kernel Harmonic chain 



We thank T. Voigtmann, T. Franosch and A. Zippelius for useful discussions. Computer simulations presented in this paper were carried out using the bwForCluster NEMO high-performance computing facility.


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Authors and Affiliations

  1. 1.Physikalisches InstitutAlbert-Ludwigs-UniversitätFreiburgGermany
  2. 2.Unit in Engineering ScienceUniversité du LuxembourgEsch-sur-AlzetteLuxembourg

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