Journal of Statistical Physics

, Volume 165, Issue 5, pp 809–844 | Cite as

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory



We consider the detailed structure of correlations in harmonic chains with pinning and a bulk velocity flip noise during the heat relaxation phase which occurs on diffusive time scales, for \(t=O(L^2)\) where L is the chain length. It has been shown earlier that for non-degenerate harmonic interactions these systems thermalize, and the dominant part of the correlations is given by local thermal equilibrium determined by a temperature profile which satisfies a linear heat equation. Here we are concerned with two new aspects about the thermalization process: the first order corrections in 1 / L to the local equilibrium correlations and the applicability of kinetic theory to study the relaxation process. Employing previously derived explicit uniform estimates for the temperature profile, we first derive an explicit form for the first order corrections to the particle position-momentum correlations. By suitably revising the definition of the Wigner transform and the kinetic scaling limit we derive a phonon Boltzmann equation whose predictions agree with the explicit computation. Comparing the two results, the corrections can be understood as arising from two different sources: a current-related term and a correction to the position-position correlations related to spatial changes in the phonon eigenbasis.


Velocity flip model Kinetic theory Boltzmann equations Lattice dynamics Diffusion processes 



We thank Giada Basile, Mario Pulvirenti, and Herbert Spohn for useful discussions on the topic, and the anonymous reviewers for their suggestions for improvements. The work has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Project 271983) and from an Academy Project (Project 258302), and partially by the French Ministry of Education through the grant ANR (EDNHS). We are also grateful to the Erwin Schrödinger Institute (ESI), Vienna, Austria for organization of a workshop and providing an opportunity for the related discussions. Alessia Nota acknowledges support also through the CRC 1060 The mathematics of emergent effects at the University of Bonn, that is funded through the German Science Foundation (DFG).


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsingin yliopistoFinland
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany

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