WIAS Preprint No. 1367, (2008)

Infinite harmonic chain with heavy mass


  • Herrmann, Michael
  • Segatti, Antonio

2010 Mathematics Subject Classification

  • 37K60 70F45


  • Harmonic lattice, atomic chain, macroscopic continuum limit




Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to $1$, except for one heavy particle which has mass $M$. We investigate the macroscopic behaviour of such a system when $M$ is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle.

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